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Hölder continuous dissipative solutions of ideal MHD with nonzero helicity

Alberto Enciso, Javier Peñafiel-Tomás, Daniel Peralta-Salas

Abstract

We prove the existence of weak solutions to the 3D ideal MHD equations, of class $C^α$ with $α=1/200$, for which the total energy and the cross helicity (i.e., the so-called Elsässer energies) are not conserved. The solutions do not possess any symmetry properties and the magnetic helicity, which is necessarily conserved for Hölder continuous solutions, is nonzero. The construction, which works both on the torus $\mathbb{T}^3$ and on $\mathbb{R}^3$ with compact spatial support, is based on a novel convex integration scheme in which the magnetic helicity is preserved at each step. This is the first construction of continuous weak solutions at a regularity level where one conservation law (here, the magnetic helicity) is necessarily preserved while another (here, the total energy or cross helicity) is not, and where the preservation of the former is nontrivial in the sense that it does not follow from symmetry considerations.

Hölder continuous dissipative solutions of ideal MHD with nonzero helicity

Abstract

We prove the existence of weak solutions to the 3D ideal MHD equations, of class with , for which the total energy and the cross helicity (i.e., the so-called Elsässer energies) are not conserved. The solutions do not possess any symmetry properties and the magnetic helicity, which is necessarily conserved for Hölder continuous solutions, is nonzero. The construction, which works both on the torus and on with compact spatial support, is based on a novel convex integration scheme in which the magnetic helicity is preserved at each step. This is the first construction of continuous weak solutions at a regularity level where one conservation law (here, the magnetic helicity) is necessarily preserved while another (here, the total energy or cross helicity) is not, and where the preservation of the former is nontrivial in the sense that it does not follow from symmetry considerations.

Paper Structure

This paper contains 25 sections, 36 theorems, 439 equations.

Table of Contents

  1. Introduction
  2. Conservation laws for the MHD system
  3. Non-conservative solutions of ideal MHD
  4. Main result and strategy of the proof
  5. Organization of the paper
  6. Overview of our convex integration scheme
  7. Basic ideas
  8. Inductive hypotheses
  9. Construction of the new subsolution
  10. Preliminaries
  11. Simplification of the Reynolds stress
  12. Definition of the diffeomorphism
  13. The new velocity and magnetic field
  14. The new Reynolds stress
  15. One final correction
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $T>0$ and let $\alpha\coloneqq \frac{1}{200}$. Given a smooth divergence-free field $\bar{B}_0\in C^\infty_c(\mathbb{R}^3,\mathbb{R}^3)$ and $\varepsilon>0$, there exists a weak solution of ideal MHD $(v,B)\in C^\alpha(\mathbb{R}^3\times[0,T])\cap C^0([0,T],L^2(\mathbb{R}^3))$ such that:

Theorems & Definitions (73)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • Definition 2.2
  • Proposition 2.3
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 63 more