Kinetic Stabilization¶of a Nonlinear Sonic Phase B

Authors: Andrea Corli Monique SabléTougeron

Publish Date: 2014/03/19

Volume: 152, Issue: 1, Pages: 1-63

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Abstract

We consider a system arising in the study of phase transitions in elastodynamics – a system of two conservation laws in a single space dimension The system has two hyperbolic regions with an elliptic zone in between A phase boundary is a strong discontinuity in a solution with left and right states belonging to different hyperbolic regions We call such a solution a phase waveWe first address the Riemann problem for initial states close to a fixed sonic phase wave in the genuinely nonlinear case This problem is naturally underdetermined We propose two essentially different types of Reimann problems a sonic one which is smooth and a kinetic one which is only Lipschitzcontinuous Both problems are well posed owing to a shared stability condition that is of a purely sonic natureIn the kinetic case we prove the global existence of solutions to the Cauchy problem for initial data having small variation and close to a sonic kinetic wave The crucial issue is the interaction of the phase boundary with a small wave of the same mode The introduction of a pertinent quantity called here detonation potential ensures a balance between ingoing and outgoing waves The proof is based on a Glimmtype scheme we define a potential which includes the detonation potential along the strong discontinuity and this potential controls the outbreak of unusual shocks

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