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

期刊论文
发表日期：

2025年4月22日

查看全文

Timothée Crin-Barat,
Ling-Yun Shou,
Jianzhong Zhang

摘要

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 $\cal{O}(\varepsilon)$ $O (ε)$ 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 example, 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}.

Strong relaxation limit and uniform time asymptotics of the Jin-Xin model in the Lp framework

摘要

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