Strong relaxation limit and uniform time asymptotics of the Jin-Xin model in the L^p framework

2023/11/07 by Crin-Barat, Timothée, Shou, Ling-Yun, Zhang, Jianzhong · 2 citations
#35K55 #35L40 #35L45 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2311.04105

Abstract

We investigate the time-asymptotic stability of the Jin-Xin model and its diffusive relaxation limit toward viscous conservation laws in ℝ^d for d≥ 1. First, we establish a priori estimates that are uniform with respect to both the time and the relaxation parameter ε>0, for initial data in hybrid Besov spaces based on L^p-norms. This uniformity enables us to derive O(ε) bounds on the difference between solutions of the viscous conservation law and its associated Jin-Xin approximation, thus justifying the strong convergence of the relaxation process. Furthermore, under an additional condition on the initial data, for instance, that the low frequencies belong to L^p/2(ℝ^d), we show that the L^p(ℝ^d)-norm of the solution to the Jin-Xin model decays at the optimal rate (1+t)^-d/2p, and the L^p(ℝ^d)-norm of its difference with the solution of the associated viscous conservation law decays at the enhanced rate ε(1+t)^-d/2p-1/2.

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• Relaxation limit and asymptotic stability for the Euler-Navier-Stokes equations
• On the decay estimates of a nonlocal convection-diffusion Hamer system

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