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Existence, Uniqueness, and Smoothing for Generalized EMHD

Chao Wu

Abstract

We study the Cauchy problem for generalized electron magnetohydrodynamics (EMHD). We establish the local existence and uniqueness of solutions in critical Sobolev spaces, as well as global existence and uniqueness for small initial data. In addition, we prove an instantaneous smoothing effect for the corresponding solutions. Finally, we derive time decay rates for the global solutions.

Existence, Uniqueness, and Smoothing for Generalized EMHD

Abstract

We study the Cauchy problem for generalized electron magnetohydrodynamics (EMHD). We establish the local existence and uniqueness of solutions in critical Sobolev spaces, as well as global existence and uniqueness for small initial data. In addition, we prove an instantaneous smoothing effect for the corresponding solutions. Finally, we derive time decay rates for the global solutions.
Paper Structure (20 sections, 5 theorems, 120 equations)

This paper contains 20 sections, 5 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Notation
  4. Littlewood-Paley Analysis
  5. Gevrey Class
  6. Commutator estimates
  7. Statement of main theorems
  8. Proof of Theorem \ref{['theorem for emhd']}
  9. Sobolev space estimates for EMHD
  10. Intermediary $L^{\frac{\kappa}{\delta}}_T\dot{H}^{\sigma_c+\delta}$-estimates
  11. $L_T^2\dot{H}^{\sigma_c+\frac{\kappa}{2}}$-estimates
  12. $L_T^{\infty}\dot{H}^{\sigma_c}$-estimates
  13. Summary of Sobolev space estimates for EMHD
  14. Gevrey class estimates for EMHD
  15. Properties of the solution to EMHD
  16. ...and 5 more sections

Key Result

Lemma 2.1

Assume that $\mathop{\mathrm{supp}}\nolimits{\hat{u}}\in\mathcal{A}_j$. Then for $1\le p<q\le\infty$, the following hold

Theorems & Definitions (8)

  • Lemma 2.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • Corollary 3.1.1
  • Theorem 4.1