Large time behaviour of solutions to the N-dimensional scalar conservation law under periodic perturbations with nonlinear degenerate viscosity

2023/12/06 by Yechi Liu, Liu, Yechi
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2312.03239

openalex publication_date 2023/12/06 · openalex created_date 2023/12/08 · openalex updated_date 2026/07/28

Abstract

In this paper, we discuss the asymptotic behaviour of the weak solution to the Cauchy problem for the scalar viscous conservation law, with nonlinear Laplacian viscosity. Firstly, we obtain the existence, uniqueness and regularity of solutions when the initial data u₀∈ C¹(\mathbb R^N)∩ W^1,∞(\mathbb R^N). Secondly, when u₀ is periodic, we prove the time-decay rate of the periodic solution and its gradient. At last, we study the long-time behaviour of perturbed solution to the Cauchy problem, in which the initial data is a N-d periodic perturbation around a planar rarefaction wave and obtain the time-decay rate of the perturbed solution approaching approximate planar rarefaction wave. The proof is given by technical energy methods and iteration technique.

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