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Conservation Laws for BeginnersIMA Workshop

Career Options for Women in Mathematical Sciences

Barbara Lee Keyfitz

The Ohio State [email protected]

March 4, 2013

Barbara Keyfitz (Ohio State) Conservation Laws for Beginners March 4, 2013 1 / 11

Outline

1 IntroductionHistoryHyperbolic PDE

2 What we know about conservation lawsOne space dimension, linear theoryOne space dimension: small data resultsOne space dimension: large dataBasics of multidimensional problemsSample results

3 What we don’t know (and would like to find out)Are multidimensional problems well-posed?Guderley Mach reflection


Introduction History

‘History of Differential Equations’

A view from the distance . . .

Euler

Weierstrass

Lie

Poincare

Computers

When we study differential equations, what are we looking for?


Introduction Hyperbolic PDE

Hyperbolic Partial Differential Equations: TRANSPORT

The idea:

ut + aux = 0 or∂u

∂t+ a

∂u

∂x= 0 vs ut + uux = 0

u = f (x − at)

characteristics dx/dt = a

weak solutions (for rough data)

solutions for all t

u=f(x-ut) (implicit)

characteristics dx/dt = u(speed = amplitude)

weak solutions always

require ut + f (u)x = 0

Discontinuousacross

characteristics

Locally smoothsolution

x

t

x

t

y

x−ut=y


What we know about conservation laws One space dimension, linear theory

One Space Dimension: Linear vs Nonlinear

Linear

First-order system, 1-D:

ut + A(x , t)ux + b(x , t) = 0

Diagonalize:

vt + Λ(x , t)vx + b(x , t) = 0

Picard method: t

x

(x0,t

0)

Nonlinear (Quasilinear) System

ut + f (u)x ≡ ut + A(u)ux

b = b + P(∂tP

−1 + A∂xP−1)

If A = A(u) then P = P(u)

So b depends on ut and ux

Unless Λ is independent of u(which is the linear case)

Mechanism is still transport, butinteractions are now nonlinear

Nonlinear effects are important,but fundamental mechanism isstill transport


What we know about conservation laws One space dimension: small data results

Approach to Nonlinear Problems

ut + A(u)ux = 0, e. v. A = {λ1(u) < λ2(u) . . . < λn(u)}

Calculate wave interactions ‘by hand’ (Glimm, Bressan)

Works but

only in total variation norm TVonly for small data: TV (u(·, t)) < ε

x

t

x

t


What we know about conservation laws One space dimension: large data

Large Data: A Mystery – Several Mysteries

Phenomena include

Explosions (L∞ growth in u)

Singular Shocks (K. & Kranzer)

Example: Marco Mazzotti’s anti-Langmuir Chromatography

∂

∂t

(ui +

aiui1− u1 + u2

)+

∂

∂xui = 0 , i = 1, 2 a1 < a2

Figure 11b944

71

Author's personal copy

M. Mazzotti et al. / J. Chromatogr. A 1217 (2010) 2002–2012 2009

Fig. 7. Effect of feed concentration on the interaction between phenetole (comp. 1) and 4-tert-butylphenol (comp. 2) in frontal analysis experiments (Zurich laboratory). (a)5 cm column, high concentration range; (b) 25 cm column, low concentration range.

Two final remarks are worth making. The first remark refers tothe shape of the peak in the experiment at 100% concentration (seeFig. 7a, inset), which is in this case clearly different from that exhib-ited by the peaks obtained at higher concentration. Both beforeand after the main sharp peak, the UV profile reaches two plateaus,which are above the feed concentrations of the two species; theyelute for a time, namely between 0.2 and 0.3 min, which is compa-rable to the elution time of the main peak itself, i.e. about 0.2 min.

We do not have an explanation for this effect, which is commonto all three columns, but is not so evident or not at all exhibited athigher concentration.

The second remark refers to the two sharp fronts exhibited byall delta-shocks’ spikes. It is well known that sharp fronts in non-linear chromatography exhibit a constant pattern behavior, whichis called shock layer, when they separate two constant states andpropagate through long enough columns [11–13]. Although there

Components phenetole (C8H10O) and 4-tert-butylphenol (C10H14O)Barbara Keyfitz (Ohio State) Conservation Laws for Beginners March 4, 2013 7 / 11

What we know about conservation laws Basics of multidimensional problems

More Than One Space Dimension:Wave Propagation Is Different

First-order system for u = u(x , y , t) ∈ Rn

With n × n flux matrices A and B, A =∂f

∂x, B =

∂g

∂y

ut + A(x , y , t, u)ux + B(x , y , t, u)uy = 0

y

t

x

Characteristics are surfaces

Waves spread

Amplitudes decay

Wave interactions are complicated

Theorem (P. Brenner): Linear systems are well-posed only in L2

Theorem (J. Rauch): The same is true in quasilinear systems

Mismatch between known 1-D and multi-D constraints

Recent results (De Lellis &Szekelyhidi) on non-uniqueness of(standard definition) weak solutions


What we know about conservation laws Sample results

Self-similar Problems: 2-D Riemann Problems (with Canic,Lieberman, Kim, Jegdic, Tesdall, Popivanov, Payne)

• Analogy with 1-D: focus on transport and wave interactions

• Occur in physically interesting problemsExample: Shock reflection by a wedge

X= tΞ

S= tΣFlowWedge

Incident Shock

ReflectedShock

t<0 t=0 t>0

• Expect to see well-posed problems (but some surprises)

• Interesting mathematics (hyperbolic + elliptic)

Approach

• Work completely in self-similar coordinates: ξ = xt , η = y

t

• Reduced eq’n (−ξ + A(U))Uξ + (−η + B(U))Uη = 0 changes type


What we know about conservation laws Sample results

Local Picture for Regular Reflection

WEAK STRONG

Incident Shock Incident Shock

Reflected

Shock Reflected

Shock

Sonic LineELLIPTIC

REGION

ELLIPTIC

REGION

FREE BOUNDARY

DEGENERACY IN ELLIPTIC EQUATION

UTSD

ut + uux + vy = 0

vx − uy = 0

NLWS

ρt + mx + ny = 0

mt + p(ρ)x = 0

nt + p(ρ)y = 0

Incident Shock

Reflected Shock

Free Boundary

Cutoff Boundary

Incident Shock

Cutoff Boundary

Reflected Shock

Free Boundary

Sonic Line

"STRONG" "WEAK"

See also results of Chen, Feldman et al on potential flow


What we don’t know (and would like to find out) Guderley MR

Guderley Mach Reflection (Hunter and Tesdall)

x/t

y/t

1.0746 1.0748 1.075 1.0752 1.0754 1.0756

0.41

0.4102

0.4104

0.4106

0.4108

Discovered in numerical simulations and verified experimentally byB. W. Skews & al. (JFM)

No theory as yet


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