Distribution of Energy and Convergence to Equilibriain Extended Dissipative Systems

Th. Gallay (Universite Joseph Fourier, Grenoble, France)

Dynamical Systems in Studies of Partial Differential Equations

IMA, University of Minnesota

September 24-28, 2012

This talk is based on ongoing work with Sinisa Slijepcevic (Zagreb, Croatia)

1

Overview

We are interested in understanding the dynamics of dissipative partial differ-ential equations on unbounded spatial domains.

• An introductory example

• The energy balance equation

• Extended dissipative systems : a tentative definition

• Further examples of extended dissipative systems

• Bounds on the integrated energy flux

• Bounds on the total energy dissipation

• Convergence on average to the set of equilibria

• Applications to the 2D vorticity equation

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Example 1 : A Reaction-Diffusion Equation

Consider the reaction-diffusion equation on RN :

∂tu(x, t) = ∆u(x, t)− V ′(u(x, t)) , (RD)

where u : RN × R+ → R and V : R → R+ is a smooth potential.

Equations of the form (RD) appear for instance

• In the theory of phase transitions (Allen & Cahn 1979)

• In population genetics (Fisher 1937, Aronson & Weinberger 1974, 1978)

Given a smooth solution of (RD), we define

• The energy density : e = 12 |∇u|2 + V (u) ,

• The (backward) energy flux : f = ut∇u ,

• The energy dissipation rate : d = u2t .

In particular, we observe that e ≥ 0 and d ≥ 0 .

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The Energy Balance Equation

The quantities e, f, d satisfy the energy balance equation :

∂te(x, t) = divxf(x, t)− d(x, t) , (EB)

which expresses the fact that energy is locally dissipated in the system.

Since V ≥ 0 we also have the following estimate on the energy flux

|f |2 = u2t |∇u|2 ≤ 2ed ,

which quantifies how much energy is dissipated during transport.

Finally one observes that

d ≡ 0 ⇒ ut ≡ 0 ,

which means that energy is dissipated by all solutions except equilibria.

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The Case of a Bounded Domain

If Eq. (RD) is considered in a bounded domain Ω ⊂ RN with (for instance)Neumann boundary conditions, then (EB) shows that the total energy

E(t) =

∫

Ω

e(x, t) dx

is a Lyapunov function of the system. Namely

E′(t) ≤ −∫

Ω

d(x, t) dx ≤ 0 ,

and E(t) is strictly decrasing outside the set of equilibria. Under naturalassumptions on the potential V , this gradient structure implies that all finite-energy solutions of (RD) converge to the set of equilibria as t → +∞ .

The situation is completely different if Ω is unbounded and if we considerinfinite energy solutions. In that case one can have travelling waves, or evennontrivial time-periodic orbits if N ≥ 3 (ThG. & Slijepcevic 2001).

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Definition of Extended Dissipative Systems

We say that a time-dependent PDE on RN is an extended dissipative system

if one can associate to each (global) solution a triple (e, f, d) with

e, d : RN × R+ → R+ and f : RN × R+ → RN ,

such that the following properties are satisfied :

i) ∂te = divxf − d

ii) |f |2 ≤ b(e)d for some increasing function b : R+ → R+

iii) d ≡ 0 only for equilibria of the system.

Instead of i), one can require that the integrated energy balance equation :∫

Ω

e(x, T2) dx−∫

Ω

e(x, T1) dx =

∫ T2

T1

∫

∂Ω

f(x, t) · ν dσ dt−∫ T2

T1

∫

Ω

d(x, t) dx dt

be satisfied for any T2 > T1 ≥ 0 and any smooth domain Ω ⊂ RN . Here ν isthe outward unit normal on ∂Ω , and dσ is the elementary surface area.

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Example 2 : The Strongly Damped Wave Equation

Given α ≥ 0 and a smooth potential V : R → R+ , we consider the equation

utt + ut − α∆ut = ∆u− V ′(u) , (DW)

where u : RN × R+ → R . If we define

e = 12u

2t +

12 |∇u|2 + V (u) ,

f = ut(∇u+ α∇ut) ,

d = u2t + α|∇ut|2 ,

then the local energy dissipation equation ∂te = divxf − d is satisfied, andd = 0 implies ut = 0 . Moreover

|f |2 ≤ Ced , where C = 2max(1, α) .

