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Global regularity for solutions of magnetohydrodynamic equations with large initial data

Xiangsheng Xu

TL;DR

The paper studies global regularity for the incompressible magnetohydrodynamic equations in $\mathbb{R}^N$ with large initial data ($N\ge3$). It develops a De Giorgi-type iteration to obtain an $L^\infty$ bound for $w=|u|^2+|b|^2$ by combining scaling, interpolation, and a Calderón–Zygmund bound on the pressure, leading to a priori estimates independent of the time horizon. A key consequence is the existence of a global-in-time strong solution and a quantitative bound $\|w\|_{\infty,Q_T}\le 32\|w(\cdot,0)\|_{\infty}+c\|w\|_{L_N,Q_T}^{s_5}$, which in particular implies global regularity in the Navier–Stokes case $b\equiv0$. The methodology provides a robust framework for proving global regularity for nonlinear parabolic systems by blending three ingredients: scaling, interpolation, and De Giorgi iteration.

Abstract

We study the regularity properties of solutions to the initial value problem for the magnetohydrodynamic equations in $\mathbb{R}^N, N\geq 3$. We obtain a global in-time strong solution without any smallness assumptions on the initial data.

Global regularity for solutions of magnetohydrodynamic equations with large initial data

TL;DR

The paper studies global regularity for the incompressible magnetohydrodynamic equations in with large initial data (). It develops a De Giorgi-type iteration to obtain an bound for by combining scaling, interpolation, and a Calderón–Zygmund bound on the pressure, leading to a priori estimates independent of the time horizon. A key consequence is the existence of a global-in-time strong solution and a quantitative bound , which in particular implies global regularity in the Navier–Stokes case . The methodology provides a robust framework for proving global regularity for nonlinear parabolic systems by blending three ingredients: scaling, interpolation, and De Giorgi iteration.

Abstract

We study the regularity properties of solutions to the initial value problem for the magnetohydrodynamic equations in . We obtain a global in-time strong solution without any smallness assumptions on the initial data.
Paper Structure (2 sections, 3 theorems, 154 equations)

This paper contains 2 sections, 3 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm']}

Key Result

Theorem 1.1

Assume that Let $(u, b)$ be a local-in-time strong solution to nsf1-nsf5. Define Then for each $L_N\geq 1$ there exist two positive numbers $c=c(N,\nu,\eta, L_N )$ and $s_5=s_5(N, L_N)$ such that

Theorems & Definitions (8)

  • Theorem 1.1
  • Lemma 1.2
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['thm']}
  • Claim 2.2
  • Claim 2.3
  • proof