Kinetic relations for undercompressive shocks.Physical models, analysis, and approximation

Philippe G. LeFloch

University of Paris 6 & CNRS

http://philippelefloch.wordpress.com

1. INTRODUCTION

Hyperbolic or hyperbolic-elliptic systems of conservation laws withsingular perturbation (diffusion, dispersion, etc)

uεt + f (uε)x = R(εuε

x , ε2uε

xx , . . .)x .

I Characterize the limit u := limε→0 uε. Admissible discontinuities ?

I Second-order: ε uεxx . Entropy conditions.

Lax, Oleinik, Volpert, Kruzkov, Dafermos, Wendroff, TP Liu.

I Add also a third order: α ε2 uεxxx , and even higher-order.

Oscillations near shocks and competition between small scales.

I Classical compressive + nonclassical undercompressive shocks (orsubsonic phase boundaries).

......... Entropy inequality supplemented with a kinetic relation.

PHYSICAL MODELS

I Continuum physics of complex flows.Van der Waals fluid, phase transitions, Nonlinear elasticity, thinliquid film, generalized Camassa-Holm, MHD with Hall effect.

MATHEMATICAL THEORY

I Nonclassical Riemann solver with entropy-compatible kinetics

I Kinetic functions determined by traveling waves

I Existence via Glimm-type scheme with generalized TV functionals

I Zero diffusion-dispersion limits

NUMERICAL APPROXIMATION

I Schemes with controled dissipation(finite differences, entropy conservative, equivalent equations)

I Computing kinetic functions

GENERALIZATION

I The case of two inflection points

I Riemann solver with kinetics and nucleation(splitting-merging phenomena)

I DLM theory – Kinetic relations for nonconservative systems(Dal Maso-LeFloch-Murat and LeFloch-TP Liu)

Typical model. Materials undergoing phase transitions

wt − vx = 0

vt − σ(w)x = ε vxx − α ε2 wxxx

v : velocity w > −1 : deformation gradient σ(w) : stressε : viscosity α ε2 : capillarity

I Slemrod (1984, etc): self-similar solutions

I Shearer (1986, etc.): Riemann problem with α = 0.

I Truskinovsky (1987, etc)

I Abeyaratne & Knowles (1990, etc): Riemann problem

I LeFloch (ARMA 1993, etc): mathematical formulation in BV, and Cauchyproblem via Glimm scheme.

Active research on undercompressive shocks.

I Collaborators. Bedjaoui, Piccoli, Shearer, Joseph, Mohamadian, Laforest,Mishra.(Former) postdocs and students. Hayes, Kondo, Baiti, Rohde, Mercier,Correia, Thanh, Chalons, Boutin.

I Related works (one- and multi-dimensional stability, thin films, trafficflows).

T.-P. Liu, Kulikovskij, Marchesin, Plohr, H.T. Fan, Benzoni,

Metivier-Williams-Zumbrun, Colombo, Bertozzi-Shearer, Corli-Tougeron.

2. PHYSICAL MODELS

Approximation by diffusion.

I Conservation law:ut + f (u)x = ε

(b(u) ux

)x

with b > 0.

I U : R → R, F (u) :=R u

f ′(v)U ′(v) dv

U(u)t + F (u)x = −ε b(u)U ′′(u) |ux |2 + Cx .

C := ε`b(u)U(u)x

´x

I Entropy inequalities:If U is convex, the formal limit u = limε→0 uε satisfies

U(u)t + F (u)x ≤ 0.

I Classical entropy solutions: entropy criterion given by Kruzkov

|u − k|t +“sgn(u − k)(f (u)− f (k))

”x≤ 0, k ∈ R.

Also equivalent to a pointwise version given by Oleinik.

Model with linear diffusion and dispersion.

I Conservation law (with β > 0)ut + f (u)x = β uxx + γ uxxx .

(Shearer et al., Hayes-PLF, Bedjaoui-PLF)

I Single entropy inequality:`u2/2

´t+ F (u)x = −D + Cx ,

D := β |ux |2 ≥ 0, C := β uux + γ`u uxx − (1/2)u2

x

´.

In the (formal) limit β, γ → 0 one gets`u2/2

´t+ F (u)x ≤ 0,

but no sign for general convex entropies!

ut + f (u)x = β uxx + γ uxxx .

Classical/nonclassical solutions:

I γ << β2 (dominant diffusion): classical entropy solutions

I γ >> β2 (dominant dispersion):high oscillations, weak convergence (Lax, Levermore)

I γ := κβ2 (balanced regime):strong convergence, mild oscillations, nonclassical, depend on κ

ut + (u3)x = 0

-4

-3

-2

-1

0

1

2

3

4

5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 , -6

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Two shocks A shock + A rarefaction

The Riemann solutions are distinct from the ones selected by Oleinik’sentropy inequalities.

Thin liquid film model.

I Conservation law (with β, γ > 0)ut + (u2 − u3)x = −γ (u3 uxxx)x

Effect of surface tension(Bertozzi-Shearer, Zumbrun, Otto-Westdickenberg, PLF-Mohamadian)

I Single entropy inequality:(u log u − u

)t+ F (u)x = −D + Cx ,

with D = γ |(u2 ux)x |2 ≥ 0, thus in the limit γ → 0(u log u − u

)t+ F (u)x ≤ 0.

x

u

0 250 500 75000.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.8

x

u

0 250 500 75000.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.8

Oscillations concentrated near shocks.

Generalized Camassa-Holm model.

I Conservation law (with β > 0)

ut + f (u)x = β uxx + γ(utxx + 2ux uxx + u uxxx

)Shallow water model for wave breaking(Bressan-Constantin, Karlsen-Coclite, Raynaud, PLF-Mohamadian)

I Single entropy inequality:`(u2 + β |ux |2)/2

´t + F (u)x =− β |ux |2 + Cx .

Here, numerical solutions are similar to, but do not coincide with, theones obtained with the linear diffusion-dispersion model.

Limiting solutions depend on the regularization.

Van der Waals fluids.

I Two conservation laws (also the energy equation may be included):

vt − ux= 0

ut + p(v)x =(β(v) ux

)x

+(γ′(v)

v2x

2−

(γ(v) vx

)x

)x

v : specific volume u : velocityβ(v): viscosity γ(v) : capillarity

p(v , T ) = RTv−b

− av2 . Hyperbolic when T sufficiently large, but of mixed

type otherwise.

I Single entropy inequality:(ε(v) +

u2

2+ γ(v)

v2y

2

)t+

(p(v) u

)x

=− β(v) u2x + Cx .

Nonclassical behavior for Van der waals fluids

0.5

1

1.5

2

2.5

3

3.5

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

lambda=1e-5lambda=1e-1lambda=0.75

,-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

lambda=1e-5lambda=1e-1lambda=0.75

Specific volume v Velocity u

Solutions depend upon the ratio λ = (viscosity)2/capillarity.

Ideal magnetohydrodynamics with Hall effect.(simplified version)

vt + ((v2 + w2) v)x = ε vxx + αεwxx

wt + ((v2 + w2) w)x = εwxx − αε vxx

(v ,w): transverse components of the magnetic field.ε: magnetic resistivity, α: Hall parameter (solar wind).

(1/2)(v2 + w2

)t+ (3/4)

((v2 + w2)2

)x

=− ε (v2x + w2

x ) + Cx .

When α = 0: Brio, Hunter, Freistuhler, Pitman, Panov, Wu, Kennel.

When α 6= 0: PLF-Mishra — Radius variable r = (v2 + w2)1/2.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−3

−2

−1

0

1

2

3

4

5

EC2:−−−−−−−−−−−−−−−−−−−−−−−−−−−

EC4:− − − − − − − − − −

EC6:− − − − − − −

EC8:o o o o o o o o

EC10:+ + + + + + + + +

Solutions depend on the (order of the) scheme.