Thus (DW) defines an extended dissipative system in our sense.

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Example 3 : The Complex Ginzburg-Landau Equation

Given α, β ∈ R we consider the complex Ginzburg-Landau equation

ut = (1 + iα)∆u+ u− (1 + iβ)|u|2u , (CGL)

where u : RN × R+ → C . We assume that α = β (‘‘gradient case’’), and weintroduce the auxiliary function

v(x, t) = u(x, t)eiαt , x ∈ RN , t > 0 ,

which satifies the evolution equation

vt = (1 + iα)(

∆v + v − |v|2v)

.

If we now define

e = 12 |∇v|2 + 1

4 (1− |v|2)2 , f = Re (vt∇v) , d =|vt|21 + α2

,

we see that ∂te = divxf − d and |f |2 ≤ Ced with C = 2(1 + α2) .

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Example 4 : The Landau-Lifshitz-Gilbert Equation

Given α ∈ R , the Landau-Lifshitz equation reads

ut = −u ∧ (u ∧∆u) + αu ∧∆u , (LLG)

where u : RN × R+ → S2 = v ∈ R

3 ; |v| = 1 . Here

• −u ∧ (u ∧∆u) = ∆u − u(u ·∆u) is the orthogonal projection of ∆u ontothe plane orthogonal to the direction u ∈ S3 ;

• u ∧∆u is the same vector rotated by π/2 in the orthogonal plane.

If we define

e = 12 |∇u|2 , f = ut∇u ≡

N∑

k=1

∂tuk∇uk , d = |u ∧∆u|2 ,

then ∂te = divxf − d and |f |2 ≤ Ced with C = 2(1 + α2) .

Thus (LLG) also defines an extended dissipative system in our sense.

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Example 5 : A Nonlinear Diffusion Equation

Given a smooth function a : R → (0,∞) , we consider the nonlinear diffusionequation

ut = div(a(u)∇u) , (NLD)

where u : RN × R+ → R , and we denote

e = 12u

2 , f = ua(u)∇u, d = a(u)|∇u|2 .

Then ∂te = divxf − d and d = 0 ⇒ ut = 0 because a(u) > 0 .

Moreover |f |2 ≤ 2a(u)ed . As a consequence, if we define

b(e) = 2e supa(u) |u2 ≤ 2e , e ≥ 0 ,

then e 7→ b(e) is increasing and we have |f |2 ≤ b(e)d by construction.

Thus (NLD) provides an example of an extended dissipative system with amore general relation between energy flux and energy disipation.

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"Example 6" : The Two-dimensional Vorticity Equation

We consider the two-dimensional incompressible Navier-Stokes equations

∂tu+ (u · ∇)u = ∆u−∇p , div u = 0 , (NS)

where u : R2 × R+ → R2 is the velocity field, and p the pressure field.

The associated vorticity distribution ω = ∂1u2 − ∂2u1 evolves according to theadvection-diffusion equation

∂tω + u · ∇ω = ∆ω , x ∈ R2 , t > 0 . (V)

If we define the enstrophy density e , the enstrophy flux f , and the enstrophydissipation rate d by the formulas

e = 12ω

2 , f = ω∇ω − 12uω

2 , d = |∇ω|2 ,

it is easy to verify that ∂te = divxf − d . But in general we do not have anupper bound of the form |f |2 ≤ b(e)d , hence it is not clear that (V) defines anextended dissipative system in our sense.

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Bounds on the Energy Flux (1)

From now on we suppose that we are given a global solution of some extendeddissipative system such that the energy density e(x, t) is uniformly bounded.We define

e0 = supx∈RN

e(x, 0) < ∞ , β = supx∈RN

supt≥0

b(e(x, t)) < ∞ ,

so that |f(x, t)|2 ≤ βd(x, t) for all x ∈ RN and all t ≥ 0 .

