Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Singularity formation for the 1D model of EMHD

Chao Wu

Abstract

In this paper, we study the singularity formation phenomenon of the 1D model of Electron Magnetohydrodynamics (EMHD). we will construct a solution whose $C^3$-norm blows up in finite time. In the end, we will show that the solution is in $C^{\infty}(\mathbb{R}\backslash \{0\})\cap C^{3,s}(\mathbb{R})\cap H^3(\mathbb{R})$ and is not asymptotically self-similar.

Singularity formation for the 1D model of EMHD

Abstract

In this paper, we study the singularity formation phenomenon of the 1D model of Electron Magnetohydrodynamics (EMHD). we will construct a solution whose -norm blows up in finite time. In the end, we will show that the solution is in and is not asymptotically self-similar.

Paper Structure

This paper contains 17 sections, 18 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. An overview
  3. The main result
  4. Technical lemmas
  5. Outline of the paper
  6. Construction of the solution
  7. The profile
  8. The form of blow-up
  9. bootstrapping ansatz
  10. Estimates
  11. Evolution of $W_{n,k}$
  12. Convergence to the limit
  13. Properties of the solution
  14. Hölder continuity
  15. Rate of blow up
  16. ...and 2 more sections

Key Result

Theorem 1.1

For any $b\neq0$ and $\epsilon>0$, there exist $s\in(0,1)$, $T>0$, $C>0$ and a solution to the 1D model of (the model) in the time interval $[-T,0)$ such that for all $t$ in this interval, the solution $B(\cdot,t)\in C^{\infty}(\mathbb{R}\backslash \{0\})\cap C^{3,s}(\mathbb{R})\cap H^3(\mathbb{R})$

Theorems & Definitions (36)

  • Theorem 1.1
  • Remark 1.1
  • Lemma 1.2
  • Remark 1.2
  • Lemma 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 26 more