stationary statistical properties of dissipative … approach continuity and perturbation results...

Statistical ApproachContinuity and perturbation results

Temporal approximation

Stationary Statistical Properties of DissipativeSystems

Wang Xiaomingwxmmathfsuedu

Florida State University

Stochastic and Probabilistic Methods in Ocean-AtmosphereDynamics Victoria July 2008

Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems



Outline

1 Statistical Approach

2 Continuity and perturbation results

3 Temporal approximation




Logistic map

T (x) = 4x(1minus x) x isin [0 1]

Figure Sensitive dependence

0 20 40 60 80 100 120 140 160 180 2000

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Orbits of the Logistic map with close ICs

Figure Statistical coherence

-02 0 02 04 06 08 1 120

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Lorenz 96 modelEdward Norton Lorenz 1917-2008

Lorenz96 duj

dt= (uj+1 minus ujminus2)ujminus1 minus uj + F

j = 0 1 middot middot middot J J = 5 F = minus12

Figure Sensitive dependence Figure Statistical coherence

minus20 minus10 0 10 200

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Lorenz96 duj




(Sensitive dependence)



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L96_demoavi
Media File (videoavi)



Statistical approach

dudt

= F(u) u isin H

Long time average

lt Φ gt= limTrarrinfin

1T

int T

0Φ(v(t)) dt

Spatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)

microt t ge 0 statistical solutions





dudt

= F(u) u isin H

Long time average


1T

int T

0Φ(v(t)) dt

Spatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)






dudt

= F(u) u isin H

Long time average


1T

int T

0Φ(v(t)) dt

Spatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)





Statistics Solutions

microt t ge 0 dvdt

= F(v) S(t) t ge 0

Pull-backmicro0(Sminus1(t)(E)) = microt(E)

Push-forward (Φ suitable test functional)intH

Φ(v) dmicrot(v) =

intH

Φ(S(t)v) dmicro0(v)

Finite ensemble example

micro0 =Nsum

j=1

pjδv0j (v) microt =Nsum

j=1

pjδvj (t)(v) vj(t) = S(t)v0j





microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1





Figure Joseph Liouville1809-1882

Figure Eberhard Hopf1902-1983




Liouville and Hopf equations

Liouville type equation

ddt

intH

Φ(v) dmicrot(v) =

intH

lt Φprime(v) F(v) gt dmicrot(v)

Φ good test functionals eg

Φ(v) = φ((v v1) middot middot middot (v vN))

Liouville equation (finite d)

part

parttp(v t) +nabla middot (p(v t)F(v)) = 0

Hopfrsquos equation (special case of Liouville type)

ddt

intH

ei(vg) dmicrot(v) =

intH

i lt F(v) g gt ei(vg) dmicrot(v)






ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH





Stationary Statistics Solutions (IM)

Invariant measure (IM) micro isin PM(H)

micro(Sminus1(t)(E) = micro(E)

Stationary statistical solutions essentiallyintH

lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ

Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)








































Figure George DavidBirkhoff 1884-1944 Definition

micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E

Theorem (Birkhoffrsquos ErgodicTheorem)

If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie

limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =

intH

ϕ(u) dmicro(u) as




Maximum entropy for conservative case

Maximum entropy principle most probable pdf maximize theShannon entropy S = minus

intp ln p

Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)

d ~Xdt

= ~F (~X ) + εd ~Wdt

Fokker-Planck equation (Kolmogorov Smoluchowski)

partppartt

+ ~F middot nabla~X p minus ε2

2∆~X p = 0

Equation for the density of Shannon entropy Q = minusp ln p

partQpartt

+nabla~X middot (~FQ)minus ε2

2∆~X Q =

ε2

2p|nablap|2

Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0






intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2





More on statistical theories for complex dynamical systems can befound

Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001




Dissipative system

Definition

A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that

A is invariant under S(t)A is compactA attracts any bounded set B in H ie

limtrarrinfin

dist(S(t)BA) = 0




IM and attractors

Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM

Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)