Further regularizations and models.

I Buckley-Leverett equation for two-phase flows in porous media(Hayes-Shearer, van Duijn, Peletier, Pop, Y. Wang)

I Quantum hydrodynamics (Marcati, Jerome)

I Phase field models

I Suliciu model (Tzavaras, Bouchut, Frid).

I Non-local, integral and fractional regularization terms (Rohde,Kissling, Karlsen, PLF).

I Discrete molecular models (Truskinovsky, Weinan E, etc).

FOR ALL THESE MODELS

I Complex wave patterns

I Different ratio/regularizations/schemes yield different solutions.

I Non-convex flux-functions. A single entropy inequality.

FOR CONVEX FLUX-FUNCTIONS

I One entropy is sufficient (Panov; Delellis, Otto, Westdickenberg)

I Shocks are regularization-independent.

I Classical entropy solutions (compressive shocks, Lax inequalities).

WHAT WE NEED

I Include macro-scale effects without resolving the small-scales.

I Encompass nonclassical entropy solutions (containingundercompressive shocks having fewer impinging characteristics).

No “universal” admissibility criterionbut “several hyperbolic theories”,each being determined by specifying a physical regularization.

.................. KINETIC RELATION

3. NONCLASSICAL RIEMANN SOLVER

For simplicity in the presentation, considerut + f (u)x = 0

I Concave-convex flux

u f ′′(u) > 0 for u 6= 0

f ′′′(0) 6= 0, limu→±∞

f ′(u) = +∞

I Tangent function ϕ\ : R → R and its inverse ϕ−\

f ′(ϕ\(u)) =f (u)− f

`ϕ\(u)

´u − ϕ\(u)

, u 6= 0

uu

!

u

ul

l

ul

l

l

l

! ( )

u! ( )

( )

! u( )

N

C RN+CN + R

r

Weak solutions.

I u ∈ L∞ in the sense of distributions∫∫ (u ϕt + f (u)ϕx

)dxdt = 0

for every smooth, compactly supported function ϕ.

I If u ∈ BVloc ∩ L∞ (bounded variation), then ut and ux are locallybounded measures and

ut + f (u)x = 0 as an equality between measures.

Shock wave solutions. For (u−, u+) ∈ R2, consider

u(t, x) =

{u−, x < λ t,

u+, x > λ t.

The Rankine-Hugoniot relation

−λ (u+ − u−) + f (u+)− f (u−) = 0

determines the shock speed

λ =f (u−)− f (u+)

u− − u+=: a(u−, u+).

Oleinik entropy inequalities for shocks.Recalled here for the sake of comparison only:

f (v)− f (u+)

v − u+≤ f (u+)− f (u−)

u+ − u−

for all v between u− and u+.

I Equivalent to the Kruzkov’s entropy inequalities.

I Equivalent also to imposing all of the entropy inequalities

U(u)t + F (u)x ≤ 0,

U ′′ > 0, F ′(u) := f ′(u) U ′(u).

A single entropy inequality.

U(u)t + F (u)x ≤ 0, U ′′ > 0, F ′(u) := f ′(u) U ′(u)

E (u−, u+) := − f (u−)− f (u+)

u− − u+

(U(u+)− U(u−)

)+ F (u+)− F (u−)

≤ 0

Observe that

E (u−, u+) = −∫ u+

u−

U ′′(v) (v−u−)( f (v)− f (u−)

v − u−− f (u+)− f (u−)

u+ − u−

)dv .

Zero entropy dissipation function ϕ[0 : R 7→ R.

E (u, ϕ[0(u)) = 0, ϕ[

0(u) 6= u ( when u 6= 0)

(ϕ[0 ◦ ϕ[

0)(u) = u.

This follows from (for u− 6= u+):

∂u+E (u−, u+) = b(u−, u+) ∂u+a(u−, u+),

b(u−, u+) := U(u−)− U(u+)− U ′(u+) (u− − u+) > 0,

∂u+a(u−, u+) =f ′(u+)− a(u−, u+)

u+ − u−.

Riemann problem. u(x , 0) =

(ul , x < 0

ur , x > 0

A single entropy inequality allows for:

I Classical compressive shocks

u− > 0, ϕ\(u−) ≤ u+ ≤ u−

satisfying Lax shock inequalities

f ′(u−) ≥ f (u+)− f (u−)

u+ − u−≥ f ′(u−).

I Nonclassical undercompressive shocks

u− > 0, ϕ[0(u−) ≤ u+ ≤ ϕ\(u−),

having all characteristics passing through

min(f ′(u−), f ′(u−)

)≥ f (u+)− f (u−)

u+ − u−.

The cord connecting u− to u+ intersects the graph of f .

I Rarefaction waves.Lipschitz continuous solutions u depending only upon ξ := x/t:

−ξ u(ξ)ξ + f (u(ξ))ξ = 0,

thusu(t, x) := (f ′)−1(x/t),

provided f ′ is invertible on the interval under consideration.

Precisely, a rarefaction consists of two constant states separated by asmooth part:

u(t, x) =

u−, x < t f ′(u−),

(f ′)−1(x/t), t f ′(u−) < x < t f ′(u−),

u+, x > t f ′(u+),

providedf ′(u−) < f ′(u+)

andf ′ is strictly monotone on the interval limited by u− and u+.

Evolutionary vs. non-evolutionary.

I Compressive shocks arise from smooth initial data.

For instance, using the method of characteristics

u(t, x) = u`0, x − t f ′(u(t, x))

´,

one sees that the implicit function theorem generally fails t is too large.

I Compressive shocks arise also from singular limits.

For instanceut + f (u)x = ε uxx .

I Undercompressive shocks arise from singular limits only.

Provided oscillations take place, for instance with

ut + f (u)x = ε uxx + αε2 uxxx .

The Riemann problem admits (up to) a one-parameter family of solutionssatisfying a single entropy inequality.

One can combine an arbitrary nonclassical shock plus a classical one.

An additional admissibility criterion is needed.

Entropy-compatible kinetic function.I A monotone decreasing, Lipschitz continuous function ϕ[ : R 7→ R

ϕ[0(u) < ϕ[(u) ≤ ϕ\(u), u > 0

I The kinetic relation u+ = ϕ[(u−) singles out one nonclassical shock.

Example. f (u) = u3, ϕ\(u) = −u/2, ϕ[0(u) = −u.

-16

-14

-12

-10

-8

-6

-4

-2

0

2 4 6 8 10 12 14

2nd order scheme4th order schemeclassical solution

TW solutionextreme nonclasssical solution

, uu

!

u

ul

l

ul

l

l

l

! ( )

u! ( )

( )

! u( )

N

C RN+CN + R

r

Observe that:

I Extremal choices:

ϕ[ = ϕ\ (classical solution, all convex entropies)ϕ[ = ϕ[

0 (dissipation-free, one entropy equality)

I Equivalently, prescribe the entropy dissipation rate.

I The property (ϕ[0 ◦ ϕ[

0)(u) = u implies the contraction property

|ϕ[(ϕ[(u)

)| < |u|, u 6= 0.

Notation: Companion (threshold) function ϕ] : R → R

Nonclassical Riemann solver. For instance, suppose ul > 0.

I ur ≥ ul : rarefaction wave.

I ur ∈ [ϕ](ul), ul): classical shock.

I ur ∈ (ϕ[(ul), ϕ](ul)): nonclassical shock

(ul , ϕ

[(ul))

+ classical

shock(ϕ[(ul), ur

).

I ur ≤ ϕ[(ul) : nonclassical shock(ul , ϕ

[(ul))

+ rarefaction wave.

