the role of characteristics in conservation laws...2012/07/09 · introduction kovalevsky and her...

The Role of Characteristics in Conservation LawsA Legacy of Sonya Kovalevsky

Barbara Lee Keyfitz

The Ohio State [email protected]

July 9 , 2012

Kovalevsky Lecture

Barbara Keyfitz (Ohio State) Role of Characteristics Kovalevsky Lecture 1 / 22

Outline1 Introduction

What are characteristics?Kovalevsky’s contributions in the context of their timeThe Cauchy-Kovalevsky theorem and the definition of characteristics

2 Characteristics in conservation laws: one space dimensionDefinitions and well-posedness theory in BVDiscontinuities in the large

3 The theory of multidimensional conservation lawsGeneral results: a theorem of Brenner and RauchCharacteristics in conservation laws: multidimensional problemsSelf-similar problems: two-dimensional Riemann problemsGuderley Mach reflection

4 Conclusions

5 Acknowledgements


Introduction What are characteristics?

Quiz: What Are Characteristics in PDE?

A linear PDE of order m: P(D)u =∑|α|≤m

cα(x)∂αu(x) = f (x)

u = u(x), x ∈ Rn, multi-index α = (α1, . . . , αn),∑αi = |α|

∂ = ∇ = (∂x1 , . . . , ∂xn), ξα = ξα11 . . . ξαn

n

What is meant by a characteristic?

2 Any property associated with this equation

2 An eigenvalue of a matrix one can form from the equation

2 A surface ϕ(x) = 0 satisfying the first-order PDE∑

cα(∇ϕ)α = 0

2 A surface S = x |ϕ(x) = 0 which is not appropriate for assigninginitial or boundary data for the PDE

2 A parameterized curve x(t) along which the solution propagates

2 None of the above

2 Several of the above



Illustration: The One- and -Two Dimensional WaveEquations

Equation: utt = c2uxx

Data: u(·, 0) = f , ut(·, 0) = g

x

t

Solution

u(x , t) =12

(f (x − ct) + f (x + ct)

)+ 1

2c

∫ x+ct

x−ctg(s) ds

Equation: utt = c2(uxx + uyy )Data: u(·, 0) = f , ut(·, 0) = g

y

t

x

Characteristic surface:ϕ2t − c2(ϕ2

x + ϕ2y ) = 0

S = t = 0 non-characteristic

S = x = 0 non-characteristic



More About Characteristics

Equation: utt = c2uxx

Data: u(·, 0) = f , ut(·, 0) = g

x

t

Solution

u(x , t) =12

(f (x − ct) + f (x + ct)

)+ 1

2c

∫ x+ct

x−ctg(s) ds

Non-char surface t = 0:Data + Equation ⇒ expressionfor utt (uνν) at surface

Char surface x = ct + c:Equation ⇒ constraint onsolution (cux − ut or cux + ut -“char vbles” - constant)

Discontinuities: on surface

ux

ut


Introduction Kovalevsky and her time

Kovalevsky’s Contributions in the Context of their Time

Dates 1850 - 1891

PhD from Gottingen, 1874

Theorem on PDE; results onrings of Saturn and on ellipticintegrals

Professor, University ofStockholm, 1889

Editor, Acta Mathematica

Prix Bordin, 1888 (Kovalevskytop)


Introduction The Cauchy-Kovalevsky Theorem

The Cauchy-Kovalevsky Theorem

Theorem

Given a fully nonlinear PDE in n independent variables,

F (x ,D, u) = 0

where F is of order m inD and real analytic, and compatible real analyticdata u,Du, . . .Dm−1u on a surface S = x |ϕ(x) = 0 which is notcharacteristic and is also real analytic, there exists a real analytic solutionu(x) in a neighborhood of S. The solution is unique in the class of realanalytic functions.

