A stability theorem for entropy solutions of initial value problems for first order quasilinear hyperbolic systems in several space variables

Abstract

In this paper we prove an uniqueness and stability theorem for the solutions of Cauchy problem for the systems $$\frac{\partial }{{\partial t}}u + \sum\limits_{i = 1}^n { \frac{\partial }{{\partial x_i }} } f^i (x,t,u) = g(x,t,u),$$ whereu is a vector function (u 1 (x, t),..., u r (x, t)),f i =(a 1 i (x, t, u),..., a r i (x, t, u)), i=1,...,n, g=(g 1 (x, t, u),...,g r (x, t, u),i G ℝ n and t≥0. We use the concept of entropy solution introduced by Kruskov and improved by Lax, Dafermos and others autors. We assume that the Jacobian matricesf u i are symmetric and the Hessian(a j i ) uu (i=1,...,n; j=1,...,r) are positive. We obtain uniqueness and stability inL loc 2 within the class of those entropy solutions which satisfy $$\frac{{u_j (---,x_i ,---,t)---u_j (---,y_i ,---,t)}}{{x_i - y_i }} \geqslant - K(t),$$ (i=1,...,n; j=1,...,r) for (−,x i ,−,t), (−,y i ,−,t) on a compact setD ⊂ ℝ n x (0, ∞) and a functionK(t)∈L loc 1 ([0, ∞)) depending onD. Here we denote by (−,x i ,−,t) and (−,y i ,−,t) two points whose coordinates only differ in thei-th space variable. At the end we relax the hypotheses of symmetry and convexity on the system and give a theorem of uniqueness and stability for entropy solutions which are locally Lipschitz continuous on a strip ℝ n x [0,T].