Proposition 1 Assume that N = 1 . For any x ∈ R and any T > 0 , we have∣

∣

∣

∣

∣

∫ T

0

f(x, t) dt

∣

∣

∣

∣

∣

≤√

βTe0 .

Thus the total energy flux through a given point over the time interval [0, T ]grows at most like T 1/2 as T → ∞ . This in particular precludes the existenceof nontrivial time-periodic orbits if N = 1 !

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Proof of Proposition 1

Given x0 ∈ R and T > 0 , we denote

F (x0, T ) =

∫ T

0

f(x0, t) dt .

We shall prove that F (x0, T ) ≤ (βTe0)1/2 (the lower bound is similar).

For any x > x0 , the energy balance equation implies

F (x, T ) = F (x0, T ) +

∫ x

x0

(

e(y, T )− e(y, 0))

dy +

∫ x

x0

∫ T

0

d(y, t) dt dy .

The strategy is to bound the right-hand side from below and to obtain anintegral inequality for the integrated flux F (x, T ) .

• Since e(y, T ) ≥ 0 and e(y, 0) ≤ e0 we have∫ x

x0

(

e(y, T )− e(y, 0))

dy ≥ −e0(x− x0) .

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• Since f(y, t)2 ≤ βd(y, t) , we also have∫ T

0

d(y, t) dt ≥ 1

β

∫ T

0

f(y, t)2 dt ≥ 1

βTF (y, T )2 .

Thus the energy balance equation implies, for all x > x0 :

F (x, T ) ≥ F (x0, T )− e0(x− x0) +1

βT

∫ x

x0

F (y, T )2 dy . (INT)

If F (x) denotes the solution of the differential equation

F ′(x) = −e0 +1

βTF (x)2 , F (x0) = F (x0, T ) , (DIFF)

it follows that F (x, T ) ≥ F (x) for x > x0 , as long as F (x) ≥ 0 .

Now, if F (x0, T ) > (βTe0)1/2 , the solution F (x) of (DIFF) is strictly increasing

and blows up at some finite point x1 > x0 , which contradicts the upper boundF (x) ≤ F (x, T ) . Thus we necessarily have F (x0, T ) ≤ (βTe0)

1/2 .

14

Bounds on the Energy Flux (2)

We now consider the higher-dimensional case N ≥ 2 . Given a radius R > 0and a time T > 0 , we define

F (R, T ) =

∫ T

0

∫

|x|=R

f(x, t) · x

|x| dσ dt .

This is the total energy entering the ball BR = x ∈ RN ; |x| ≤ R over thetime interval [0, T ] . Since |f |2 ≤ βd , we have

|F (R, T )|2 ≤ TωNRN−1

∫ T

0

∫

|x|=R

|f(x, t)|2 dσ dt

≤ βTωNRN−1

∫ T

0

∫

|x|=R

d(x, t) dσ dt ,

where ωN =2πN/2

Γ(N/2)is the Euclidean measure of the unit sphere in RN .

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If R > R0 > 0 , the analog of the integral equation (INT) is :

F (R, T ) ≥ F (R0, T )− e0ωN

N(RN − RN

0 ) +1

βTωN

∫ R

R0

F (r, T )2

rN−1dr ,

As a comparison, we use the differential equation

F ′(R) = −e0ωNRN−1 +1

βTωN

F (R)2

RN−1,

with initial data F (R0) = F (R0, T ) .

To eliminate all parameters, we set

F (R)

ωNRN−1=

√

βTe0 H

(

R

√

e0βT

)

, R > R0 .

and we arrive at the simpler ODE

H ′(r) +N−1

rH(r) = −1 +H(r)2 .

16

Threshold Behavior for a Nonlinear ODE

Lemma. For any N ≥ 2 , the differential equation

h′(r) +N−1

rh(r) = −1 + h(r)2 , r > 0 , (ODE)

has a unique positive solution hN : (0,+∞) → (0,+∞) . This solution is strictlydecreasing and satisfies

hN (r) = 1 +N−1

2r+O

( 1

r2

)

as r → +∞ ,

and

hN (r) ∼

1

r log(1/r)if N = 2 ,

N−2

rif N ≥ 3 ,

as r → 0 .

