Criteria for the $a$-contraction and stability for the piecewise-smooth solutions to hyperbolic balance laws

We show uniqueness and stability in $L^2$ and for all time for piecewise-smooth solutions to hyperbolic balance laws. We have in mind applications to gas dynamics, the isentropic Euler system and the full Euler system for a polytropic gas in particular. We assume the discontinuity in the piecewise-smooth solution is an extremal shock. We use only mild hypotheses on the system. Our techniques and result hold without smallness assumptions on the solutions. We can handle shocks of any size. We work in the class of bounded, measurable solutions satisfying a single entropy condition. We also assume a strong trace condition on the solutions, but this is weaker than $BV_\mathrm{loc}$. We use the theory of $a$-contraction (see Kang and Vasseur [Arch. Ration. Mech. Anal., 222(1):343–391, 2016]) developed for the stability of pure shocks in the case without source.

Keywords

system of conservation laws, compressible Euler equation, Euler system, isentropic solutions, generalized Riemann problem, piecewise-smooth solutions, Rankine–Hugoniot discontinuity, shock, stability, uniqueness

2010 Mathematics Subject Classification

Primary 35L65. Secondary 35A02, 35B35, 35D30, 35L45, 35L67, 35Q31, 35Q35, 76L05, 76N10, 76N15.

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This work was partially supported by NSF Grant DMS-1614918. The author was also partially supported by NSF-DMS Grant 1840314.

Received 11 September 2019

Accepted 14 March 2020

Published 4 November 2020