Strong relaxation limit and uniform time asymptotics of the Jin-Xin model in the $L^{p}$ framework
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908441129254912 |
|---|---|
| author | Crin-Barat, Timothée Shou, Ling-Yun Zhang, Jianzhong |
| author_facet | Crin-Barat, Timothée Shou, Ling-Yun Zhang, Jianzhong |
| contents | We investigate the time-asymptotic stability of the Jin-Xin model and its diffusive relaxation limit toward viscous conservation laws in $\mathbb{R}^d$ for $d\geq 1$. First, we establish a priori estimates that are uniform with respect to both the time and the relaxation parameter $\varepsilon>0$, for initial data in hybrid Besov spaces based on $L^{p}$-norms. This uniformity enables us to derive $\mathcal{O}(\varepsilon)$ 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 $L^{p/2}(\mathbb{R}^{d})$, we show that the $L^{p}(\mathbb{R}^d)$-norm of the solution to the Jin-Xin model decays at the optimal rate $(1+t)^{-d/{2p}}$, and the $L^{p}(\mathbb{R}^d)$-norm of its difference with the solution of the associated viscous conservation law decays at the enhanced rate $\varepsilon(1+t)^{-d/{2p}-1/2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_04105 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Strong relaxation limit and uniform time asymptotics of the Jin-Xin model in the $L^{p}$ framework Crin-Barat, Timothée Shou, Ling-Yun Zhang, Jianzhong Analysis of PDEs 35L40, 35L45, 35K55 We investigate the time-asymptotic stability of the Jin-Xin model and its diffusive relaxation limit toward viscous conservation laws in $\mathbb{R}^d$ for $d\geq 1$. First, we establish a priori estimates that are uniform with respect to both the time and the relaxation parameter $\varepsilon>0$, for initial data in hybrid Besov spaces based on $L^{p}$-norms. This uniformity enables us to derive $\mathcal{O}(\varepsilon)$ 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 $L^{p/2}(\mathbb{R}^{d})$, we show that the $L^{p}(\mathbb{R}^d)$-norm of the solution to the Jin-Xin model decays at the optimal rate $(1+t)^{-d/{2p}}$, and the $L^{p}(\mathbb{R}^d)$-norm of its difference with the solution of the associated viscous conservation law decays at the enhanced rate $\varepsilon(1+t)^{-d/{2p}-1/2}$. |
| title | Strong relaxation limit and uniform time asymptotics of the Jin-Xin model in the $L^{p}$ framework |
| topic | Analysis of PDEs 35L40, 35L45, 35K55 |
| url | https://arxiv.org/abs/2311.04105 |