Decay Rates of global weak solutions for the MHD equations in $\dot{\mbox{\boldmath{$H$}}}^{s}(\mathbb{R}^n)$

TL;DR

This paper proves that the decay rates of global weak solutions to the incompressible MHD equations in certain Sobolev spaces tend to zero as time approaches infinity, extending understanding of long-term behavior.

Contribution

It establishes new decay rate results for Leray solutions of the MHD equations in Sobolev spaces for dimensions 2 to 4, which was previously not well understood.

Findings

Decay of Sobolev norm scaled by t^{s/2} tends to zero as t→∞.

L^q norms of solutions decay to zero for all q between 2 and ∞.

Results hold for dimensions 2 through 4.

Abstract

We show that $t^{s/2} \Vert (\mbox{\boldmath $u$},\mbox{\boldmath $b$})(.,t)\Vert_{\dot{H}(\mathbb{R}^{n})} \rightarrow 0,$ as $t\rightarrow \infty$ for Leray solutions $(\mbox{\boldmath $u$}, \mbox{\boldmath $b$})(.,t)$ of the incompressible MHD equations, where $2 \leq n \leq 4$ and $s \geq 0.$ As a corollary of main result described previously we have also that $\lim_{t\rightarrow\infty} t^{\frac{n}{2} - \frac{n}{2q}} \Vert(\mbox{\boldmath $u$},\mbox{\boldmath $b$})(.,t)\Vert_{L^{q}(\mathbb{R}^{n})} = 0, 2\leq q\leq \infty.$