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