Generalized Persistence of Entropy Weak Solutions for System of Hyperbolic Conservation Laws?

Yi ZHOU

Abstract Let u(t,x) be the solution to the Cauchy problem of a scalar conservation law in one space dimension. It is well known that even for smooth initial data the solution can become discontinuous in finite time and global entropy weak solution can best lie in the space of bounded total variations. It is impossible that the solutions belong to, for example, H1 because by Sobolev embedding theorem H1 functions are Hlder continuous.However, the author notes that from any point (t,x), he can draw a generalized characteristic downward which meets the initial axis at y =α(t,x). If he regards u as a function of (t,y), it indeed belongs to H1 as a function of y if the initial data belongs to H1. He may call this generalized persistence (of high regularity) of the entropy weak solutions.The main purpose of this paper is to prove some kinds of generalized persistence (of high regularity) for the scalar and 2×2 Temple system of hyperbolic conservation laws in one space dimension.

Keywords Quasilinear hyperbolic system,Cauchy problem,Entropy weak solution,Vanishing viscosity method

1 Introduction

The Cauchy problem for system of conservation laws in one space dimension takes the form

Here u = (u1,··· ,un) is the vector of conserved quantities, while the components of f =(f1,··· ,fn) are the fluxes. We assume that the flux function f :Rn→Rnis smooth and that the system is hyperbolic, i.e., at each point u the Jacobian matrix A(u) = ?f(u) has n real eigenvalues

and a bases of right and left eigenvectors ri(u), li(u), normalized so that

where δijstands for Kroneker’s symbol. We make an assumption that all the eigenvalues and eigenvectors are smooth functions of u, which in particular holds when the eigenvalues are all distinct, i.e., the system is strictly hyperbolic.

It is well known that the solution can develop singularities in finite time even with smooth initial data,see Lax[11],John[8]and Li[12]. Therefore,global solutions can only be constructed within a space of discontinuous functions. Global weak solutions to the Cauchy problem is a subject of a large literature,notably,Lax[10],Glimm[7],DiPerna[6],Dafermos[4],Bressan[2],Bianchini and Bressan [1]. We refer to the classical monograph of Dafermos [5] for references.

In the classical paper of Kruzkov [9], global entropy weak solutions to a scalar equation are constructed by a vanishing viscosity method. That is, the entropy weak solutions of the hyperbolic equation actually coincide with the limits of solutions to the parabolic equation

by letting the viscosity coefficients ε →0. The same result is also proved for n×n strictly hyperbolic systems in a celebrated paper of Bianchini and Bressan[1] for small BV initial data.

Although the one dimension theory of systems of hyperbolic conservation laws has by now quite matured, the multi-dimensional problem is still very challenging except for the scalar case. In recent years, much progress has been made to understand the formation of shocks for the compressible Euler equations for small initial data, see Sideris [14] and Christodoulou[3]. However, the problem of constructing entropy weak solutions beyond the time of shock formation is still largely open and even so in the radial symmetric case. The main difficulty is that on one hand the weak solution can best lie in BV, on the other hand, the BV space is not a scaling invariant space for the system. Especially, in n space dimensions, ˙W1,nis the critical space. This motivates us to study systems of hyperbolic conservation laws in one space dimension for W1,p(1

Let u(t,x) be the solution to the Cauchy problem of a scalar conservation law in one space dimension. It is well known that even for smooth initial data the solution can become discontinuous in finite time and global entropy weak solution can best lie in the space of bounded total variations. It is impossible that the solution belongs to W1,p(1 < p < +∞) because by Sobolev embedding theorem W1,pfunctions are Hlder continuous. However,we note that from any point(t,x) we can draw a generalized characteristic downward which meets the initial axis at y = α(t,x). If we regard u as a function of (t,y), it indeed belongs to W1,pas a function of y if the initial data belongs to W1,p. We may call this generalized persistence (of high regularity) of the entropy weak solutions. The main purpose of this paper is to prove some kind of generalized persistence of W1,pregularity of entropy weak solutions for 2×2 Temple system of hyperbolic conservation laws in one space dimension. Some interesting hyperbolic system arising in applications which satisfies the Temple condition can be found in Serre [13].

Our main theorem can be stated as follows.

Theorem 1.1Consider the Cauchy problem (1.1)–(1.2) for systems of two conservation laws. Suppose that the 2×2 matrix A(u) is hyperbolic, smoothly depending on u and possessing a complete sets of smooth eigenvalues and eigenvectors as well as two global Riemann invariants.Suppose that the Temple condition

is satisfied. Suppose furthermore that

and there exists 1

Then, the Cauchy problem (1.1)–(1.2) admits a global entropy weak solution which can be represented as

where U is a smooth function of Riemann invariants W1, W2, α1(t,x), α2(t,x) are locally bounded monotone increasing function of x and W1(t,α), W2(t,α) are Hlder continuous functions of α, moreover,

Theorem 1.1 will be proved by a vanishing viscosity approach.

