$L^2$-Contraction and Asymptotic Stability of Large Shock for Scalar Viscous Conservation Laws
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.
Fields of Study & Subjects
Advanced Mathematical Physics Problems
Navier-Stokes equation solutions
Stability and Controllability of Differential Equations
Publication details
- Journal / source
- Communications in Mathematical Analysis and Applications
- Volume
- 5
- Issue
- 1
- Pages
- 35–64
- Publication date
- 15 January 2026
- Language
- EN