Abstract: This paper is devoted to the study of linear and nonlinear stability of undercompressive shock waves for first order systems of hyperbolic conservation laws in a multidimensional space. We first recall the framework proposed by Freist\"uhler to extend Majda's work on classical shock waves to undercompressive shock waves. Then we show how the so-called uniform stability condition yields a linear stability result in terms of a maximal $L^2$ estimate. We follow Majda's strategy on shock waves with several improvements and modifications inspired from M\'etivier's work. The linearized problems are solved by duality and the nonlinear equations by means of a Newton type iteration scheme. Finally, we show how this work applies to phase transitions in an isothermal van der Waals fluid.
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