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

u

-2

-1.5

-1

-0.5

0

0.5

1

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

u

Conclusion.Given a kinetic function ϕ[ compatible with an entropy, the Riemannproblem admits a unique solution, satisfying :

I hyperbolic conservation law with Riemann initial data

I single entropy inequality plus a kinetic relation u+ = ϕ[(u−)

L1 continuous dependence property.

‖u(t)− v(t)‖L1(K) ≤ C (T ,K ) ‖u(0)− v(0)‖L1(K)

for all t ∈ [0,T ] and all compact set K ⊂ R.

But, the solutions contain “spikes” – some states are discontinuous.

Generalization (piecewise smooth solutions to the Riemann problem).

I 2× 2 isentropic Euler equations and nonlinear elasticity or phasetransition system

– uniqueness if hyperbolic

– non-uniqueness if hyperbolic-elliptic

(Slemrod, Truskinovsky, Shearer et al., PLF-Thanh,

Hattori, Mercier-Piccoli, Corli-Tougeron)

I N × N strictly hyperbolic systems of conservation laws.Hayes-PLF (SIAM J. Math. Anal., 2000).

4. KINETIC FUNCTIONS ASSOCIATED WITH TRAVELING WAVES

TW analysis. For simplicity in the presentation, consider

ut + f (u)x = β(|ux |p ux

)x

+ uxxx

f concave-convex, β > 0, p ≥ 0.

Second-order ODE:

u(x , t) = u(y), y = x − λ t

− λ (u − u−) + f (u)− f (u−) = β |u′|p u′ + u′′

with boundary conditions

limy→±∞

u(y) = u±

and prescribed data u±, λ satisfying the Rankine-Hugoniot relation.

Remark. The small parameter ε has been removed by scaling. Theproper scaling is ε2 for the diffusion and ε for the dispersion.

Main issues. (LeFloch-Bedjaoui, 2001)

I existence of classical / nonclassical traveling waves

I Kinetic function ϕ[ associated to this model ?

I Monotonicity ?

I Behavior near u = 0 ?

Earlier result for the cubic flux. Explicit formulas

I p = 0 : Shearer et al. (1995)

I p = 1 : Hayes - PLF (1997). Here, ϕ[′(0) = ϕ[0(0) = −1.

Traveling wave analysis.

I Phase plane analysis:

I First-order system in the plane (u, u′).

I The equilibria are the solutions u1 < u2 < u3 = u− to

−λ (u − u−) + f (u)− f (u−) = 0.

For λ and u− fixed, there exist two non-trivial solutions.

I Equilibria are saddle points (two real eigenvalues with opposite signs)or nodes (two eigenvalues with same sign).

I Existence of:saddle-node connections from u− to u2 (classical shocks)andsaddle-saddle connections from u− to u1 (nonclassical shocks).

I Dependence upon u−, λ, β, p.

Admissible shocks

S(u−) :=˘u+ / there exists a TW connecting u±

¯Theorem. (Bedjaoui - PLF, 2001 & 2004).(i) Kinetic function ϕ[ : R → R, Lipschitz continuous, strictly decreasing,

S(u) ={ϕ[(u)

}∪

(ϕ](u), u

], u > 0

ϕ[0(u) < ϕ[(u) ≤ ϕ\(u), u > 0

(ii) Threshold function A\ such that

I 0 ≤ p ≤ 1/3 :

A\ : R → [0,∞) Lipschitz continuous, A\(0) = 0

ϕ[(u) = ϕ\(u) iff β ≥ A\(u)

I p > 1/3 :ϕ[(u) 6= ϕ\(u) (u 6= 0)

(iii) Behavior of infinitesimally small shocks:

I p = 0: ϕ[′(0) = ϕ\′(0) = −1/2

A\(0) = 0, A\′(0±) 6= 0

I 0 < p ≤ 1/3 : ϕ[′(0) = −1/2

A\(0) = 0, A\′(0±) = +∞

I 1/3 < p < 1/2 : ϕ[′(0) = −1/2

I p = 1/2 : ϕ[′(0) ∈(ϕ−[

0

′(0),−1/2

)= (−1,−1/2)

limβ→0+

ϕ[′(0) = −1, limβ→+∞

ϕ[′(0) = −1/2

I p > 1/2 : ϕ[′(0) = −1

Remark. Useful in the BV existence theory for the initial value problem.

Remark.Explicit formula for the cubic flux f (u) = u3.Recall that ϕ\(u) = −u/2, ϕ[

0(u) = −u.

I p = 1/2 : Linear kinetic function

ϕ[(u) = −cβ u, cβ ∈ (1/2, 1).

Conclusion.To the augmented model one can associate a unique kinetic functionwhich is monotone and satisfies all the assumptions required in thetheory of the Riemann problem.

Generalizations.

I 2× 2 Nonlinear elasticity/Euler equations (non-nec. monotone)(PLF-Bedjaoui, Truskinovsky, Benzoni, Shearer)

I 2×2 Van de Waals model (two inflection points) (multiple solutions)(Bedjaoui-Chalons-Coquel-PLF).

Partial results (on traveling waves).

I Thin liquid film model(Bertozzi, Shearer, Munch).

I Generalized Camassa-Holm model(Constantin, Strauss, Lenells).

5. EXISTENCE THEORY for NONCLASSICAL ENTROPY SOLUTIONS

For simplicity, consider the conservation law with concave-convex flux

∂tu + ∂x f (u) = 0

u(x , 0) = u0(x) u0 ∈ BV (R),

f concave-convex.

That is, u0,x is a bounded measure and the total variation TV (u0) is themass of this measure.

Nonclassical Rieman solver based on a kinetic function ϕ[

Dafermos front tracking method.

Piecewise constant approximations

uh : R+ × R → R.

I u0 replaced by a piecewise constant approximation.

I At t = 0, solve a Riemann problem at each jump point.

I Replace rarefaction waves by several small fronts, traveling with theRankine-Hugoniot speed.

I Solve a new Riemann problem at each wave interaction.

Convergence ? Main difficulties:

I Show that the number of waves and interaction points remain finite.(For scalar equations, at most two waves in each Riemann solution.)

I Lack of monotonicity and possible increase of the (standard) totalvariation.

I For systems, lack of regularity of the wave curves.

Assumptions on the kinetic function.

I ϕ[ : R → R: Lipschitz continuous, monotone decreasing.

I The second iterate of ϕ[ is a strict contraction: for K ∈ (0, 1)∣∣ϕ[ ◦ ϕ[(u)∣∣ ≤ K |u|, u 6= 0.

Remark. Since ∣∣ϕ[ ◦ ϕ[(u)∣∣ < |u|

for u 6= 0, this is only a condition at u = 0, about nonclassical shockswith infinitesimally small strength.

Notion of generalized wave strength (Laforest-LeFloch, 2009).

σ(u−, u+) = |ψ(u−)− ψ(u+)|, ψ(u) =

{u, u > 0,

ϕ[0(u), u < 0.

Properties.

I Compare states with the same sign.

I “Equivalence” with the standard strength:

C |u− − u+| ≤ σ(u−, u+) ≤ C |u− − u+|.

I Continuity as u+ crosses ϕ](u−) during a transition from a singlecrossing shock to a two wave pattern:

σ(u−, ϕ](u−)) =

∣∣u− − ϕ[0 ◦ ϕ](u−)

∣∣=

∣∣u− − ϕ[0 ◦ ϕ[(u−)

∣∣ +∣∣ϕ[

0 ◦ ϕ[(u−)− ϕ[0 ◦ ϕ](u−)

∣∣= σ(u−, ϕ

[(u−)) + σ(ϕ[(u−), ϕ](u−)).