Nonlinear equations included (∑

cα(∇ϕ)α 6= 0 @ S)Elliptic equations have no real characteristic surfacesTheorem can be stated for systemsNothing about well-posednessProof: Method of Majorants


Conservation laws in one space dimension Definitions and well-posedness

Conservation Laws: One Space Dimension

Hyperbolic Conservation Laws:

Ut +∑

∂iFi (U) = 0, U = U(x , t) ∈ Rm, x ∈ Rn

Hyperbolic: det(I τ +∑

Aiξi ) = 0 ⇒ τj(ξ) real

Govern physical processes like compressible fluid flow, free surfaceflow, chemical deposition processes (chromatography)

First-order equation model omits dissipation (viscosity), dispersion,kinetic effects

‘Mono-scale’ physics

Interesting mathematical property: effect in the large of dependenceof characteristics on the state variable U



The Simplest Example

Burgers Equation ut + uux = 0Characteristic surface (curve): ϕt + uϕx = 0 or dx

dt = uSolution of equation

u(y + u(y)t, t) = u(y)

Converging characteristics: form shock, weak solution(weak solution:

∫uφt + f (u)φx = 0 or s[u] = [f (u)])

Diverging characteristics: form rarefaction

x

t

y

x−ut=y

Conclusions:

CL live in world of weak solutions

Shocks replace characteristics as locusof discontinuities

Nonlinear quantities replace char vbles



Rarefactions: Another Type of Nonlinear Behavior

Appearance of rarefaction waves

u=0

u=1

u=x/t

x

t

Example (Burgers again: ut + uux = 0)

u(x , t) =

u = 0, x < 0u = x/t, x < tu = 1, x > t

(Ir)regularity: ux(·, t) not bdd in Lp, p > 1



One Space Dimension: Riemann Problems forStrictly Hyperbolic Systems

One space dimension ut + f (u)x = ut + A(u)ux = 0, u ∈ Rn

Self-similar solutions: ξ = xt

Riemann Data

u(x , 0) =

u`, x < 0ur , x ≥ 0 u

0

u1

u2

un

x

t

In each characteristic family (λj of A)

s[u] = [f (u)], where s ∼ λi is speed (dx/dt) of discont, or

x/t = λi (u), u = Ri (u), or

Linear (characteristic) discontinuity

(Linear analogue: 1-D char decomposition of a discontinuity)



Random Choice and Front Tracking: BV Solutions

Theorem (Glimm, Bressan)

Assume ut + f (u)x = 0 is strictly hyperbolic and each characteristic familyis either linearly degenerate or genuinely nonlinear in a set Ω. Ifu(·, 0) ⊂ Ω and TV (u·, 0) is sufficiently small, then ∃! sol’n in BV .

Riemann solutions give approximationsGlimm’s random choice (1965) Risebro-Bressan’s WF tracking

x

t

x

t

Var uh(·, t) ≤ M,∫|uh(t, ·)− uh(s, ·)| ≤ L|t − s|

Helly’s theorem ⇒ subsequence cvges ptwise to BV soln.Bressan: SRS ⇒ uniqueness, well-posedness, & regularity (continuousexcept for countable set of shock curves & interaction points)


Conservation laws in one space dimension Discontinuities in the large

Nonlinear Characteristics (Shock Surfaces) in the Large

We know:

Solutions of nonlinear hyperbolic conservation laws are discontinuousDiscontinuities are confined to characteristic surfaces, or theirnonlinear analogues, shock surfacesAcross shock surfaces, linear constraints on characteristic variables arereplaced by nonlinear constraints (Rankine-Hugoniot relations)What happens when this fails?