Remark 1.1Theorem 1.1 is also true for the initial boundary value problems with periodic boundary conditions, with (1.8) replaced by

The same proof applies.

Remark 1.2With additional assumption (1.8), Theorem 1.1 gives an alternative proof of global existence of entropy weak solution for 2×2 Temple system without using the so called compensated compactness method.

This paper is organized as follows: In Section 2, we will discuss generalized persistence of a scalar conservation law in one space dimension in various high regularity spaces. In Section 3, we will discuss related problem for a scalar conservation law in multi-dimensions. Finally, in Section 4, we will discuss the 2×2 Temple system in one space dimension and prove our main result.

Notations: Let f(x) be a scalar or vector function of x ∈R, we denote

2 Scalar Equation in One Space Dimension

We consider the following Cauchy problem for a scalar conservation law in one space dimension:

where u0is a suitably smooth function. It is well known that the global solution is the limit of the viscous approximations

By maximum principle, we have

We write

for simplicity of notation,here we denote Uε(t,αε(t,x))just by U(t,α(t,x)). Substitubing (2.6)to (2.3), we get

We take αε(t,x) to be the solution to the following Cauchy problem

then, Uε(t,α) will satisfy

Then by maximum principle, Θεis a positive function, and moreover, by (2.8)–(2.9),

Then, it is easy to get the following series of estimates

and for any 1 ≤p ≤∞,

Upon taking a subsequence,Uε(t,α)converges to U(t,α)and αε(t,x)converges to α(t,x). Then u(t,x) = U(t,α(t,x)) is the solution to the Cauchy problem (2.1)–(2.2). We see immediately that U(t,α)αis a function of bounded total variation for the variable α provided thatis a function of bounded total variation and U(t,α)αis an Lp(1

Theorem 2.1Let

Then the global entropy solution to system (2.1)–(2.2) can be represented as

where α(t,x) is a locally bounded monotone increasing function of x representing the generalized characteristics and U(t,α) as a function of α satisfies

and for any 1

provided that the left-hand side of the inequality is finite, i.e., u0is suitable smooth. In particular, U(t,α) is Hlder continuous.

3 Scalar Conservation Law in Multi-Dimensions

In this section, we consider the initial boundary value problem with periodic boundary conditions of a scalar conservation law in multi-dimensions

As always, u is the limit of its viscous approximations:

where ?is the Laplacian operator in Tn.

Due to the multi-dimensional nature of the problem,there no longer exists a transformation y = α(t,x) like that in one space dimensions. Therefore, in this section, we are limited to discuss regularity properties of solutions of the viscous approximations.

Let Θε(t,x) be the solution to the initial boundary value problem with periodic boundary conditions of the following equation

By maximum principle, Θεis a positive function, moreover, integrating (3.5) in x yields

Let

Then differentiating (3.1) with respect to x1yields

A simple computation shows

Then by maximum principle, we get

uniformly for all (t,x). In a same way, we have

uniformly for all (t,x).

Let 1

where we write vεas v for simplicity of notation. Similar equality holds forwe integrate the above equality to yield

(3.12), (3.14) give a kind of LP(1 < p ≤∞) bound for the derivatives of the solution. We notice that by Hlder’s inequality and (3.7), for any 1 ≤p1

4 2×2 Temple System in One Space Dimension

We consider the viscous approximations

with initial conditions

where Jεis the Friedriches mollifier. We assume that there exist two Riemann invariants= R1(uε),= R2(uε), and we assume that we can also write uε= U(. In the following calculations, we just denote uεby u andby Riwhen there is no confusion of notation. Taking li(u)=?uRi(u), we get

Thus, we get

Thus, taking inner product of (4.1) with liand noting the Temple condition, we get

where Dijare smooth functions of uε. Thus,

where

Then, by maximum principle

Thus, we have

We write

then (4.6) becomes

Thus, we get

We impose the following initial conditions

We start the proof of Theorem 1.1 with the parabolic estimate. By (4.1)–(4.2), we get

where

is the heat kernel. We take δ = δ(D) to be a constant depending only on D and δ ?D. We consider first the equation on the time interval t ∈[0,εδ], by (4.18), we have

Therefore, we get

Thus, we have

which implies

provided that δ is taken to be small enough. Now, let t ≥εδ, we consider the equation on the time interval s ∈[t ?εδ,t], we have

and thus,

Therefore, we get

Thus, we have

which implies

provided that δ is taken to be sufficiently small. Take s=t, we get

By (4.23) and (4.29), we finally arrive at

because the left-hand side is a subsolution and the right-hand side is a supsolution. Thus,there exists a subsequence such that(t,x) converges to some αi(t,x) all most everywhere.By (4.17), there exists a further subsequence such thatweakly converges to some Wi(t,α)in1,p, by Sobolev embedding Theorem,strongly converges to Wi(t,α) in Hlder space.Therefore, taking limit in

we conclude the proof of Theorem 1.1.