Generalized TV functional.

For a piecewise constant function u = u(t, ·) made of shock orrarefaction fronts (uα

−, uα+), the generalized total variation

V(u(t)

):=

∑α

σ(uα−, u

α+)

is “equivalent” to the total variation

TV(u(t)

):=

∑α

∣∣uα− − uα

+

∣∣.

Classification of wave interaction patterns.

When two fronts (ul , um) and (um, ur ) meet........................... 20 cases.

Case CR-4. (C↓±R↓−)–(N↓

±′C↑−′)

Monotone to non-monotone:

ϕ[(ul) < ur < ϕ](ul) < um ≤ 0 < ul .

The standard total variation TV(uh(t)

)increases, but V

(uh(t)

)decreases.

Case NC. (N↓±C↑)–(C↓′)

Non-monotone to monotone:

um = ϕ[(ul) and ϕ](ul) < ur < ϕ](um) < ul .

Both the standard total variation TV(uh(t)

)and the generalized one

V(uh(t)

)decrease.

Proposition (Laforest - PLF, 2009). The generalized total variationfunctional V = V

(uh(t)

)is non-increasing along a sequence of front

tracking approximations.

More precisely, at each interaction

[V

]≤

−2σ(R in), Cases RC-1, RC-3, CR-1, CR-2, CR-4,

−C σ(R in), Cases RC-2, RN,

−2(σ(R in)− σ(Rout)

), Case CR-3,

0, all other cases,

where R in and Rout denote the incoming/outgoing rarefactions, and

C :=Lip

((ϕ[

0)−1 ◦ (id− ϕ[)−1

)Lip(ϕ[

0)) .

Theorem [Existence of nonclassical entropy solutions, Baiti-PLF-Piccoli(2001), Laforest -PLF (2009)].

‖uh(t)‖L∞(R) . ‖u0‖L∞(R),

TV (uh(t)) . TV (u0),

‖uh(t)− uh(s)‖L1(R) . |t − s|,

and uh converge in L1 to a weak solution satisfying the entropy inequalityand the kinetic relation.

Remarks.I Compactness follows from Helly’s compactness theorem.

I Behavior near u = 0 important to prevent blow-up of the TV.

I Assumption ϕ[′(0−) ϕ[′(0+) < 1, satisfied by kinetic functions generatedby nonlinear diffusion-dispersion

α`b(u, ux) |ux |p ux

´x+

`c1(u) (c2(u) ux)x

´x

provided p < 1/2.

I Counter-example of blow-up of TV if ϕ[(0) = −1.

Alternative strategy of proof (Baiti-PLF-Piccoli, 2001).

However:

I Applies only to scalar equations.

I Requires stronger conditions on the kinetic function.

Technique:

I Decomposition into intervals where uh(t, ·) is alternativelyincreasing/decreasing.

I Use Fillipov-Dafermos’s theory of generalized characteristics to trackthe maxima and minima.

I Compute the total variation.

Wave interaction potential.

Q(u(t)

):=

∑α,β approaching

σ(uα−, u

α+)σ(uβ

−, uβ+),

provided not two of them being rarefactions.

It is bounded by C TV(u(t)

)2.

Remark. For classical entropy solutions, t 7→ Q(uh(t)) decreasing along a

sequence of front-tracking solutions.

Theorem (Nonclassical entropy solutions).1. t 7→ Q(uh(t)) decreasing along a sequence of front-tracking solutions,except at interactions ...→ N + ...

2. The potential Q(uh(t)) is globally decreasing in the case of a“splitting-merging” pattern (C → N + C , but later N + C → C ).

Generalization.

I Kinetics and nucleation: PLF-Shearer (2005).

I Hyperbolic systems:

I Perturbation of a nonclassical wavePLF (1993), Corli-Tougeron (2002), Colombo-Corli (2002), Hattori

(2003), Laforest-PLF (2009).

I The new functional opens the way to further investigations.

Uniqueness.

I Classical setting: Bressan-LeFloch (1997, tame variation),extended by Bressan with Goatin and Lewicka.

I Nonclassical setting:Baiti-PLF-Piccoli (JDE, 2001)

L1 continuous dependence.

I Classical setting : Bressan et al., LeFloch et al., Liu-Yang.

I Nonclassical setting: open problem.

Further reading. Lect. in Math., ETH Zurich, Birkhauser.Download at http://www.ann.jussieu.fr/˜lefloch

6. ZERO DIFFUSION-DISPERSION LIMITS

I Tartar’s compensated compactness method.

I 1D scalar equations: Schonbek (1982)Hayes - PLF (1997)

PLF - Natalini (1999)I 2× 2 elasticity system: Hayes - PLF (2000)I Camassa-Holm equation : Coclite - Karlsen (2006)

I DiPerna’s measure-valued solutions.

I Multidimensional conservation laws: Correia - PLF (1999)Kondo - PLF (2001)

I With discontinuous flux: Holden - Karlsen - Mitrovic (2009)

I Lions-Perthame-Tadmor’s kinetic formulation.

I Multidimensional conservation laws: Hwang - Tzavaras (2002)Hwang (2004)

Kissling - PLF - Rohde (2009) (non-local regularization)

7. SCHEMES WITH CONTROLED DISSIPATION

For simplicity in the presentation, consider

∂tu + ∂x f (u) = ε uxx + α ε2 uxxx

I uα: the limit when ε→ 0.

I ϕ[α: the associated kinetic function.

Can we design a scheme converging to uα ?

Glimm scheme and front tracking schemes.

I Theoretical convergence results (Baiti - PLF - Piccoli)

I Numerical experimentsChalons and LeFloch, Interfaces and Free Boundaries (2003).

Level set techniques

I Hou - PLF - Rosakis (JCP, 1999). Later extended by Merkle -Rohde (2006).

I Nonlinear elasticity model, with trilinear law in two spatialdimensions.

I Complex interfacial structure. Needles attached to the boundary.

Combination of differences and interface tracking

I Hou - PLF- Zhang (JCP, 1996)

I Boutin, Coquel, Lagoutiere, PLF (2008)

These methods ensure that the interface is sharp and (almost) exactlypropagated.

Finite difference schemes.Hayes-LeFloch (SINUM, 1998).

I u∆xα : numerical solution

vα := lim∆x→0 u∆xα : the limit of the scheme.

ψ[α: the numerical kinetic function.

I Observation:vα 6= uα, ψ[

α 6= ϕ[α

Even if the scheme is “conservative”, “consistent”, ”high-order”, etc.Small scale effects play a critical role in the selection of shocks.The discrete dissipation 6= discrete dissipation.

Proposed criterion: ψ[α should be a good approximation of ϕ[

α.

Schemes with controled dissipation.

I High-order accurate, hyperbolic flux

I High-order discretization of the augmented terms (diffusion,dispersion)

I Equivalent equation coincide with the augmented physical model, upto a sufficiently high order of accuracy. For instance, for

∂tu + ∂x f (u) = ε uxx + α ε2 uxxx

we require

∂tu + ∂x f (u) = ∆x uxx + α (∆x)2 uxxx + O(∆x)p,

for p ≥ 3 at least.

References.

I Hayes - PLF (SINUM, 1998) : scalar conservation laws

I PLF - Rohde (SINUM, 2000) : third and fourth order

I Chalons - PLF (JCP, 2001) : van der Waals fluids

I PLF - Mohamadian (JCP, 2008) : very high-order schemes

Conjecture on the equivalent equation.

I PLF : As p →∞ the kinetic function ψ[α,p associated with a scheme

with equivalent equation

∂tu + ∂x f (u) = ∆x uxx + α (∆x)2 uxxx + O(∆x)p

converges to the exact kinetic function ϕ[α

limp→∞

ψ[α,p = ϕ[

α.

