Global existence and uniqueness of weak solutions for the MHD equations with large $L^3$-initial values

TL;DR

The paper analyzes the global existence and uniqueness of weak solutions to the 3D incompressible MHD equations with large initial data in $L^3(\mathbb{R}^3)$. To overcome limitations arising from slip boundary conditions for the magnetic field, the authors combine Leray's approximation with perturbation theory to construct global weak $L^3$-solutions to the Cauchy problem, and they obtain a conditional uniqueness result. The analysis hinges on the caloric decomposition $v=v_1+v_2$, $H=H_1+H_2$, with $v_1=e^{t\Delta}v_0$, $H_1=e^{t\Delta}h_0$, and a perturbation system for $(v_2,H_2)$ that satisfies global and local energy inequalities; uniform a priori estimates and a compactness framework (Aubin–Lions) enable the passage to the limit. Uniqueness is established under smallness assumptions on the $L^\infty(0,T_1;L^3)$-norms of the perturbations, providing a robust, self-contained weak $L^3$-theory for the MHD system and yielding a byproduct route to weak $L^3$-theory for the Navier–Stokes equations.

Abstract

This paper is concerned with the weak solution theory for the MHD system with large $L^3$-initial data. Due to the fact that the natural boundary condition on the magnetic field $H$ is the slip boundary condition, the Leray-Schauder fixed-point theorem, which have used to investigate the weak solution theory of the Navier-Stokes system, becomes invalid. To address such difficulty, we will invoke the Leray's approximation technique and the perturbation theory to seek a global weak solution to the Cauchy problem for MHD equations with large $L^3$-initial data. Our strategy provides a simple alternative (self-contained) proof of weak $L^3$-solution theory of incompressible Navier-Stokes system. Moreover, this weak solution is unique under some restrictions.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Lemma 2.2: Fixed point theorem
• proof
• Lemma 2.3: 3-D ns equations
• Lemma 2.4