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arXiv 2302.02046math.AP
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具有分数阶扩散的Stokes-Magneto系统弱解的整体存在唯一性

Global existence and uniqueness of weak solutions of a Stokes-Magneto system with fractional diffusions

Hyunseok Kim, Hyunwoo Kwon

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40 pages
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摘要：

本文考虑$\mathbb{R}^d$（$d\geq 2$）中的Stokes-磁电机系统，速度$\boldsymbol{u}$和磁场$\boldsymbol{b}$分别具有分数扩散$Λ^{2α}\boldsymbol{u}$和$Λ^{2β}\boldsymbol{b}$.这里，$α,β$是正常数，而$Λ^s = (-Δ)^{s/2}$是$s$阶的分数拉普拉斯算子。当$α$，$β$满足$1/2<α<(d+1)/2$，$β>0$，和$\min\{α+β,2α+β-1\}>d/2$时，我们建立了$L_{2}$中任意初始值的Stokes-Magneto方程组整体弱解的存在性.此外，还证明了当$β\geq 1$和$\min \{α+β,2α+β-1\}\geq d/2+1$满足时，弱解是唯一的.

英文摘要：

We consider a Stokes-Magneto system in $\mathbb{R}^d$ ($d\geq 2$) with fractional diffusions $Λ^{2α}\boldsymbol{u}$ and $Λ^{2β}\boldsymbol{b}$ for the velocity $\boldsymbol{u}$ and the magnetic field $\boldsymbol{b}$, respectively. Here $α,β$ are positive constants and $Λ^s = (-Δ)^{s/2}$ is the fractional Laplacian of order $s$. We establish global existence of weak solutions of the Stokes-Magneto system for any initial data in $L_{2}$ when $α$, $β$ satisfy $1/2<α<(d+1)/2$, $β>0$, and $\min\{α+β,2α+β-1\}>d/2$. It is also shown that weak solutions are unique if $β\geq 1$ and $\min \{α+β,2α+β-1\}\geq d/2+1$, in addition.

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