I PLF - Mohamadian (JCP, 2008) : Conjecture establishednumerically for several models ! See below.

The role of entropy conservative schemes

I Higher order accurate, entropy conservative, discrete flux.I Discrete version of the physically relevant entropy inequality. Hence,

preserve exactly (and globally in time) an approximate entropybalance

Entropy variable.

System of conservation laws endowed with an entropy pair:

∂tu + ∂x f (u) = 0, ∂tU(u) + ∂xF (u) ≤ 0.

I v(u) = ∇U(u): entropy variable. Suppose that U strictly convex or,more generally, f (u) is a function of v .

I Set f (u) = g(v), F (u) = G (v), B(v) = Dg(v).

I B(v) is symmetric, since Dg(v) = Df (u)D2U(u)−1. So, there existsψ(v) such that g = ∇ψ. In fact

ψ(v) = v · g(v)− G (v).

Consider (2p + 1)-point, conservative, semi-discrete schemes

d

dtuj = −1

h

(g∗j+1/2 − g∗j−1/2

),

I uj = uj(t) is an approximation of u(xj , t), and h > 0 is the meshlength

I The discrete flux

g∗j+1/2 = g∗(vj−p+1, · · · , vj+p), vj = ∇U(uj)

must be consistent with the exact flux g

g∗(v , . . . , v) = g(v).

Theorem (Second-order, Tadmor, 1984). Two-point numerical flux

g∗(v0, v1) =

∫ 1

0

g(v0 + s (v1 − v0)) ds, v0, v1 ∈ RN ,

where v is the entropy variable associated with a strictly convex entropy.

I Entropy conservative scheme, satisfying

d

dtU(uj) +

1

h

(G∗

j+1/2 − G∗j−1/2

)= 0,

with

G∗(v0, v1) =1

2(G (v0) + G (v1)) +

1

2(v0 + v1) g∗(v0, v1)

− 1

2

(v0 · g(v0) + v1 · g(v1)

).

I Second-order accurate, with (conservative) equivalent equation

∂tu + ∂x f (u) =h2

6∂x

(− g(v)xx +

1

2vx · ∂xDg(v)

).

Theorem. (Third-order, PLF - Rohde, 2000) Given any symmetric N ×Nmatrices B∗(v−p+2, · · · , vp), the (2p + 1)-point scheme associated with

g∗(v−p+1, · · · , vp) =

∫ 1

0

g(v0 + s (v1 − v0)) ds

− 1

12

((v2 − v1) · B∗(v−p+2, · · · , vp)

− (v0 − v−1) · B∗(v−p+1, · · · , vp−1))

is entropy conservative, with entropy flux

G∗(v−p+1, · · · , vp) =1

2(v0 + v1) · g∗(v−p+1, · · · , vp)

− 1

2

(ψ∗(v−p+2, · · · , vp)+ψ

∗(v−p+1, · · · , vp−1)).

When p = 2 and B∗(v , v , v) = B(v)(

= Dg(v)), this five-point scheme

is third-order, at least.

Generalization: arbitrarily high order LeFloch-Mercier-Rohde (SINUM,2002).

8. COMPUTING KINETIC FUNCTIONS. Cubic conservation law

ut + (u3)x = ε uxx + α ε2 uxxx

Relatively small α:Kinetic function Scaled entropy dissipation

φ(s)/s2 (versus shock speed s)

uL

u M

2 4 6 8 10 12 14 16 18 20

-18

-16

-14

-12

-10

-8

-6

-4

-2 Forth orderSixth orderEighth orderTenth orderExact

Shock speed

Scaledentropydissipation

100 200 300

-0.5

-0.4

-0.3

-0.2

-0.1

Forth orderSixth orderEighth orderTenth orderExact

Cubic conservation law with (relatively) large diffusion

uL

uM

2 4 6 8 10 12 14 16 18 20

-18

-16

-14

-12

-10

-8

-6

-4

-2Forth orderSixth orderEighth orderTenth orderExact

Shock speed

Scaledentropydissipation

100 200 300

-0.2

-0.15

-0.1

-0.05

Forth orderSixth orderEighth orderTenth orderExact

Kinetic function Scaled entropy dissipation

8. COMPUTING KINETIC FUNCTIONS. Camassa-Holm model

ut + (u3)x = ε uxx + α ε2(utxx + 2ux uxx + u uxxx

)Theory.

I Well-posedness for the initial-value problem

Bressan, Constantin, Karlsen, Coclite, Raynaud.

I Kinetic relations via traveling wave analysis: open problem.

Numerical investigation

I Existence of a kinetic function ? Globally monotone ?

I Relation with the linear diffusive-dispersive model ?

Shocks with moderate strength.

uL0.5 0.75 1 1.25 1.5

-1.6-1.5-1.4-1.3-1.2-1.1-1

-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2 Fourth order

Sixth orderEighth orderTenth orderEntropy bounds

The kinetic functionsfor the linear diffusion-dispersion and Camassa-Holm modelsessentially coincide for shocks with moderate strength.

Shocks with large strength.

uL50 100 150 200-200

-175

-150

-125

-100

-75

-50

-25

Fourth orderSixth orderEighth orderTenth orderEntropy bounds

uL50 100 150 200-200

-175

-150

-125

-100

-75

-50

-25

Fourth orderSixth orderEighth orderTenth orderEntropy bounds

Linear diffusion-dispersion model Camassa-Holm model

8. COMPUTING KINETIC FUNCTIONS. Van der Waals fluids.

Complex wave structure.

Initial data τL = 0.8, τR = 2, uR = 1 with variable left-hand data uL.

x

v

0 0.25 0.5 0.75 10.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

uL=1.5

x

v

0 0.25 0.5 0.75 10.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

uL=0.5

x

v

0 0.25 0.5 0.75 10.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5uL=0.2

x

v

0 0.25 0.5 0.75 10.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

uL=0.95

x

v

0 0.25 0.5 0.75 10.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

uL=0.7

x

v

0 0.25 0.5 0.75 10.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

uL=1.4

x

v

0 0.25 0.5 0.75 10.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

uL=.6

x

v

0 0.25 0.5 0.75 10.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

uL=1.1

x

!

0 0.25 0.5 0.75 10.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

uL=1.3

!

Better described... with the kinetic function.

Kinetic function.

For τ near to 1: existence and monotonicity.

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

0.65 0.655 0.66 0.665 0.67 0.675 0.68 0.685

lambda=0.4lambda=0.5lambda=0.7

Maxwell curve

u-

u +

0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.91.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

Fourth orderSixth orderEighth orderTenth order

!+

!"

Varying the capillarity coefficient Varying the order of the discretization.

8. COMPUTING KINETIC FUNCTIONS.Magnetohydrodynamics with Hall effect.

LeFloch - Mishra (2009)Kinetic function Scaled entropy dissipation

2 4 6 8 10 12 14 16 18 20−15

−10

−5

0

EC2−−−−−−−−−−−−−−−−−−−−−−−−−

EC4:−o−o−o−o−o−o−o−o−o−o−o

EC6:−#−#−#−#−#−#−#−#−#−#

EC8:−x−x−x−x−x−x−x−x−x

EC10:−+−+−+−+−+−+−+

,

0 50 100 150 200 250 300 350−0.76

−0.74

−0.72

−0.7

−0.68

−0.66

−0.64

−0.62

−0.6

−0.58

−0.56

EC2:−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

EC4:−o−o−o−o−o−o−o−o−o−o−o−o

EC6:−#−#−#−#−#−#−#−#−#−#−#

EC8:−x−x−x−x−x−x−x−x−x−x

EC10:−+−+−+−+−+−+−+−+

Summary: schemes with controled dissipation.

