Continuous Glimm functionals and uniqueness of solutions of the Riemann problem. (English) Zbl 0579.35053

Schatzman, Michelle

Indiana Univ. Math. J. 34, 533-589 (1985).

The following conservation laws are studied: \[ u_ t+f(u)_ x=0,\quad x\quad in\quad R,\quad t>0,\quad u(x,0)=u_ 0(x). \] f is assumed in \(C^ 3\) on an open set U of \(R^ n\) to \(R^ n\). It is assumed that Df(u) has n distinct eigenvalues for each u in U.
This paper contains a construction of functionals \({\mathcal L}\) and Q on functions of bounded variation with values in U such that (roughly) if u is a solution satisfying an entropy condition of Lax and its initial total variation is sufficiently small then there is a constant M such that (\({\mathcal L}+MQ)(u(t))\) decreases in time. Also, \({\mathcal L}+MQ\) is an estimate of the spatial variation of u.
An application to the Riemann problem (i.e., where \(u(x,0)=u^{\ell}\) if \(x<0\), \(u(x,0)=u^ r\) if \(x>0)\) is that there is a unique solution in the class of ”well behaved” functions if \(u^{\ell}\) and \(u^ r\) are close enough.

Reviewer: E.Barron

MSC:

35L65	Hyperbolic conservation laws
35A05	General existence and uniqueness theorems (PDE) (MSC2000)

Keywords:

uniqueness; conservation laws; functionals; functions of bounded variation; entropy condition; decreases; Riemann problem