Preprint 1997-036

Continuous Dependence in the Large for Some Equations of Gas Dynamics

Rinaldo M. Colombo and Nils Henrik Risebro

Abstract: The main result of this paper is the construction of the {\it Standard Riemann Semigroup} for a class of $2 \times 2$ Conservation Laws without the requirement that the total variation of the initial data be {\it small}. More precisely, fix a positive $M$. For a class of equations of the form $$ \quad\quad\quad u_t + \left[f(u)\right]_x =0 \quad \hbox{ with } \quad t\geq 0 \, , \ x \in \reali \, , \ u \in \reali^2 \quad\quad\quad\hfill{(\star)} $$ it is shown that there exists an $\L1$--Lipschitz continuous semigroup $S$ defined on functions of bounded variation, such that the trajectories of $S$ are weak entropy solutions to~{$(\star)$}. This class includes the classical and relativistic isentropic Euler equations and all the $2\times 2$ Temple class systems.

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Continuous dependence in the large for some equations of gas dynamics
Rinaldo M. Colombo,
Nils Henrik Risebro, <>
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<> December 11 1997.

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