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On compressible magnetic relaxation in planar symmetry

Taehun Kim

TL;DR

This work studies a compressible MHD relaxation model under planar symmetry on the torus, reducing to a 1D system for the density \rho and magnetic field component B with a nonzero background field. The authors prove local well-posedness with a uniform no-vacuum property by constructing diagonalizing transforms W and Z that satisfy parabolic-type equations, enabling a maximum-principle bound on ρ and B. They further establish magnetic relaxation to constant steady states: for small perturbations around (\\bar{ρ},\\bar{B}) the solution exists globally and decays exponentially in time, with distinct analyses for the cases \bar{B}=0 and \bar{B}≠0, using decoupled linear combinations of ρ and B. The results provide a rigorous mathematical validation of magnetic relaxation in the compressible MRE setting and lay groundwork for future numerical schemes and higher-dimensional analyses. Key contributions include the construction of W and Z, the a priori bounds preventing vacuum, and the exponential relaxation proofs for both zero and nonzero mean magnetic-field scenarios.

Abstract

We consider the compressible Magnetic Relaxation Equations on the three-dimensional torus $\mathbb{T}^{3}$. The system is derived from compressible magnetohydrodynamics (MHD) by replacing the acceleration term with a Darcy-type friction. Under planar symmetry, we establish three main results: (1) local well-posedness for smooth initial data, (2) magnetic relaxation for smooth perturbations of constant steady states, and (3) the absence of vacuum states or implosions prior to and at the time of a potential singularity.

On compressible magnetic relaxation in planar symmetry

TL;DR

This work studies a compressible MHD relaxation model under planar symmetry on the torus, reducing to a 1D system for the density \rho and magnetic field component B with a nonzero background field. The authors prove local well-posedness with a uniform no-vacuum property by constructing diagonalizing transforms W and Z that satisfy parabolic-type equations, enabling a maximum-principle bound on ρ and B. They further establish magnetic relaxation to constant steady states: for small perturbations around (\\bar{ρ},\\bar{B}) the solution exists globally and decays exponentially in time, with distinct analyses for the cases \bar{B}=0 and \bar{B}≠0, using decoupled linear combinations of ρ and B. The results provide a rigorous mathematical validation of magnetic relaxation in the compressible MRE setting and lay groundwork for future numerical schemes and higher-dimensional analyses. Key contributions include the construction of W and Z, the a priori bounds preventing vacuum, and the exponential relaxation proofs for both zero and nonzero mean magnetic-field scenarios.

Abstract

We consider the compressible Magnetic Relaxation Equations on the three-dimensional torus . The system is derived from compressible magnetohydrodynamics (MHD) by replacing the acceleration term with a Darcy-type friction. Under planar symmetry, we establish three main results: (1) local well-posedness for smooth initial data, (2) magnetic relaxation for smooth perturbations of constant steady states, and (3) the absence of vacuum states or implosions prior to and at the time of a potential singularity.
Paper Structure (17 sections, 5 theorems, 81 equations, 1 figure)

This paper contains 17 sections, 5 theorems, 81 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Compressible Magnetic Relaxation
  3. Idea of the proof
  4. Acknowledgments
  5. Local well-posedness and absence of vacuum
  6. A-priori estimate
  7. Boundedness of $\rho, \rho^{-1}$, and $B$
  8. Definition of new variables $W$ and $Z$
  9. Equations satisfied by $W$ and $Z$
  10. Proof that $\rho$, $\rho^{-1}$, and $|B|$ are bounded from above
  11. Proof of Theorem \ref{['20260110thm1']}
  12. Relaxation to constant steady state
  13. Decay of $H^{s}$ norm
  14. Proof of Theorem \ref{['20260110thm2']} for the case $\overline{B}=0$
  15. Decay of coupled $H^{s}$ norm
  16. ...and 2 more sections

Key Result

Theorem 1

Assume that $1<\gamma<2$ and $B_{0}\neq 0$. Consider smooth initial data $\rho_{t=0}, B_{t=0} \in C^{\infty}(\mathbb{T})$. Then there exists a maximal existence time $T=T(\gamma, B_{0}, \|\rho_{t=0}\|_{H^{3}}, \|B_{t=0}\|_{H^{3}}, \|\rho_{t=0}^{-1}\|_{L^{\infty}})>0$ and unique smooth solutions $\rh

Figures (1)

  • Figure 1: Level curves of $W$ and $Z$ drawn for $\gamma=1.5, B_{0}=1$. As one can see in the small scale plots, there is a singularity near $(\rho, B)=(B_{0}^{\frac{2}{\gamma}}, 0)$. The shaded regions are the set $\{W \le W_{0}\}$, $\{Z \le Z_{0}\}$, for some constants $W_{0} \in \mathbb{R}$ and $Z_{0}<1$.

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Remark
  • Remark
  • Lemma 3: A-priori weighted $L^{2}$ estimate
  • proof : Proof of Lemma \ref{['20260111lem1']}
  • Lemma 4: A-priori weighted $H^{s}$ estimate
  • proof : Proof of Lemma \ref{['20260110lem1']}
  • Corollary 5: A-priori $H^{s}$ estimate
  • proof : Proof of Corollary \ref{['20260110cor1']}