On the structure of solutions of multidimensional conservation laws with discontinuous flux

On the structure of solutions of multidimensional conservation laws with discontinuous flux

Graziano Crasta

Universit`a di Roma “La Sapienza”

11th Meeting on Nonlinear Hyperbolic PDEs and Applications On the occasion of the 60th birthday of Alberto Bressan

SISSA, Trieste, June 13–17, 2016

Based on joint works with V. De Cicco, G. De Philippis, F. Ghiraldin

Outline

1 Introduction to conservation laws with discontinuous flux

2 Assumptions on the flux

3 Weak entropy solutions (WES)

4 Existence of (generalized) traces

5 Generalized Kato inequality

Introduction to conservation laws with discontinuous flux

Conservation laws with discontinuous flux

. Scalar multi-dimensional conservation law with discontinuous flux:

(*) ut+ divxF (t, x , u) = 0, (t, x ) ∈ (0, +∞) × R^N.

The vector fieldF : (0, +∞) × R^N× R → R^N ispossibly discontinuous in (t, x ).

. Motivations: vehicular traffic models, flows in porous media, etc.

. Our results: structure of solutions of (*) and applications to uniqueness of solutions to the Cauchy problem

(u_t+ div_xF (t, x , u) = 0, (t, x ) ∈ (0, +∞) × R^N, u(0, x ) = u₀(x ), x ∈ R^N.

. Beware: in this talk we do not address the existence issue!

Introduction to conservation laws with discontinuous flux

Basic model

. Most studied: one space variable (N = 1), one point of discontinuity at the origin:

F (t, x , u) = F⁻(u) 1_{x<0}+ F⁺(u) 1_{x>0}, with F^± regular functions.

. The flux is regular in the two regionsΩ^±:= {(t, x ) : t > 0, ±x > 0}.

Discontinuities are located along the interfaceN := (0, +∞) × {0}.

. A typical definition of entropy solution requires that:

(i) the restrictions of u toΩ^± are Kruˇzkov entropy solutions;

(ii) along the interfaceN the strong tracesu^±of u satisfy some additional condition.

Introduction to conservation laws with discontinuous flux

Consequences of the first requirement:

(i) the restrictions of u toΩ^± are Kruˇzkov entropy solutions.

. Existence of strong traces ofu alongN follows from genuine nonlinearity of F^± (Panov, Vasseur, ...)

. Ifu andv are two bounded distributional solutions satisfying (i), then Z

|u(T ) − v (T )| dx ≤ Z

|u₀− v₀| dx + Z

N_T

W (u^±, v^±) d H^N where N_T := N ∩ ((0, T ) × R^N).

. In this Kato–like inequality, the additional term concentrated onN can be characterized in terms of the traces u^±, v^±:

W (u^±, v^±) := q⁺(u⁺, v⁺) − q⁻(u⁻, v⁻), where q^± are the Kruˇzkov entropy fluxes

q^±(u, v ) := sign(u − v ) [F^±(u) − F^±(v )].

Introduction to conservation laws with discontinuous flux

• Kato–like inequality

• W ≤ 0 on N







=⇒ contractivity of the semigroup and uniqueness

The additional condition in (ii) is used to show that, if u,v are entropy solutions, thenW (u^±, v^±) ≤ 0 along the interfaceN.

. It can be coded algebraically using the notion of germ (Andreianov, Karlsen, Risebro 2011) or of dissipative Riemann solver (Garavello, Natalini, Piccoli, Terracina 2007), or of transmission map (Andreianov, Canc`es 2015).

Introduction to conservation laws with discontinuous flux

Main objectives

. Notion ofweak entropy solution (WES) (including requirement (i) as a special case) for fluxes with a general set of discontinuities in several space variables.

. Structure resultsfor WES, including:

• existence of traces u^± of a WES u on the setN of discontinuities of the fluxF;

• Kato–like contraction inequality: if u andv are WES, then Z

|u(T ) − v (T )| dx ≤ Z

|u₀− v₀| dx + Z

N_T

W (u^±, v^±) d H^N where

• NT := N ∩ ((0, T ) × R^N)

• W (u^±, v^±) := q⁺(u⁺, v⁺) − q⁻(u⁻, v⁻)

• q^±(u, v ) := sign(u − v ) hF^±(t, x , u) − F^±(t, x , v ), ν(t, x )i, andν is the normal vector field toN.

Assumptions on the flux

Assumptions on the flux

Our assumptions on F are mainly modeled on the case F (t, x , u) = bF (h(t, x ), u), with F : R × R → Rb ^N regular,h in SBV.

. The jump setN ⊂ (0, +∞) × R^N ofF (·, ·, u) is independent ofu. . N is a countably H^N–rectifiable set.

