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Half Strong Ill-Posedness of $2 \frac{1}{2}$D Electron Magnetohydrodynamics with Fractional Resistivity

Xiaotong, Yang, Haoming Zhu

Abstract

We study the $2\frac{1}{2}$D electron magnetohydrodynamics (MHD): the electron MHD system that has $3$D magnetic field but is independent of $z$-variable. We establish a "half" strong ill-posedness result in $2\frac{1}{2}$D electron MHD with fractional resistivity $(-Δ)^α$ in the supercritical Sobolev space $H^β\times H^{β-1}$ for $3<β<4-2α$. Specifically, we construct small initial data $(a_0,b_0)$ whose solution develops a norm inflation in $a$ but the norm of $b$ remains small.

Half Strong Ill-Posedness of $2 \frac{1}{2}$D Electron Magnetohydrodynamics with Fractional Resistivity

Abstract

We study the D electron magnetohydrodynamics (MHD): the electron MHD system that has D magnetic field but is independent of -variable. We establish a "half" strong ill-posedness result in D electron MHD with fractional resistivity in the supercritical Sobolev space for . Specifically, we construct small initial data whose solution develops a norm inflation in but the norm of remains small.
Paper Structure (6 sections, 6 theorems, 87 equations)

This paper contains 6 sections, 6 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Initial data, approximate solution, and approximate norm inflation
  4. Error equation and estimate of error
  5. Proof of Theorem \ref{['thm:main']}
  6. References

Key Result

Theorem 1.1

Let $3<\beta<4-2\alpha$ and $0\le\alpha<1/2$, then for any small $\epsilon>0$, any large $\Lambda>0$, and any small time $t_*>0$, there exist smooth initial data $(a_0,b_0)$ such that, such that the generalized $2\frac{1}{2}$D electron MHD system eq:resistive-25d has a solution whose norm inflates while the norm of $b$ remains small:

Theorems & Definitions (16)

  • Theorem 1.1: Half strong ill-posedness in generalized $2\frac{1}{2}$D electron MHD
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 3.1: Initial data bounds
  • proof
  • Lemma 3.2: Approximate solution inflation
  • proof
  • Lemma 3.3: Smallness of $\bar{u}$
  • proof
  • ...and 6 more