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The magnetohydrodynamical system in bounded non smooth domains of dimension $n\ge3$

Sylvie Monniaux

Abstract

Existence of mild solutions for the magnetohydrodynamical system in C 1 domains is established in critical spaces in dimension n $\ge$3. The proof relies on recent regularity results on the Stokes operator in C 1 domains and a Leibniz-like formula for differential forms proved here in Section 2.

The magnetohydrodynamical system in bounded non smooth domains of dimension $n\ge3$

Abstract

Existence of mild solutions for the magnetohydrodynamical system in C 1 domains is established in critical spaces in dimension n 3. The proof relies on recent regularity results on the Stokes operator in C 1 domains and a Leibniz-like formula for differential forms proved here in Section 2.

Paper Structure

This paper contains 13 sections, 16 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Functional analytic tools
  3. The Dirichlet-Stokes operator in ${\mathscr{C}}^1$ bounded domains
  4. Differential forms and Hodge-Laplacians in ${\mathscr{C}}^1$ bounded domains
  5. Formulation of \ref{['mhd']} in dimension $n\ge 3$
  6. The magic formula
  7. Existence results in critical spaces
  8. Mild solutions
  9. Global existence
  10. Local existence
  11. Deferred proofs
  12. About initial conditions
  13. The $B$ operators

Key Result

Theorem 1

For $u_0,b_0$ in a suitable functional space embedded into $L^n$, there is a global mild solution of mhd,bc if $u_0,b_0$ are small enough in $L^n$ and a local mild solution if no size restrictions are imposed on the initial data.

Theorems & Definitions (31)

  • Theorem
  • Theorem : Theorem 1.1 & Corollary 1.3 in GS25
  • Corollary 2.1
  • proof
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 21 more