IM and attractors






IM and time averages

Theorem (IM and time averages W 08)

Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that

LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH

ϕ(u) dmicro(u)forallϕ isin C(H)

Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)




Ergodicity and extremal points

Theorem (Ergodicity and extremal points W 08)

Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM

Other versions with group (ODE) assumption is well-known




Regular perturbation

Theorem (Conv of IM regular version W 08)

Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =

⋃0lt|ε|leε0

Aε

2 (finite time u-conv)limεrarr0 supuisinAε

S(t ε)uminus S(t 0)uH = 0forallt ge 0

Thenlimεrarr0 microε = micro0 microε isin IMε micro0 isin IM0

Beyond continuity linear response







⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε








Two time scale set-up

Two-time-scale problem X1 X2 Hilbert spaces

ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20

Limit problem ( ε = 0)

0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)




Conv of stat prop 2 time scale case

Theorem (Conv of IM singular version W 08)

Assume1 (uniform dissipativity) pre-compactness of K =

⋃0ltεltε0

Aε

2 (dissipativity of the limit system) A0 in X23 (conv of the slow variable)

limεrarr0 supu2isinP2AεP2S(t ε)(F1(u2 0) u2)minus S(t 0)u2X2 =

0 forall t ge 0

4 (smallness of the perturbation)limεrarr0 sup(u1u2)isinAε

ε( du1dt + g(u1 u2))X1 = 0

5 (continuity of the slave relation) u1 = F1(u2 y)

Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0




Rayleigh and Beacutenard

Figure Lord Rayleigh (JohnWilliam Strutt) 1842-1919

Figure Henri Beacutenard (left)1874-1939




Application to RBC W CPAM 07

Boussinesq system for Rayleigh-Beacutenard convection

1Pr

(partupartt

+ (u middot nabla)u) +nablap = ∆u + Ra kθ nabla middot u = 0 u|z=01 = 0

partθ

partt+ u middot nablaθ minus u3 = ∆θ θ|z=01 = 0

Infinite Prandtl number model for convection

nablap0 = ∆u0 + Ra kθ0 nabla middot u0 = 0 u0|z=01 = 0

partθ0

partt+ u0 middot nablaθ0 minus u0

3 = ∆θ0 θ0|z=01 = 0

X1 = H X2 = L2 ε = 1Pr and F1(θ y) = Ra Aminus1(kθ)minus Aminus1(y)






1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0





RBC set-up and numerics

Figure RBC set-up Figure Numerical simulation

(infin Pr simulation)


m1mpg
Media File (videompeg)



Possible deficiency of classical schemes

dudt

= F(u) u isin H

classical scheme of order m

u(n∆t)minus un le C(n∆t)∆tm

Dependence on TC(T ) = exp(αT )

Error in approximation of long time averages

| lim supNrarrinfin

1N

Nsumn=1

(Φ(u(n∆t))minus Φ(un))|

le c lim supNrarrinfin

∆tm exp((N + 1)α∆t)minus exp(α∆t)exp(α∆t)minus 1

= infin

Classical schemes may not be able to capture the climatealthough they may work very well for weather





dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin





Difficulty with large Rayleigh number

Infinite Pr number model

partθ0

partt+ Ra Aminus1(kθ0) middot nablaθ0 minus Ra Aminus1(kθ0)3 = ∆θ0

A Stokes operatorAlternative form with s = Ra t

partθ0

parts+ Aminus1(kθ0) middot nablaθ0 minus Aminus1(kθ0)3 =

1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0




Theorem (General result W 08)

Assume1 (Uniform dissipativity) K =

⋃0ltklek0

Ak

2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn

k uminus S(nk)u rarr 0 as k rarr 0

3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0

Then1 (Conv of stationary stat prop)

microk micro microk isin IMk micro isin IM

2 (Conv of attractors)

limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0




Application to infin Pr model

infin Prandtl number model (alternative form)

partθ

partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0

Semi-implicit scheme

θn+1 minus θn

k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1

Equivalent form (T = θ + 1minus z)