Moreover, any solution of (ODE) above hN blows up in finite "time", and anysolution below hN cannot stay positive.

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Bounds on the Energy Flux (3)

Proposition 2. Assume that N ≥ 2 . For any radius R > 0 and any timeT > 0 , the integrated energy flux satisfies

F (R, T )

ωNRN−1≤

√

βTe0 hN

(

R

√

e0βT

)

, (∗)

where hN is the unique positive solution of (ODE).

Remark. It is easy to verify that h1(r) = 1 and h3(r) = 1 + 1/r .

It is instructive to study the behavior of (∗) as T → ∞ :

√

βTe0 hN

(

R

√

e0βT

)

∼

2βT

R log(

βTe0

R2) if N = 2 ,

(N − 2)βT

Rif N ≥ 3 .

In particular nontrivial time-periodic orbits cannot exist if N = 2 .

18

Bounds on the Total Energy Dissipation

Proposition 3. Given any solution of an extended dissipative system withuniformly bounded energy density, the following estimates hold for all R > 0 :

1) If N = 1 ,

lim supT→∞

1√T

∫ T

0

∫

BR

d(x, t) dx dt ≤ 2√

βe0 .

2) If N = 2 ,

lim supT→∞

log T

T

∫ T

0

∫

BR

d(x, t) dx dt ≤ 4πβ .

3) If N ≥ 3 ,

lim supT→∞

1

T

∫ T

0

∫

BR

d(x, t) dx dt ≤ β(N − 2)ωNRN−2 .

Remark. We recall that e0 = supx e(x, 0) and |f |2 ≤ βd .

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Function Spaces and Topologies

For definiteness, we assume from now on that our extended dissipative systemis a PDE of the form

∂tu = F(u) , (PDE)

and that the Cauchy problem for (PDE) is globally well-posed for positive timesin some Banach space X . In addition to the uniform topology T defined bythe norm, we suppose that X is equipped with a weaker topology Tloc suchthat :

H1 : All solutions of (PDE) which are bounded in X are relatively compact withrespect to the topology Tloc ;

H2 : For any R > 0 and any T > 0 , the integrated energy flux and the totalenergy dissipation depend continuously on the initial data in the topology Tloc .

We denote by Xloc the space X equipped with the topology Tloc .

Typical example : Xloc = Hkul(R

N ) equipped with the topology of Hkloc(R

N ) .

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The Time Spent Near a Nonequilibrium Point

Proposition 4. Assume that N ≤ 2 and let u(t)t≥0 be a solution of (PDE)which is bounded in X . If u ∈ X is not an equilibrium point, then u has aneighborhood V in Xloc such that :

If N = 1 ,

lim supT→∞

1√T

∫ T

0

1V(u(t)) dt < ∞ ,

If N = 2 ,

lim supT→∞

log(T )

T

∫ T

0

1V(u(t)) dt < ∞ .

Here 1V denotes the characteristic function of V ⊂ X .

Remark. In [ThG. & Slijepcevic 2001] we obtained the following weaker result :

limT→∞

1

T

∫ T

0

1V(u(t)) dt = 0 , if N ≤ 2 .

21

Omega-Limit Sets

If u(t)t≥0 is a solution of (PDE) that is bounded in X , hence compact inXloc , the omega-limit set is defined by

ω =

u ∈ X∣

∣

∣∃tn → +∞ such that u(tn) → u in Xloc

⊂ X .

Then ω is nonempty, compact in Xloc , and the solution u(t) converges towardω in Xloc as t → +∞ . Under natural additional assumptions, one can alsoprove that ω is connected in Xloc and fully invariant under the semiflowgenerated by (PDE). But in general ω is not contained in the set of equilibria.

