Continuous Dependence in Conservation Laws with Phase Transitions

Abstract

This paper is concerned with systems of 2 X 2 conservation laws \def\theequation{$\star$}% \begin{equation} \partial_t u + \partial_x \left[f(u)\right]=0, \quad\quad t \geq 0 \, ,\ x \in \R \, , \ u \in \R^2, \end{equation} \def\theequation{\thesection.\arabic{equation}}% developing phase transitions, as happens in models related to elastodynamics or to van der Waals fluids, for instance. In the present paper, a definition of $\Psi$-admissible solution to ($\star$) is given which comprises the various definitions in the current literature. Furthermore, the $\Psi$-admissible Riemann semigroup ($\Psi$RS) generated by ($\star$) is introduced and constructed by means of a wave-front tracking algorithm. Uniqueness and continuous dependence for $\Psi$-admissible solutions to ($\star$) thus follow.