$L^2$-Contraction and Asymptotic Stability of Large Shock for Scalar Viscous Conservation Laws

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Abstract

We investigate $L^2$-contraction and time-asymptotic stability of large shock for scalar viscous conservation laws with polynomial flux. For the flux $f(u) = u^p (2 ≤ p ≤ 4)$ in the regime of its strict convexity, we can prove $L^2$-contraction and time-asymptotic stability of arbitrarily large viscous shock profile in $H^1$-framework by using a-contraction method with time-dependent shift and suitable weight function, which answers a question in [Blochas and Cheng, arXiv2501.01537, 2025]. Additionally, if the initial perturbation belongs to $L^1$ , then $L^2$ time-asymptotic decay rate $t^{−1/4}$ can be obtained.

Keywords:

$a$-contraction asymptotic stability large shock

Author Biographies

  • Alexis F. Vasseur

    Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, USA

  • Yi Wang

    State Key Laboratory of Mathematical Sciences and Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100190, P. R. China

    School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 101408, P. R. China

  • Jian Zhang

    School of Mathematical Sciences, Chengdu University of Technology, Chengdu 610059, P. R. China

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DOI

10.4208/cmaa.2025-0021