I No convergence to the analytical solution.

I Practically useful schemes based on an analysis of the equivalentequation.

The numerical kinetic function approaches the exact kinetic function.

I Kinetic functions exist and are monotone for large classes ofphysically relevant models:

I thin liquid films, generalized Camassa-Holm, nonlinear phasetransitions, van der Waals fluids (small shocks),magnetohydrodynamics.

I Computing the kinetic function.Useful to investigate:

I Effects of the diffusion/dispersion ratio, regularization, order ofaccuracy of the schemes.

I Efficiency of the schemes.I Compare several physical models.

Kinetic functions associated with schemes.Hayes - LeFloch (1997)

I Beam-Warming scheme (for concave-convex flux) producesnon-classical shocks.

I No such shocks observed with the Lax-Wendroff scheme.

All of this depends crucially on the sign of the numerical dispersioncoefficient !

Approximation of nonlinear hyperbolic systems in nonconservative form:

– Hou and PLF, Why nonconservative schemes converge to wrongsolutions. Error analysis, Math. of Comput. (1994).

– Berthon, Coquel, and PLF, unpublished notes (2003).

– Castro, PLF, Munoz-Ruiz, and Pares, J. Comput. Phys. (2008).

http://philippelefloch.wordpress.com

9. THE CASE OF TWO INFLECTION POINTS

Van der Waals fluids with viscosity and capillarity

τt − ux = 0

ut + p(τ)x =(α(τ) |τx |q ux

)x− τxxx

I τ : specific volume u : velocityI α : viscosity/capillarity, q ≥ 0

I Convex/concave/convex pressure law :

p′′(τ) ≥ 0, τ ∈ (0, a) ∪ (c ,+∞)

p′′(τ) ≤ 0, τ ∈ (a, c), p′(a) > 0

!

p

1 2 3

0.8

1

1.2

1.4

1.6

1.8

2

Traveling wave analysis.

2-wave issuing from (τ0, u0) at −∞ and with speed λ > 0:

λ (τ − τ0) + u − u0 = 0

λ (u − u0)− p(τ) + p(τ0) = −α(τ)|τ ′|qu′ + τ ′′.

Phase plane analysis in the plane (τ, τ ′):

I Second-order differential equation + an algebraic equation.

I Fix a left-hand state τ0 and a speed λ within the interval wherethere exist three other equilibria τ1, τ2, τ3.

-4

-2

0

2

4

6

8

10

0 2 4 6 8 10 12

Pressure pLine d with lambda = 0.80Line d with lambda = 0.85Line d with lambda = 0.90

Entropy inequality.

−λ U ′ + F ′ = −α(τ)|τ ′|q(u′)2 < 0

with entropy

U := −∫ τ

p(s) ds +u2

2+

(τ ′)2

2

and entropy flux

F := u p(τ) + λ (τ ′)2 + u τ ′′ − u α(τ) |τ ′|q u′.

Lemma. (Classification of the equilibrium points.)

I For all q ≥ 0, the equilibria (τ0, 0) and (τ2, 0) are saddle points (tworeal eigenvalues with opposite signs).

I For q = 0 and i = 1, 3, the point (τi , 0) is :

I a stable node (two negative eigenvalues) if p′(τi ) + λ2 ≤ (λα(τi ))2/4

I a stable spiral (two eigenvalues with the same negative real part andwith opposite sign and non-zero imaginary parts) ifp′(τi ) + λ2 > (λα(τi ))

2/4.

I For q > 0 the equilibria (τ1, 0) and (τ3, 0) are centers (two purelyimaginary eigenvalues).

Theorem. (Bedjaoui, Chalons, Coquel, PLF, 2005). There exists adecreasing sequence of diffusion/dispersion ratio αn = αn(τ0, λ) → 0 forn ≥ 0 such that:

I α(τ0) = αn (nonclassical) TW with n oscillations connecting τ0 toτ2.

I α(τ0) ∈ (α2m+2, α2m+1) ∪ (α0,+∞)(classical) TW connecting τ0 to τ1.

I α(τ0) ∈ (α2m+1, α2m) (classical) TW connecting τ0 to τ3.

Remark.In the case of a single inflection point one would have a single criticalvalue α0(τ0, λ), only.

Nonclassical trajectories. Infinitely many, associated to a sequenceαn → 0:

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.5 1 1.5 2 2.5 3

Trajectory connecting tau0 and tau2 without oscillation

,-4

-3

-2

-1

0

1

2

3

4

0 2 4 6 8 10 12 14

Trajectory connecting tau0 and tau2 with one oscillation (increasing part)Trajectory connecting tau0 and tau2 with one oscillation (decreasing part)

-4

-3

-2

-1

0

1

2

3

4

0 2 4 6 8 10 12 14

Trajectory connecting tau0 and tau2 with two oscillations (first increasing part)Trajectory connecting tau0 and tau2 with two oscillations (decreasing part)

Trajectory connecting tau0 and tau2 with two oscillations (second increasing part)

, 0

2

4

6

8

10

12

14

-5 0 5 10 15 20 25

Trajectory connecting tau0 and tau2 without oscillationTrajectory connecting tau0 and tau2 with one oscillation

Trajectory connecting tau0 and tau2 with two oscillations

New features found with the van der Waals model.

I Non-classical trajectories may be non-monotone.

I Several kinetic functions: u+ = ϕ[α(u−), that is, the right-hand side

is not unique.

I Non-uniqueness for the Riemann problem.

10. RIEMANN SOLVER WITH KINETICS AND NUCLEATION

Joint work with M. Shearer (scalar equations) and M. Laforest (systems).

Available shocks after imposing a kinetic relation:

I All nonclassical shocks (u−, u+) satisfy the kinetic relationu+ = ϕ[(u−).

I All classical shocks : ϕ\(u−) ≤ u+ ≤ u−.

Riemann problem. Given data ul , ur with ul > 0 there are actually stilltwo solutions for every ur < ϕ](ul):

I Classical Riemann solution:

I Single shock (if ur > ϕ\(ul)),I or (right-characteristic) shock plus rarefaction (if ur < ϕ\(ul)).

I Nonclassical Riemann solution:Undercompressive shock

(ul , ϕ

[(ul))

and faster wave(ϕ[(ul), ur

),

which is

I either a classical shock (if ur > ϕ[(ul))I or a rarefaction (if ur < ϕ[(ul)).

Nucleation criterion.

I Nucleation threshold:Lipschitz continuous function ϕN satisfyingϕ\(ul) ≤ ϕN(ul) ≤ ϕ](ul) for ul > 0.

I Impose the condition

If ur ≤ ϕN(ul), the solution is nonclassical;

it is classical, otherwise.

Remarks.

I A large initial jump always “nucleates”.Similar criterion proposed in material science: nucleation in

austenite-martensite materials (Abeyaratne - Knowles)

I Extremal choices :ϕN = ϕ\ (fully classical)ϕN = ϕ] (fully nonclassical)

Nonclassical Riemann solver with kinetics and nucleation.

I Solution that satisfies the entropy inequality, the kinetic relation,and the nucleation criterion.

I Admissible waves:Waves satisfying the entropy inequality, the kinetic relation and thenucleation criterion.

Difficulties.

I Nucleation solver not uniquely characterized by the set of admissiblewaves.

I Some Riemann problems have two solutions.

I Nucleation solver not continuous in L1 w.r.t. the initial data (unlessϕN ≡ ϕ]). At the transition value ur = ϕN(ul).

Relevance of the Riemann solver with nucleation.

I Models with second-order diffusion/third-order dispersion.Lax shocks lose TW profiles precisely when an undercompressiveshock admits a TW, so ϕN = ϕ].