T n+1 minus T n

k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1






partθ



θn+1 minus θn



T n+1 minus T n





Nusselt number (recast)

For the infinite Prandtl number model

Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ

|nablaT (x s)|2 dxds

= 1 + Ra supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ

Aminus1(kT (x s))3T (x s) dxds


lim suptrarrinfin

1tLxLy

int t

0

intΩ

Aminus1(kθ(x s))3θ(x s) dxds

For the scheme

Nuk = 1 + Ra supθ0isinL2

lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ

Aminus1(kθn(x))3θn(x) dx






Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ





Nusselt number limit

Convergence of Nusselt number

lim supkrarr0

Nuk le Nu

Complements variational approach (Constantinamp Doering)






lim supkrarr0

Nuk le Nu





Summary

Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal




Summary





Summary





Summary





Summary





Questions

Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems




Questions





Questions





Questions





Questions





Questions





Questions





Questions



Statistical Approach

Continuity and perturbation results




Outline

1 Statistical Approach

2 Continuity and perturbation results

3 Temporal approximation




Logistic map



0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07

08

09

1



-02 0 02 04 06 08 1 120

100

200

300

400

500

600





Lorenz96 duj





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Lorenz96 duj







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L96_demoavi




dudt

= F(u) u isin H

Long time average


1T

int T

0Φ(v(t)) dt

Spatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)






dudt

= F(u) u isin H

Long time average


1T

int T

0Φ(v(t)) dt

Spatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)






dudt

= F(u) u isin H

Long time average


1T

int T

0Φ(v(t)) dt

Spatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1












ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH






















































limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =

intH

ϕ(u) dmicro(u) as






intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions








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Lorenz96 duj





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Lorenz96 duj







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L96_demoavi




dudt

= F(u) u isin H

Long time average


1T

int T

0Φ(v(t)) dt

Spatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)






dudt

= F(u) u isin H

Long time average


1T

int T

0Φ(v(t)) dt

Spatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)






dudt

= F(u) u isin H

Long time average


1T

int T

0Φ(v(t)) dt

Spatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1












ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH






















































limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =

intH

ϕ(u) dmicro(u) as






intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions









Lorenz96 duj





2000

4000

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Lorenz96 duj







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L96_demoavi




dudt

= F(u) u isin H

Long time average


1T

int T

0Φ(v(t)) dt

Spatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)






dudt

= F(u) u isin H

Long time average


1T

int T

0Φ(v(t)) dt

Spatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)






dudt

= F(u) u isin H

Long time average


1T

int T

0Φ(v(t)) dt

Spatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1












ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH






















































limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =

intH

ϕ(u) dmicro(u) as






intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions









Lorenz96 duj







2000

4000

6000

8000

10000

12000

14000


2000

4000

6000

8000

10000

12000

14000


2000

4000

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10000

12000

14000


2000

4000

6000

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10000

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14000


2000

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10000

12000

14000


L96_demoavi




dudt

= F(u) u isin H

Long time average


1T

int T

0Φ(v(t)) dt

Spatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)






dudt

= F(u) u isin H

Long time average


1T

int T

0Φ(v(t)) dt

Spatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)






dudt

= F(u) u isin H

Long time average


1T

int T

0Φ(v(t)) dt

Spatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1












ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH






















































limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =

intH

ϕ(u) dmicro(u) as






intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions









dudt

= F(u) u isin H

Long time average


1T

int T

0Φ(v(t)) dt

Spatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)






dudt

= F(u) u isin H

Long time average


1T

int T

0Φ(v(t)) dt

Spatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)






dudt

= F(u) u isin H

Long time average


1T

int T

0Φ(v(t)) dt

Spatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1












ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH






















































limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =

intH

ϕ(u) dmicro(u) as






intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions









dudt

= F(u) u isin H

Long time average


1T

int T

0Φ(v(t)) dt

Spatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)






