Generalized persistence of entropy weak solutions for system of hyperbolic conservation laws

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 ,$H^1$ because by Sobolev embedding theorem $H^1$ functions are H$\mathrm{\ddot{o}}$lder continuous. However, we note that from any point $(t,x)$ we can draw a generalized characteristic downward which meets the initial axis at $y=\alpha (t,x)$. if we regard $u$ as a function of $(t,y)$, it indeed belongs to $H^1$ as a function of $y$ if the initial data belongs to $H^1$. We 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 $2\times 2$ Temple system of hyperbolic conservation laws in one space dimension .