2023/12/06 by Lingjun Liu, Shu Wang, Liu, Lingjun +3
Engineering · Mathematics · Physics and Astronomy · #35K15 #35L65 #35L67 #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #Cosmology and Gravitation Theories #FOS: Mathematics #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2312.03553
openalex publication_date 2023/12/06 · openalex created_date 2023/12/08 · openalex updated_date 2026/07/28
In this paper, we study the time-decay rate toward the planar viscous shock wave for multi-dimensional (m-d) scalar viscous conservation law. We first decompose the perturbation into zero and non-zero mode, and then introduce the anti-derivative of the zero mode. Though an Lp estimate and the area inequality introduced in \citeDHS2020, we obtained the decay rate for planar shock wave for n-d scalar viscous conservation law for all n≥1. The initial perturbations we studied are small, i.e., ‖Φ0‖H2\bigcap‖Φ0‖Lp≤ ε, where Φ0 is the anti-derivative of the zero mode of initial perturbation and ε is a small constant, see \crefantiderivative. It is noted that there is no additional requirement on Φ0, i.e., Φ0(x1) only belongs to H2(\R). Thus, there are essential differences from previous results, in which the initial data is required to belong to some weighted Sobolev space, cf.\citeGoo1989,KM1985. Moreover, the exponential decay rate of the non-zero mode is also obtained.