In the following, we consider the more general equation

div_zA(z, u) = 0, z ∈ Rⁿ (n = N + 1, z = (t, x ), A = (u, F )).

Assumptions on the flux

div_zA(z, u) = 0, z ∈ Rⁿ.

• A(·, v ) ∈ SBV (Rⁿ, Rⁿ),A(z, ·) ∈ C¹(R, Rⁿ).

Notation: DzA(z, v ) = ∇A(z, v ) Lⁿ+ D^jA(z, v )

• |∂_vA(z, v )| ≤ M for every (z, v ), and∃ ω modulus of continuity s.t.

|∂_vA(z, v ) − ∂_vA(z, u)| ≤ ω(|v − u|), ∀z, u, v .

• ∃ g ∈ L₁(Rⁿ) s.t. |∇A(z, u) − ∇A(z, v )| ≤ g (z) |u − v |, ∀z, u, v .

• The following measure is finite:

σ := _

u∈R

|D_zA(·, u)|

(σ is the smallest measure greater thansup_u|D_zA(·, u)|.)

Assumptions on the flux

The last assumption permits to define a universal (in u) jump set forA:

N :=n

z ∈ Rⁿ: lim inf

r →0

σ(B_r(z)) rⁿ⁻¹ > 0o

. σ finite =⇒ N countablyHⁿ⁻¹ rectifiable set.

For Hⁿ⁻¹–a.e. z ∈ N and every v ∈ R the traces A^± ofA anda^± of a := ∂vA are well defined.

Finally, we assume that

• Ais genuinely nonlinear, i.e.

L¹ {v : a^±(z, v ) · ξ = 0}
= 0 for every ξ ∈ Sⁿ⁻¹ and forHⁿ⁻¹–a.e. z ∈ N.

Weak entropy solutions (WES)

Weak entropy solutions (WES) – 1D model

F (t, x , u) = F⁻(u) 1_{x<0}+ F⁺(u) 1_{x>0}

. A bounded distributional solutionu satisfies (i), i.e. its restrictions to Ω^± are Kruˇzkov entropy solutions, iff

(i’) ∃ C ≥ 0s.t. for every k ∈ R,

∂t|u − k| + ∂_x sign(u − k)[F (t, x , u) − F (t, x , k)]
≤ µ where µ(t, x ) := C δ0(x ) is a measure concentrated on the interfaceN. . IfA = (u, F ) andz = (t, x ), this condition gives

div_z

sign(u − k)[A(z, u) − A(z, k)]

≤ µ.

. We shall call weak entropy solutions the bounded distributional solutions satisfying (i’).

Weak entropy solutions (WES)

Weak entropy solutions (WES) – General case

u ∈ L^∞(Rⁿ)is a weak entropy solution (WES) of

(*) div_zA(z, u) = 0, z ∈ Rⁿ

if u is a distributional solution of (*) and for everyk ∈ R it holds (EC) div_z

sign(u − k)[A(z, u) − A(z, k)]

+ sign(u − k) div^a_zA(z, k) ≤ µ, where µis a non-negative Radon measure independent ofk and such that µ(Rⁿ\ N ) = 0.

Remark: div^a_zA(z, k) = tr∇A(z, k) is the absolutely continuous part of divzA(·, k).

Weak entropy solutions (WES)

Comparison with other entropy conditions

. Kruˇzkov: Aregular in both variables divz



sign(u − k)[A(z, u) − A(z, k)]

+ sign(u − k) divzA(z, k) ≤ 0

=⇒ existence and uniqueness for evolutionary equation . Panov: div_zA(·, k)locally finite Borel measure divz



sign(u−k)[A(z, u)−A(z, k)]



+sign(u−k) div^a_zA(z, k) ≤ | div^s_zA(z, k)|

=⇒ existence for evolutionary equation

. Vanishing viscosity(Diehl, Andreianov–Mitrovic, ...): ∃ ˆu everywhere defined representative of u s.t.

divz



sign(u − k)[A(z, u) − A(z, k)]



+ sign(u − k) div^a_zA(z, k)

≤ sign( ˆu − k) div^s_zA(z, k)

Existence of (generalized) traces

Generalized traces

Definition (Traces)

u ∈ L^∞(Rⁿ),J ⊂ Rⁿ Hⁿ⁻¹–rectifiable set oriented by ν.

Set of traces of u atz0 ∈ J: Γu,J(z0) :=

n

(c⁻, c⁺) : ∃rk ↓ 0 : u_z0,rk → c⁻1_H⁻+ c⁺1_H⁺ in L¹_loc o

, where u_z0,r(z) := u(z₀+ rz),H^±:= {z ∈ Rⁿ: ±hz − z₀, ν(z₀)i ≥ 0}.

