Global Fujita-Kato solutions of the incompressible inhomogeneous magnetohydrodynamic equations
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arXiv
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| Natura: | Preprint |
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2025
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| author | Li, Fucai Ni, Jinkai Shou, Ling-Yun |
| author_facet | Li, Fucai Ni, Jinkai Shou, Ling-Yun |
| contents | We investigate the incompressible inhomogeneous magnetohydrodynamic equations in $\mathbb{R}^3$, under the assumptions that the initial density $ρ_0$ is only bounded, and the initial velocity $u_0$ and magnetic field $B_0$ exhibit critical regularities. In particular, the density is allowed to be piecewise constant with jumps. First, we establish the global-in-time well-posedness and large-time behavior of solutions to the Cauchy problem in the case that $ρ_0$ has small variations, and $u_0$ and $B_0$ are sufficiently small in the critical Besov space $\dot{B}^{3/p-1}_{p,1}$ with $1<p<3$. Moreover, the small variation assumption on $ρ_0$ is no longer required in the case $p=2$. Then, we construct a unique global Fujita-Kato solution under the weaker condition that $u_0$ and $B_0$ are small in $\dot{B}^{1/2}_{2,\infty}$ but may be large in $\dot{H}^{1/2}$. Additionally, we show a general uniqueness result with only bounded and nonnegative density, without assuming the $L^1(0,T;L^{\infty})$ regularity of the velocity. Our study systematically addresses the global solvability of the inhomogeneous magnetohydrodynamic equations with rough density in the critical regularity setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_06543 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global Fujita-Kato solutions of the incompressible inhomogeneous magnetohydrodynamic equations Li, Fucai Ni, Jinkai Shou, Ling-Yun Analysis of PDEs 76D03, 35Q30, 35Q35 We investigate the incompressible inhomogeneous magnetohydrodynamic equations in $\mathbb{R}^3$, under the assumptions that the initial density $ρ_0$ is only bounded, and the initial velocity $u_0$ and magnetic field $B_0$ exhibit critical regularities. In particular, the density is allowed to be piecewise constant with jumps. First, we establish the global-in-time well-posedness and large-time behavior of solutions to the Cauchy problem in the case that $ρ_0$ has small variations, and $u_0$ and $B_0$ are sufficiently small in the critical Besov space $\dot{B}^{3/p-1}_{p,1}$ with $1<p<3$. Moreover, the small variation assumption on $ρ_0$ is no longer required in the case $p=2$. Then, we construct a unique global Fujita-Kato solution under the weaker condition that $u_0$ and $B_0$ are small in $\dot{B}^{1/2}_{2,\infty}$ but may be large in $\dot{H}^{1/2}$. Additionally, we show a general uniqueness result with only bounded and nonnegative density, without assuming the $L^1(0,T;L^{\infty})$ regularity of the velocity. Our study systematically addresses the global solvability of the inhomogeneous magnetohydrodynamic equations with rough density in the critical regularity setting. |
| title | Global Fujita-Kato solutions of the incompressible inhomogeneous magnetohydrodynamic equations |
| topic | Analysis of PDEs 76D03, 35Q30, 35Q35 |
| url | https://arxiv.org/abs/2501.06543 |