Example (joint with Charis Tsikkou)

Gas dynamics: u = velocity; v = entropy

ut + ((3− γ)

2u2 − v)x = 0

vt + [(2− γ)(5− 3γ)

6u3 + (γ − 1)uv ]x = 0

1 < γ < 5/3

B u

v

R2

R1

(u0,v

0)

S1

S2

S1

S2



“Singular Shocks” (with Herbert Kranzer)

Weak solutions of very low regularity (not L∞)

Singularity is concentrated at a discontinuity (that is, superimposedon a shock)

Can be described in terms of measures or weighted measures

Can be approximated in a number of ways, all consistent

Dirac superimposed on shock (Sever)Space of weighted measures (K. & Kranzer)Colombeau generalized distributions (Shelkovich, Nedelykov, et al.)Geometric singular perturbation theory (Schecter, Krupa)

Spent 25 obscure years in the annals of pathology

Made a recent appearance in a chemical engineering application(chromatography) with experimental verification



Marco Mazzotti’s anti-Langmuir Chromatography

‘Generalized Langmuir’ in chromatography (Mazzotti, ETH)

∂

∂t

(ui +

aiui

1− u1 + u2

)+

∂

∂xui = 0 , i = 1, 2 a1 < a2

u

u

1

2

NH

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)


Multidimensional conservation laws General results

Quasilinear Systems in Multidimensional SpaceQuasilinear systems in more than one space dimension

ut +n∑1

Aj(u)uxj + b(u) = 0

Linear & semilinear : W s,2 theory for smooth data, s ≥ n+12

Quasilinear : short time well-posedness in W s,2 pre-shock

Theorem (Rauch, following P. Brenner)

No BV bounds. For C∞ data, if∫Rn

|∇xu(x , t)| dx ≤ C

∫Rn

|∇xu(x , 0)| dx

then AjAk = AkAj ∀j , k.

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


Multidimensional conservation laws Multidimensional characteristics

Characteristic Surfaces in Space

Recall:Equation: utt = c2(uxx + uyy )Data: u(·, 0) = f , ut(·, 0) = g

y

t

x

Characteristic surface:c2t2 = x2 + y2

Dual surface (normal cone):τ2 = c2(ξ2 + η2)

Nondegeneracy vs AjAk = AkAj

Proof of Brenner’s Theorem.

Fourier multipliers in Lp

Nonplanar normal cones ⇒instability (lack of continuity ofmultipliers) in Lp for p 6= 2

Planar solutions to characteristicequation ⇒ commoneigenvectors

Common eigenvectors ⇒commuting matrices

Reminder: Connection betweenhyperbolicity and well-posedness forlinear equations


Multidimensional conservation laws Self-similar problems

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

• Analogy with 1-D

• 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

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


Multidimensional conservation laws Self-similar problems

Local Picture for Regular Reflection

WEAK STRONG

Incident Shock Incident Shock

Reflected

ShockReflected

Shock

Sonic Line

ELLIPTIC

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


Multidimensional conservation laws 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


Conclusions Learning from characteristics

Conclusions

We have seen

Characteristics introduced in Cauchy-Kovalevsky theorem to proveexistence of solutions

Relation between hyperbolicity and well-posedness in linear PDEcame later

In non-linear hyperbolic PDE the theory is still incompleteRecent advances include

Well-posedness in 1-D & understanding of regularity of solutionsProgress in other directions (IV-BV problems in 1-D)Numerical simulations and studiesRelation of HCL to other models (viscous, kinetic, statistical)Balance laws and reacting flows (1 space dimension)Pathologies associated with large data

First steps in a multi-dimensional theory

Characteristics are not the whole story, but neither have we yetobtained all that it is possible to learn from them


Acknowledgements To co-authors, family, friends and mentors

Acknowledgements

Herbert C. Kranzer

Milton Lopes

Karen Ames

Suncica Canic

Gary Lieberman

Eun Heui Kim

Michael Sever

Richard Sanders

Fu Zhang

Katarina Jegdic

Allen Tesdall

Mary Chern

Charis Tsikkou

Marty Golubitsky

Elizabeth Golubitsky

Alexander Golubitsky

Nathan Keyfitz

Beatrice Keyfitz

W. Kahan

Chandler Davis

Peter Lax

Cathleen Morawetz

Olga Oleinik

Nancy Kopell

Karen Uhlenbeck

Garry Etgen


the role of characteristics in conservation laws...2012/07/09 · introduction kovalevsky and her...

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