If N ≤ 2 , we introduce a smaller omega-limit set defined as follows :

ω =

u ∈ X∣

∣

∣lim supT→∞

ΨN (T )

T

∫ T

0

1V(u(t)) dt = ∞ for all V ∈ Tloc with V ∋ u

,

where Ψ1(T ) =√T and Ψ2(T ) = log(T ) . In other words u ∈ ω if u(t) comes

"sufficiently often" in any neighborhood of u . In particular ω ⊂ ω .

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Convergence on Average to the Set of Equilibria

Proposition 5. Assume that N ≤ 2 and that u(t)t≥0 is a bounded solutionof (PDE) in X . Then the set ω is nonempty, compact in Xloc , and containedin the set of equilibria of (PDE). Moreover, if V is any neighborhood of ω inXloc , then

lim supT→∞

ΨN (T )

T

∫ T

0

1Vc(u(t)) dt < ∞ ,

where Ψ1(T ) =√T and Ψ2(T ) = log(T ) .

Remark. One can also show that ω is fully invariant under the semiflow of(PDE), but ω is usually not connected in Xloc .

Example. For the Allen-Cahn equation ut = uxx + u − u3 , Eckmann andRougemont constructed an even solution for which infinitely many pairs ofkinks and anti-kinks annihilate at the origin. In that example ω = +1,−1 ,but the omega-limit set ω also includes heteroclinic connections between bothequilibria.

23

Application to the 2D Vorticity Equation

We consider the incompressible Navier-Stokes equation in the infinite cylinderΩ = R× T , where T = R/Z . The system reads

∂tu+ (u · ∇)u = ∆u−∇p , divu = 0 ,

where u : Ω×R+ → R2 is the velocity field, and p : Ω×R+ → R the pressure.

Without loss of generality, we can assume that

u(x1, x2, t) =

(

0m(x1, t)

)

+

(

u1(x1, x2, t)u2(x1, x2, t)

)

,

where m = 〈u2〉 , namely m(x1, t) =

∫

T

u2(x1, x2, t) dx2 .

The corresponding vorticity field reads

ω(x1, x2, t) = ∂1m(x1, t) + ω(x1, x2, t) ,

where ∂1m = 〈ω〉 and ω = ∂1u2 − ∂2u1 .

24

The Biot-Savart Formula

The oscillating part u of the velocity field is given by the Biot-Savart formula :

u(x1, x2, t) =

∫

R

∫

T

∇⊥K(x1 − y1, x2 − y2) ω(y1, y2, t) dy2 dy1 ,

where ∇⊥ = (−∂2, ∂1) and

K(x1, x2) =1

4π

log(

cosh(2πx1)− cos(2πx1))

− |x1|

.

It follows that

‖u‖L∞(Ω) ≤ C‖ω‖L∞(Ω) , and ‖v‖L∞(Ω) ≤ C‖ω‖L∞(Ω) ,

where v ∈ L∞(Ω) is such that u1 = −∂2v .

However the average speed m(x1, t) cannot be fully reconstructed from thevorticity, and we only know that ∂1m = 〈ω〉 .

25

The Enstrophy Balance Equation

The Navier-Stokes equation on the infinite cylinder Ω = R×T is equivalent tothe system

∂tm+ ∂1〈u1u2〉 = ∂21m , x1 ∈ R ,

∂tω + u · ∇ω = ∆ω , (x1, x2) ∈ R× T .(SYS)

Given a solution of (SYS) we define

e(x1, t) =1

2

∫

T

ω(x1, x2, t)2 dx2 ,

f(x1, t) =1

2

∫

T

(

∂1ω2 − u1ω

2)

(x1, x2, t) dx2 ,

d(x1, t) =

∫

T

|∇ω(x1, x2, t)|2 dx2 .

Then ∂te = ∂1f − d and |f |2 ≤ C(1 + ‖e‖L∞)ed , thus (SYS) defines aone-dimensional extended dissipative system.

26

Thank you for your attention...

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... and more importantly :

Many thanks to Peter, Eugene, and Chongchun for organizing that fantasticworkshop !

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