Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the well-posedness of the Hall-MHD system in a critical setting of Besov-Morrey type

Lucas C. F. Ferreira, Rafael P. da Silva

Abstract

In this paper, we address the 3D incompressible Hall-magnetohydrodynamic system (Hall-MHD). Our objective is to provide local and global well-posedness results for initial velocity $u_{0}$, magnetic field $B_{0}$ and the current $J_{0}:=\nabla\times B_{0}$ in a new critical framework, namely critical Besov-Morrey spaces. These spaces combine typical characteristics from both Besov and Morrey spaces allowing a broader framework that encompasses the regularity properties inherent in Besov spaces with the Morrey space structure. Compared to previous works in Sobolev and Besov spaces, our approach accommodates a broader class of initial data, ensuring the construction of a unique solution over time.

On the well-posedness of the Hall-MHD system in a critical setting of Besov-Morrey type

Abstract

In this paper, we address the 3D incompressible Hall-magnetohydrodynamic system (Hall-MHD). Our objective is to provide local and global well-posedness results for initial velocity , magnetic field and the current in a new critical framework, namely critical Besov-Morrey spaces. These spaces combine typical characteristics from both Besov and Morrey spaces allowing a broader framework that encompasses the regularity properties inherent in Besov spaces with the Morrey space structure. Compared to previous works in Sobolev and Besov spaces, our approach accommodates a broader class of initial data, ensuring the construction of a unique solution over time.

Paper Structure

This paper contains 4 sections, 14 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local-in-time well-posedness
  4. Global-in-time well-posedness

Key Result

Theorem 1.1

Let $1\leq q\leq p<\infty$, $u_{0},B_{0}\in\mathcal{N}_{p,q,1}^{\frac{3}{p}-1}$ with $\mathop{\mathrm{div}}\nolimits u_{0}=\mathop{\mathrm{div}}\nolimits B_{0}=0,$ and $J_{0}=\nabla\times B_{0}\in\mathcal{N}_{p,q,1}^{\frac{3}{p}-1}$. Then, $\exists\delta=\delta(p,q,\mu,\nu)>0$ and an existence time then system (MHD) admits a unique local-in-time solution $(u,B)\in X_{p}(T)\times X_{p}(T)$ with $J

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 10 more