Remark: if

u_z0,r → c⁻1_H⁻+ c⁺1_H⁺ in L¹_loc, for r ↓ 0,

then Γ_u,J(z₀) = {(c⁻, c⁺)}where(c⁻, c⁺) are the usual strong traces.

Existence of (generalized) traces

Existence of (generalized) traces

Theorem (Existence of traces)

Ifu is a WES, then for Hⁿ⁻¹ almost everyz0 ∈ N Γ_u,N(z₀) 6= ∅.

. If(c⁻, c⁺) ∈ Γ_u,N(z₀) satisfiesc⁻6= c⁺ then the traces are unique:

Γ_u,N(z₀) = {(c⁻, c⁺)}.

. OtherwiseΓu,N(z0) = {(v , v ) : v ∈ [a, b]} for somea ≤ b.

. The Rankine–Hugoniot condition holds:

A⁻(z0, c⁻) · ν(z0) = A⁺(z0, c⁺) · ν(z0) ∀ (c⁻, c⁺) ∈ Γu,N(z0).

Existence of (generalized) traces

Sketch of the proof

Kinetic formulation

. For everyk ∈ R, the distribution η_k := div_z

sign(u − k)[A(z, u) − A(z, k)]

+ sign(u − k) div^a_zA(z, k) is a Radon measure in Rⁿ (recall η_k ≤ µ), and

C_c^∞(Rⁿ× R) 3 Φ 7→ hη, Φi :=

Z Z

Rⁿ×R

Φ(z, k) d ηk(z) dk is a Radon measure in Rⁿ⁺¹.

. The function(k, z) 7→ χ(k, u(z)) := sign(u(z) − k) is a distributional solution of the kinetic equation

div_z[χ(k, u)∂_vA(z, k)] − ∂_k[χ(k, u) div^a_zA(z, k)] = −∂_kη in D⁰(Rⁿ⁺¹).

Existence of (generalized) traces

Blow-up in kinetic equation

ur(z) := u(z0+ rz), Ar(z, v ) := A(z0+ rz, v ), η_k,r(V ) := η_k(z0+ rV ) rⁿ⁻¹ . Strong pre-compactness of blow-ups: divzAr(z, ur) = 0 =⇒ (ur)r is pre-compact in L¹(B₁) (Panov).

. u^∞ cluster point of (ur): from kinetic equation we get

div_zχ(k, u^∞)∂_vA_z0(z, k) = −∂_k h(z₀) λ_z0(k) Hⁿ⁻¹ ∂H⁺) where∂vAz0(z, k) = a⁺(k) 1_H⁺(z) + a⁻(k) 1_H⁻(z),a^±(k) := ∂vA^±(z0, k).

. InH⁺ this is the transport equation a⁺(k) · ∇zχ(k, u^∞) = 0.

By GNL =⇒ u^∞= const =: u⁺ in H⁺. Similarly, u^∞= const =: u⁻ inH⁻.

Generalized Kato inequality

Generalized Kato inequality

Theorem (Generalized Kato Inequality)

Let u and v be WES. Then there exists a Borel function ρ : N → R such that the following Kato inequality holds true:

div_z

sign(u − v )[A(z, u) − A(z, v )]

≤ ρ Hⁿ⁻¹ N .

. Representation ofρ: for Hⁿ⁻¹ a.e. z ∈ N₁ := {z ∈ N : ρ(z) 6= 0},u and v admit unique traces at z and

ρ = W (u^±, v^±) := q⁺(u⁺, v⁺) − q⁻(u⁻, v⁻).

Generalized Kato inequality

Sketch of the proof

1. Doubling of variables Weak entropy condition

(EC) div_z

sign(u − k)[A(z, u) − A(z, k)]

+ sign(u − k) div^a_zA(z, k) ≤ µ, for u(z) with k = v (z⁰)and forv (z⁰) withk = u(z).

Use test function Φ(z, z⁰) = δε(z − z⁰) ϕ(z + z⁰),ϕ ∈ C_c¹(Rⁿ), δ_ε(ζ) = _ε¹nψ(^ζ_ε) standard family of mollifiers, and sum up:

[All that you expect at l.h.s. from doubling of variables]

≥ −2 Z Z

Φ(z, z⁰) dz⁰d µ(z) i.e.

I₁^ε− I₂^ε+ I₃^ε≥ −2 Z Z

Φ(z, z⁰) dz⁰d µ(z).