I Models for which TW analysis does not lead to a unique Riemannsolver and instabilities and complex large-time behavior are observed.

I Hyperbolic-elliptic systems modeling phase transitions.In a range of data, the TW analysis allows for two distinct solutionscontaining zero or two propagating phase boundaries.

I Higher-order regularizations: thin liquid films, Camassa-Holm, etc.

Applications to the thin liquid film model.

I Make direct numerical simulation of the augmented model withdiffusion and dispersion.

I The nucleation function could be determined numerically. Find arange where both classical and nonclassical connections are available.

Work in progress. To what extend the hyperbolic theory with kinetics andnucleation is valid ?

I Instabilities (one-wave / two-wave patterns) observed numerically.

I Difficult to sort out the competition between viscosity, capillarity,time and space step sizes, perturbation, etc.

I Hyperbolic model valid in a range of parameters only.

Existence results.

I Front tracking scheme applies with the solver with kinetics andnucleation.

I Same TV estimate. Same convergence theory.

I Qualitative properties of solutions ? Quite different when thenucleation criterion is effective.

Observation.

I Instability phenomena.

I Limiting front-tracking solution depends on the approximation of theinitial data u0.

Perturbations of Riemann data.

limx→−∞

u0(x) = ul , limx→+∞

u0(x) = ur = ϕN(ul).

Two possible asymptotic patterns as t → +∞ :

a single shock or a two-wave solution ?

Example 1. Two possible time-asymptotic behaviors.

ur ∈ (ϕN(ul), ϕ](ul)), u

(1)m ∈ (ϕ\(ul), ϕ

N(ul)), u(2)m ∈ (ϕN(ul), ur )

xx

tt

u u

u

u u

u u

u

u

u u

u

mm

u00

(1)

um(1)

(2)

um(2)

u! ( )N

u

f f

u! ( )N

u! ( )

u! ( )

u! ( )

=

l

l l

l

ll

ll

l

r

rr

r

C-C −→ C N-C-C −→ N-C

Example 2. Order of wave interactions matters.

u1 = ϕ[(ul), u2 = ϕ[(u1), ϕN(ul) < ur < ϕ](ul).

uul

ul

ulul

uu u

l

l

u u

ru

uu

ru

ru

u

ru

1

u1

u1u1

u1

2

f

u! ( )= u2

u2

u2

u2

u1! ( )=

aa bb x

x

x

x

00

t > 0

t = 0

r

(i) a large, b − a small. (ii) a small, b − a large.N-N-C −→ N-C N-N-C −→ C

Example 3. Splitting-merging pattern.

ϕ[(ul) < u2 < ϕN(ul) < u1 < ϕ](ul) < ur < ϕN(ϕ[(ul)) < ul

x

t

u uu

u

u

0

uu

1

u1

2

u2

u

u! ( )N

u! ( )

l

l

l

l

r

ru =

C

C

C

C

N

C-R-C −→ N-C-C −→ N-C −→ C

Splitting-merging initial data.

I Fix u∗ > 0 and u0(x) = uN0 (x) + v0(x)

uN0 (x) :=

{u∗, x < 0

ϕN(u∗), x > 0I Two parameters in the problem : η << ε

TV (v0) < ε << 1η := ϕ](u∗)− ϕN(u∗) << 1

I The initial data uN0 gives rise to two distinct solutions made of

admissible waves.

Example. v0(x) =

{0, x < 0

δ, x > 0

I When δ > 0, a single classical shock C↓:

u↓(x , t) :=

{u∗, x < s t

ϕN(u∗) + δ, x > s t

I When δ < 0, a two-wave solution u↓↑ consisting of anundercompressive shock N↓ plus a classical shock C↑.

Splitting/merging structure.

I One or two big waves at each time t > 0.

I Solution close to a one-wave solution u↓ or a two-wave solution u↓↑

I Notation: x = y(t) : locus of big shocks N↓ and C↓

x = z(t) : locus of the big shock C↑

Main issue. Generalized TV functional for front tracking solutionsuh = uh(t, x)

x

t

0

C

C

C

C

C

N

N

x =

x = y(t)

z ( t )

Generalized wave strength.

V (t) :=∑

jumps(u−,u+)

σ(u−, u+)

I Generalized strengths of the big increasing classical shock located atz = z(t)

σC (u−, u+) := ϕ[(u−)− ϕ[(u+) > 0.

I Nonclassical shock at y = u(t)

σNC (u) := (u − ψ(u))− (ϕ[ ◦ ψ(u)− ϕ[ ◦ ϕ[(u))

I Standard definition for the big decreasing classical shock.

Properties.

I Continuity/decreasing properties above, since for u > 0

ψ(u) < ϕ[ ◦ ψ(u) < ϕ[ ◦ ϕ[(u) < u.

I The generalized strength σ(u) is strictly positive.

Notation.

I Total strength of small waves in each region :

V hleft(t) := TV

yh(t)−∞ (uh(t)), V h

middle(t) := TVzh(t)yh(t)

(uh(t))

V hright(t) := TV +∞

yh(t)(uh(t))

V h(t) = V hleft(t) + κ0 V h

middle(t) + κ0 V hright(t)

I Total strength of big waves : W h(t)

Theorem [LeFloch - Shearer]

I Front tracking solutions uh = uh(x , t) have the splitting/mergingstructure, with

V h(t) + κ2 W h(t) ≤ V h(0) + κ2 W h(0)

V hleft(t) ≤ V h

left(0),

V hright(t) ≤ V h

right(0)

V hleft(t) + κ1 V h

middle(t) ≤ V hleft(t) + κ1 V h

middle(t)

I The limit h → 0 yields an exact, splitting/merging solutionu = u(x , t) made of admissible waves, only.

I At each splitting, V h(t) + κ2 W h(t) decreases by at leastψ(u∗)− ϕN(u∗).At each merging, it decreases by at least ϕ](u∗)− ψ(u∗).

I If ϕN(u∗) 6= ϕ](u∗), only finitely many mergings/splittingsand the solution eventually settles to a solution having a specified(one-wave or two-wave) structure.

I When ϕN(u∗) = ϕ](u∗), the splittings/mergingsmay continue for all times.

Generalization to strictly hyperbolic systems. Laforest - PLF.

11. DLM THEORY – Kinetic relations for nonconservative systems

I Physical models for fluid mixtures (see below). Need averagingprocedure and simplifying assumptions.

I Underline the importance of small-scale phenomena for formulatinga well-posed hyperbolic theory.

I Notion of family of paths proposed by Dal Maso - PLF - Murat(1990, 1995): Φ : [0, 1]× RN × RN × RN

I Φ(·; u−, u+) is a path connecting u− to u+

Φ(0; u−, u+) = u−, Φ(1; u−, u+) = u+,

TV[0,1]

`Φ(·; u−, u+)

´. |u+ − u−|.

I Φ is Lipschitz continuous in the graph distance

dist“Φ(·; u−, u+), Φ(·; u′−, u′+)

”. |u− − u′−|+ |u+ − u′+|.

Definition. Given u ∈ BV (R,RN) and g a Borel function, there exists a

unique measure called the nonconservative product µ =[g(u) ∂xu

]Φ

such that:

I If B is a Borel subset of C(u), then µ(B) :=∫B

g(u) ∂xu.

I At a point of jump x of u, setting u± := u±(x)

µ({x}) :=

∫[0,1]

g(Φ(·; u−, u+)) ∂sΦ(·; u−, u+).

Remark.

I A definite concept of weak solutions of nonconservative systems,once a family of paths is prescribed.

I In the conservative case, this is consistent with the distributional

definition[∇h(u) ∂xu

]Φ

= ∂xh(u).

