Abstract
We study the nonlinear stability of shock waves for viscous conservation laws. Our approach is based on a new construction of a fundamental solution for a linearized system around a shock profile. We obtain, for the first time, the pointwise estimates of nonlinear wave interactions across a shock wave. Our results apply to all ranges of weak shock waves and small perturbations. In particular, our results reduce to the time-asymptotic behavior of constant state perturbation, uniformly as the strength of the shock wave tends to zero.
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Communicated by P. Constantin
The research of the first author was partially supported by NSC Grant 96-2628-M-001-011 and NSF Grant DMS-0709248.
The research of the second author was partially supported by NSF Grant DMS-0207154 and UAB Advance Program, sponsored by NSF.
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Liu, TP., Zeng, Y. Time-Asymptotic Behavior of Wave Propagation Around a Viscous Shock Profile. Commun. Math. Phys. 290, 23–82 (2009). https://doi.org/10.1007/s00220-009-0820-6
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DOI: https://doi.org/10.1007/s00220-009-0820-6