Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2403.05203 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914708060110848 |
|---|---|
| author | Zhao, Youyi |
| author_facet | Zhao, Youyi |
| contents | We revisit the global existence of solutions with some large perturbations to the incompressible, viscous, and non-resistive MHD system in a three-dimensional periodic domain, where the impressed magnetic field satisfies the Diophantine condition, and the intensity of the impressed magnetic field, denoted by $m$, is large compared to the perturbations. It was proved by Jiang--Jiang that the highest-order derivatives of the velocity increase with $m$, and the convergence rate of the nonlinear system towards a linearized problem is of $m^{-1/2}$ in [F. Jiang and S. Jiang, Arch. Ration. Mech. Anal., 247 (2023), 96]. In this paper, we adopt a different approach by leveraging vorticity estimates to establish the highest-order energy estimate. This strategy prevents the appearance of terms that grow with $m$, and thus the increasing behavior of the highest-order derivatives of the velocity with respect to $m$ does not appear. Additionally, we use the vorticity estimate to demonstrate the convergence rate of the nonlinear system towards a linearized problem as time or $m$ approaches infinity. Notably, our analysis reveals that the convergence rate in $m$ is faster compared to the finding of Jiang--Jiang. Finally, a key contribution of our work is the identification of an integrable time-decay of the lower dissipation, which can replace the time-decay of lower energy in closing the highest-order energy estimate. This finding significantly relaxes the regularity requirements for the initial perturbations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05203 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global Solutions and Asymptotic Behavior for the Three-dimensional Viscous Non-resistive MHD System with Some Large Perturbations Zhao, Youyi Analysis of PDEs We revisit the global existence of solutions with some large perturbations to the incompressible, viscous, and non-resistive MHD system in a three-dimensional periodic domain, where the impressed magnetic field satisfies the Diophantine condition, and the intensity of the impressed magnetic field, denoted by $m$, is large compared to the perturbations. It was proved by Jiang--Jiang that the highest-order derivatives of the velocity increase with $m$, and the convergence rate of the nonlinear system towards a linearized problem is of $m^{-1/2}$ in [F. Jiang and S. Jiang, Arch. Ration. Mech. Anal., 247 (2023), 96]. In this paper, we adopt a different approach by leveraging vorticity estimates to establish the highest-order energy estimate. This strategy prevents the appearance of terms that grow with $m$, and thus the increasing behavior of the highest-order derivatives of the velocity with respect to $m$ does not appear. Additionally, we use the vorticity estimate to demonstrate the convergence rate of the nonlinear system towards a linearized problem as time or $m$ approaches infinity. Notably, our analysis reveals that the convergence rate in $m$ is faster compared to the finding of Jiang--Jiang. Finally, a key contribution of our work is the identification of an integrable time-decay of the lower dissipation, which can replace the time-decay of lower energy in closing the highest-order energy estimate. This finding significantly relaxes the regularity requirements for the initial perturbations. |
| title | Global Solutions and Asymptotic Behavior for the Three-dimensional Viscous Non-resistive MHD System with Some Large Perturbations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2403.05203 |