Theorem (Riemann problem, Dal Maso - PLF - Murat, 1990).Given a nonconservative, strictly hyperbolic, genuinely nonlinearsystem and a family of paths Φ, the Riemann problem admits anentropy solution (in the DLM sense) satisfying Lax shock inequalities.

Properties.

I Generalized Hugoniot jump relations

−λu (u+ − u−) +

Z 1

0

A(Φ(·; u−, u+)) ∂sΦ(·; u−, u+) = 0.

I Wave curves Lipschitz continuous at the origin

I In contrast with Lax’s standard C 2 regularity result. See Bianchini -Bressan, Iguchi - PLF, and Liu - Yang.

Theorem (Existence result, PLF - Liu, Forum Math. 1992).Glimm scheme for nonconservative, strictly hyperbolic, genuinelynonlinear system:

I uh = uh(t, x) have uniformly bounded TV

I converge to an entropy solution u in the DLM sense

I and for all but countably many times[A(uh) ∂xu

h]Φ(t) ⇀

[A(u) ∂xu

]Φ(t).

Remark.

I The theory of nonconservative systems / the theory of conservativesystems.

I Bianchini-Bressan’s theory via the vanishing viscosity method.

Paths subordinate to traveling waves

I The DLM family of paths encodes the information required for thehyperbolic theory.

I No canonical choice. Need an augmented model

∂tuε + A(uε) ∂xu

ε = ∂xR[ε ∂xu

ε, ε2 ∂xxuε, . . .

]

Definition [PLF, IMA Preprint 1989]. A DLM family of paths Φ issubordinate to the family of traveling waves if, whenever u−, u+ areconnected by a traveling wave u, then Φ(·; u−, u+) ∼ u.

Theorem. [PLF, IMA Preprint 1989]. If Φ is a DLM family subordinateto TW, and if uε = u((x − λ t)/ε) is a TW, converging to

u(t, x) := u− for x < λ t, u+ for x > λ t,

then A(uε) ∂xuε(t) ⇀

[A(u) ∂xu

]Φ(t) weak-star, and u is a weak

solution in the DLM sense.

Remark. Existence of traveling waves : Sainsaulieu (1995), Schecter(2000), Bianchini-Bressan (2002)

Example. One-dimensional nozzle flows.

I Evolution equations

∂t(aρ) + ∂x(aρv) = 0

∂t(aρv) + ∂x(aρv2 + a p(ρ))− p(ρ) ∂xa = 0

a : R → R: piecewise Lipschitz continuous. Set u := (aρ, aρv).

I Solutions obey the entropy inequality

∂tU(u, a) + ∂xF (u, a) ≤ 0

U(u, a) = a2ρu2

2+ aρ e(ρ), F (u, a) =

(U(u, a) + p(ρ)

)u

I See contributions by Bouchut; Gallouet, Herard, and Seguin; PLFand Thanh. For similar systems: Amadori, Gosse, Guerra, Jin, etc.

Example. Shallow water equations with topography.

I Evolution equations

∂tρ+ ∂x(ρv) = 0,

∂t(ρv) + ∂x

(ρv2 + g

ρ2

2

)− gρ ∂xZ = 0,

ρ: mass density; v : velocity of the fluid. Set u := (ρ, ρv).

I Prescribed topography function Z : R → R, depending on x , solelypiecewise Lipschitz continuous. g : gravity constant.

I Entropy inequality

∂tU(u, a) + ∂xF (u, a) ≤ 0,

U(u, a) := ρE (v) + ρZ , e′(ρ) =p(ρ)

ρ2,

F (u, a) := ρv3

2+ ρv e(ρ) + p(ρ)v + ρv Z .

Example. Two-fluid mixtures.

I α1: fraction of the fluid 1. α2 = 1− α1: fraction of the fluid 2

I Evolution equations with stiff source-terms

∂tα1 + VI∂xα1 = λ(p2 − p1)

∂t(α1ρ1) + ∂x(α1ρ1 u1) = 0

∂t(α1ρ1 u1) + ∂x(α1ρ1 u21 + α1 p1)− PI∂xα1 = λ (u2 − u1) + ε∂x(µ1∂xu1)

∂t(α2ρ2) + ∂x(α2ρ2 u2) = 0,

∂t(α2ρ2u2) + ∂x(α2ρ2u22 + α2p2)− PI∂xα2 = −λ(u2 − u1) + ε∂x(µ2∂xu2)

I Pressure law pi = pi (ρi ) satisfying p′i (ρi ) > 0.

I Relaxation parameter λ > 0 (large).Inverse of Reynolds number ε (small).

I Ransom and Hicks, Baer and Nunziato.Investigated by Berthon, Coquel, Gallouet, Herard, Nkonga, Seguin.

I Constitutive functions:VI : interfacial velocity. PI : interfacial pressure.For instance, Ransom and Hicks imposes

VI :=1

2(u1 + u2), PI :=

1

2(p1 + p2).

I Independently of this choice and provided the non-resonancecondition holds

|VI − ui | 6= ci (ρi ), i = 1, 2,

then the system is admits five real eigenvalues

VI , ui ± ci (ρi ),

with c2i (ρi ) := p′(ρi ) > 0, as well as a basis of right eigenvectors.

I Key issue: Closure laws for VI and PI

I Entropy balance law ?Compatibility condition

VI (p2 − p1) + PI (u2 − u1) = p2u1 − p1u2

ensuring that

U := α1ρ1E1 + α2ρ2E2, Ei :=u2

i

2+ ei (ρi )

is a mathematical entropy

∂tU + ∂xF = −λ(u2 − u1)2 − λ(p2 − p1)

2 −D,U := (α1ρ1E1 + α2ρ2E2)

F :=((α1ρ1E1 + α1p1)u1 + (α2ρ2E2) + α2p2)u2

),

D := εµ1(∂xu1)2 + εµ2(∂xu2)

2 − ε ∂x(µ1α1∂xu1 + µ2α2∂xu2).

A class of nonconservative systems endowed with an entropy.Collaboration with C. Berthon and F. Coquel

I A class of nonconservative hyperbolic models with singularsource-term

∂tv + ∂x f (v , a) = g(v , a) ∂xa.

a: given, piecewise Lipschitz continuous function of x

I Endowed with an entropy pair U,F satisfying the entropy inequality

∂tU(v , a) + ∂xF (v , a) ≤ 0.

I Reformulation as a nonconservative system: following PLF (PreprintIMA, 1989) for the nozzle flow equations. we introduce the extendedvariable u := (v , a)

∂tv + ∂x f (v , a)− g(v , a) ∂xa = 0,

∂ta = 0,

I Away from resonance, PLF-Liu existence theory applies.

Kinetic relations for nonconservative systems.

I Impose a kinetic relation for shock (u−, u+) with speed Λ

−Λ (U(u+)− U(u−)) + F (u+)− F (u−) = Φ(u−,Λ).

I Typical example: Φ ≡ 0 for standing waves and, for other waves, Φcoincides with the entropy dissipation of the given conservation laws(with a constant).

In progress.

I Riemann problem for classes of nonconservative systems withimposed kinetic relations

I Application and analysis of traveling waves to the two-fluid model.

I Numerical discretization that “tune” the entropy dissipation rate.

I Numerical experiments with the two-fluid model and plot the kineticfunction.

Some directions of research

I Physics of phase transitions and multi-fluid phenomena

I Liquid-vapor mixturesI Solid-solid phase transitions (smart shape-memory materials,

martensite-austenite, Cu-Al-Ni alloys)I Better description of the internal structure. Hidden variables.

Discrete models. Integral terms.

I Mathematical issues

I Stability of multi-dimensional interfaces

I Numerical approximation

I Physically realistic situationsI Multi-scale flows and multi-fluid mixtures