dudt

= F(u) u isin H

Long time average


1T

int T

0Φ(v(t)) dt

Spatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1












ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH






















































limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =

intH

ϕ(u) dmicro(u) as






intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions









dudt

= F(u) u isin H

Long time average


1T

int T

0Φ(v(t)) dt

Spatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1












ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH






















































limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =

intH

ϕ(u) dmicro(u) as






intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions









microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1












ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH






















































limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =

intH

ϕ(u) dmicro(u) as






intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions









microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1












ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH






















































limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =

intH

ϕ(u) dmicro(u) as






intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions









microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1






microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1












ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH






















































limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =

intH

ϕ(u) dmicro(u) as






intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions









microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1












ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH






















































limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =

intH

ϕ(u) dmicro(u) as






intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions















ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH






















































limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =

intH

ϕ(u) dmicro(u) as






intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH






















































limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =

intH

ϕ(u) dmicro(u) as






intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH







ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH






















































limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =

intH

ϕ(u) dmicro(u) as






intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










ddt

intH

Φ(v) dmicrot(v) =

intH





part



ddt

intH

ei(vg) dmicrot(v) =

intH






















































limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =

intH

ϕ(u) dmicro(u) as






intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions

























































limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =

intH

ϕ(u) dmicro(u) as






intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2







intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










intp ln p


d ~Xdt

= ~F (~X ) + εd ~Wdt


partppartt


2∆~X p = 0


partQpartt


2∆~X Q =

ε2

2p|nablap|2










Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions













Dissipative system

Definition



limtrarrinfin

dist(S(t)BA) = 0




IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions








IM and attractors






IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions








IM and attractors









LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions











LIMTrarrinfin1T

int T

0ϕ(S(t)u0) dt =

intH
















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions


















⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions











⋃0lt|ε|leε0

Aε











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions











⋃0lt|ε|leε0

Aε










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










ε(du1

dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10

du2

dt= f2(u1 u2) u2(0) = u20


0 = f1(u01 u0

2)

du02

dt= f2(u0

1 u02) u0

2(0) = u20

y = f1(u1 u2) hArr u1 = F1(u2 y)







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions











⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions











⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions











⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions











⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions











⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0







⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions











⋃0ltεltε0

Aε



0 forall t ge 0


ε( du1dt + g(u1 u2))X1 = 0


Thenmicroε Lmicro0












1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions
















1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0







1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










1Pr

(partupartt


partθ




partθ0


3 = ∆θ0 θ0|z=01 = 0









m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions












m1mpg




dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions









dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions









dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions









dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin






dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions









dudt

= F(u) u isin H





| lim supNrarrinfin

1N

Nsumn=1




= infin







partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










partθ0



partθ0


1Ra

∆θ0






partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










partθ0



partθ0


1Ra

∆θ0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










⋃0ltklek0

Ak







limkrarr0

dist(Ak A) = 0






partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










partθ



θn+1 minus θn



T n+1 minus T n







partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










partθ



θn+1 minus θn



T n+1 minus T n







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










Nu = supθ0isinL2

lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ



lim suptrarrinfin

1tLxLy

int t

0

intΩ


For the scheme


lim supNrarrinfin

1NLxLy

Nsumn=1

intΩ







lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










lim supkrarr0

Nuk le Nu







lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions










lim supkrarr0

Nuk le Nu





Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions








Summary





Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions








Summary





Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions








Summary





Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions








Summary





Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions








Summary





Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions








Questions





Questions





Questions





Questions





Questions





Questions





Questions





Questions








Questions





Questions





Questions





Questions





Questions





Questions





Questions








Questions





Questions





Questions





Questions





Questions





Questions








Questions





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stationary statistical properties of dissipative … approach continuity and perturbation results...

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