Generalized Kato inequality

(z⁰ = z − εw) I₁^ε=

Z Z

ψ(w )∇ϕ(2z − εw ) sign (u(z) − v (z − εw )
×

×n

A(z, u(z)) + A(z − εw , u(z))

− A(z, v (z − εw )) − A(z − εw , v (z − εw ))o dwdz, I₂^ε=

Z Z

ϕ(2z − εw ) sign u(z) − v (z − εw )
×

×n

∇ψ(w )A(z − εw , u(z)) − A(z, u(z)) ε

− ψ(w ) div^a_zA(z − εw , u(z))o dwdz, I₃^ε=

Z Z

ϕ(2z + εw ) sign u(z + εw ) − v (z)

×n

∇ψ(w )A(z, v (z)) − A(z + εw , v (z)) ε

Generalized Kato inequality

2. Limit as ε → 0

I₁^ε= Z Z

ψ(w )∇ϕ(2z − εw ) sign (u(z) − v (z − εw )
×

×n

A(z, u(z)) + A(z − εw , u(z))

− A(z, v (z − εw )) − A(z − εw , v (z − εw ))o dwdz.

Not difficult to show that

I₁^ε→ 2 Z

∇ϕ(2z) sign (u(z) − v (z)

A(z, u(z)) − A(z, v (z))
dz.

Generalized Kato inequality

. To pass to the limit inI₂^ε andI₃^ε we need a modification ofAmbrosio’s lemma on differential quotients:

For every w ∈ Rⁿ there exists a null setD ⊂ Rⁿ s.t.

A(z + εw , v ) − A(z, v )

ε = A¹_ε(z, v ) + A²_ε(z, v ) where A¹_ε andA²_ε satisfy the following properties:

(i) lim

ε↓0A¹_ε(z, v ) = ∇zA(z, v ) · w , ∀v ∈ R and z ∈ Rⁿ\ D;

(ii) h_ε(z) := |w | sup

v ∈R

A¹_ε(z, v )

 is equi-integrable;

(iii) For every compact set K ⊂ Rⁿ we have Z

K

sup

v ∈R

A²_ε(z, v )

 dz ≤ |w | σ^s(K + B_{ε|w |}).

Generalized Kato inequality

I₂^ε= Z Z

ϕ(2z − εw ) sign u(z) − v (z − εw )
×

×n

∇ψ(w )A(z − εw , u(z)) − A(z, u(z)) ε

− ψ(w ) div^a_zA(z − εw , u(z)) o

dwdz Ambrosio’s lemma:

A(z − εw , u(z)) − A(z, u(z))

ε = A¹_ε,w(z) + A²_ε,w(z) where

A¹_ε,w(z) ^L

1

−→ −∇A(z, u(z)) · wloc

and Z

|A²_ε,w(z)|ϕ(z) dz ≤ |w | kϕk∞|σ^s|(spt ϕ + B_{ε|w |}) .

Generalized Kato inequality

Hence

I₂^ε= Z Z

ϕ(2z − εw ) sign u(z) − v (z − εw )

× − ∇ψ(w )∇A(z, u(z)) · w − ψ(w ) div^a_zA(z, u(z)) dw dz + R^ε, where

lim sup

ε→0

|R^ε| ≤ C (ψ)kϕk_∞|σ^s|(spt ϕ).

Passing to the limit:

Z Z . . . →

Z Z

ϕ(2z) sign u(z) − v (z)
×

×∇ψ(w )∇A(z, u(z)) · w + ψ(w ) div^a_zA(z, u(z)) dwdz

= [integrate by parts in w ] = 0.

Generalized Kato inequality

3. Conclusion

. Same estimate forI₃^ε. . Putting all together:

div_z

sign (u(z) − v (z)

A(z, u(z)) − A(z, v (z))


≤ 2µ + C |σ^s| =: β

. Sinceβ(Rⁿ\ N ) = 0, the vector field B(z) := sign (u(z) − v (z)

A(z, u(z)) − A(z, v (z))

has divergence–measure, div B  Hⁿ⁻¹ and its generalized traces on N are

B^±(z) = sign u^±(z) − v^±(z)

A^±(z, u^±(z)) − A^±(z, v^±(z))

Generalized Kato inequality

Open problems

. Well posedness: existence and uniqueness of solutions in the general setting.

(Partial well posedness results are known in the case of the vanishing viscosity germ and flux with a regular discontinuity set.)

. Relax assumptions on the flux A:

divzA(·, k) measure ∀k σ := _

k∈R

| div_zA(·, k)| finite measure, N rectifiable.

Generalized Kato inequality

References:

. L. Ambrosio, G. Crasta, V. De Cicco, and G. De Philippis, A

nonautonomous chain rule in W^1,p andBV, Manuscripta Math. 140 (2013), pp. 461–480.

. B. Andreianov, K.H. Karlsen, and N.H. Risebro, A theory of

L¹-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Rational Mech. Anal. 201 (2011), pp. 27–86.

. G. Crasta, V. De Cicco, G. De Philippis and F. Ghiraldin, Structure of solutions of multidimensional conservation laws with discontinuous flux and applications to uniqueness, Arch. Rational Mech. Anal. 221 (2016), pp. 961-985.

Thank you for your attention and

Happy birthday Alberto!