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

$L^2$-Contraction and Asymptotic Stability of Large Shock for Scalar Viscous Conservation Laws Global Science Press 02/25/2026 10688270 10.4208/cmaa.2025-0021 Communications in Mathematical Analysis and Applications 2790-1920 5 1 Alexis F VasseurYi WangJian Zhang 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.

]]>