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Wellposedness of the electron MHD without resistivity for large perturbations of the uniform magnetic field

In-Jee Jeong, Sung-Jin Oh

Abstract

We prove the local wellposedness of the Cauchy problems for the electron magnetohydrodynamics equations (E-MHD) without resistivity for possibly large perturbations of nonzero uniform magnetic fields. While the local wellposedness problem for (E-MHD) has been extensively studied in the presence of resistivity (which provides dissipative effects), this seems to be the first such result without resistivity. (E-MHD) is a fluid description of plasma in small scales where the motion of electrons relative to ions is significant. Mathematically, it is a quasilinear dispersive equation with nondegenerate but nonelliptic second-order principal term. Our result significantly improves upon the straightforward adaptation of the classical work of Kenig--Ponce--Rolvung--Vega on the quasilinear ultrahyperbolic Schrödinger equations, as the regularity and decay assumptions on the initial data are greatly weakened to the level analogous to the recent work of Marzuola--Metcalfe--Tataru in the case of elliptic principal term. A key ingredient of our proof is a simple observation about the relationship between the size of a symbol and the operator norm of its quantization as a pseudodifferential operator when restricted to high frequencies. This allows us to localize the (non-classical) pseudodifferential renormalization operator considered by Kenig--Ponce--Rolvung--Vega, and produce instead a classical pseudodifferential renormalization operator. We furthermore incorporate the function space framework of Marzuola--Metcalfe--Tataru to the present case of nonelliptic principal term.

Wellposedness of the electron MHD without resistivity for large perturbations of the uniform magnetic field

Abstract

We prove the local wellposedness of the Cauchy problems for the electron magnetohydrodynamics equations (E-MHD) without resistivity for possibly large perturbations of nonzero uniform magnetic fields. While the local wellposedness problem for (E-MHD) has been extensively studied in the presence of resistivity (which provides dissipative effects), this seems to be the first such result without resistivity. (E-MHD) is a fluid description of plasma in small scales where the motion of electrons relative to ions is significant. Mathematically, it is a quasilinear dispersive equation with nondegenerate but nonelliptic second-order principal term. Our result significantly improves upon the straightforward adaptation of the classical work of Kenig--Ponce--Rolvung--Vega on the quasilinear ultrahyperbolic Schrödinger equations, as the regularity and decay assumptions on the initial data are greatly weakened to the level analogous to the recent work of Marzuola--Metcalfe--Tataru in the case of elliptic principal term. A key ingredient of our proof is a simple observation about the relationship between the size of a symbol and the operator norm of its quantization as a pseudodifferential operator when restricted to high frequencies. This allows us to localize the (non-classical) pseudodifferential renormalization operator considered by Kenig--Ponce--Rolvung--Vega, and produce instead a classical pseudodifferential renormalization operator. We furthermore incorporate the function space framework of Marzuola--Metcalfe--Tataru to the present case of nonelliptic principal term.
Paper Structure (42 sections, 38 theorems, 373 equations)

This paper contains 42 sections, 38 theorems, 373 equations.

Table of Contents

  1. Introduction
  2. Main ideas
  3. Basic setup and local smoothing estimate
  4. Previous approaches
  5. New ideas in this work
  6. Discussions
  7. Structure of the paper
  8. Preliminaries
  9. Notation and conventions
  10. Function spaces
  11. Pseudo- and paradifferential operators
  12. Classical symbols and quantizations
  13. Paradifferential operators
  14. Symbolic calculus for pseudodifferential operators
  15. $L^{2}$-boundedness, positivity and high frequency Calderón--Vaillancourt trick
  16. ...and 27 more sections

Key Result

Theorem 1.1

Let $s > \frac{7}{2}$ and consider a vector field ${\bf B}_{0} : \mathbb R^{3} \to \mathbb R^{3}$ satisfying $\nabla \cdot {\bf B}_{0} = 0$. Assume furthermore that ${\bf B}_{0}$ satisfies the following properties: Then the Cauchy problem for eq:e-mhd with ${\bf B}(t=0) = {\bf B}_{0}$ is locally wellposed.

Theorems & Definitions (72)

  • Theorem 1.1: Main theorem, simple version
  • Corollary 1.2: Small data local wellposedness
  • Remark 1.3: Extension to the Hall-MHD system
  • Remark 1.4: Extension to quasilinear ultrahyperbolic Schrödinger equations PiTa
  • Remark 1.5: Possible relaxations of assumptions
  • Remark 1.6: Extension to quasilinear ultrahyperbolic Schrödinger equations PiTa
  • Lemma 2.1: Slow variance of $X_{k}$ and $Y_{k}$
  • Lemma 2.2
  • Proposition 2.3
  • Proposition 2.4
  • ...and 62 more