Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The three dimensional magneto-hydrostatic equations with Grad-Rubin boundary value

Diego Alonso-Orán, Daniel Sánchez-Simón del Pino, Juan J. L. Velázquez

Abstract

In this work, we study the well-posedness of the three dimensional magneto-hydrostatic equation under Grad-Rubin boundary value conditions. The proof relies on a fixed point argument to construct solutions to an elliptic-hyperbolic problem in a perturbative regime by means of pseudo-differential operators with symbols with limited regularity in Hölder spaces. As a byproduct, the employed technique in this work is more flexible and simplifies the arguments of the proof for the previous two-dimensional setting.

The three dimensional magneto-hydrostatic equations with Grad-Rubin boundary value

Abstract

In this work, we study the well-posedness of the three dimensional magneto-hydrostatic equation under Grad-Rubin boundary value conditions. The proof relies on a fixed point argument to construct solutions to an elliptic-hyperbolic problem in a perturbative regime by means of pseudo-differential operators with symbols with limited regularity in Hölder spaces. As a byproduct, the employed technique in this work is more flexible and simplifies the arguments of the proof for the previous two-dimensional setting.
Paper Structure (35 sections, 30 theorems, 383 equations)

This paper contains 35 sections, 30 theorems, 383 equations.

Table of Contents

  1. Introduction and prior results
  2. Notation
  3. The main result: novelties and strategy towards the proof
  4. Plan of the paper
  5. Preliminaries: functional setting and auxiliary results
  6. Littlewood-Paley theory and Besov spaces
  7. Symbols, pseudo-differential operators and adjoints
  8. Transferring properties to periodic pseudo-differential operators
  9. Classical Mikhlin-Hörmander multipliers in $\mathbb{R}^{d}$ and $\mathbb{T}^{d}$
  10. Hölder estimates for the div-curl system and transport type problem
  11. Hölder estimates for the div-curl system.
  12. Hölder estimates for the transport problem
  13. Transport divergence condition along characteristics
  14. The linearized problem
  15. The non-linear setting: the integral equation for the current
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $\alpha\in (0,1)$, $\Omega=\mathbb{T}^2\times [0,L]$, and denote $\partial\Omega_-=\mathbb{T}^2\times \{0\}$. There exists a (small) constant $M=M(\alpha,L)$ such that for every $f\in C^{2,\alpha}(\partial\Omega)$ and $g\in C^{2,\alpha}(\partial\Omega_-)$ satisfying and there exists a solution $(B,p)\in C^{2,\alpha}(\Omega)\times C^{2,\alpha}(\Omega)$ for the problem where $n$ equals the ou

Theorems & Definitions (60)

  • Theorem 1.1
  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Corollary 2.6
  • Lemma 2.7
  • proof : Proof of Lemma \ref{['periodico']}
  • Definition 2.8
  • ...and 50 more