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Large bulk viscosity limit for compressible MHD equations in critical Besov spaces

Gennaro Ciampa, Donatella Donatelli, Giada Pellecchia

TL;DR

This work addresses the global behavior of the compressible MHD system with large bulk viscosity in scaling-critical Besov spaces, proving global strong solutions for large initial data and quantitative convergence to incompressible MHD as $\lambda\to\infty$. The authors employ a stability-first strategy around an incompressible reference flow, decomposing the velocity via Leray projections and deriving coupled Besov-energy estimates that can be closed in the large-$\lambda$ regime. They extend the analysis to $L^p$-based Besov spaces, preserving global well-posedness and convergence with explicit rates, and apply the results to construct compressible MHD solutions that exhibit magnetic reconnection, transferring the reconnection mechanism from the incompressible setting. The findings provide a rigorous framework for singular limits in MHD and demonstrate that reconnection phenomena persist in the compressible regime under large bulk viscosity, with potential implications for modeling highly viscous plasmas.

Abstract

We study the large bulk viscosity limit for the compressible magnetohydrodynamics (MHD) equations in two and three dimensions. For arbitrarily large initial data in critical Besov spaces, we prove the global well-posedness of strong solutions and establish their convergence, with explicit quantitative rates, to solutions of the incompressible MHD system, as the bulk viscosity parameter tends to infinity. As an application of this singular-limit analysis, we construct global smooth solutions to the compressible MHD equations whose magnetic field undergoes reconnection, thereby extending to the compressible regime the reconnection scenarios previously identified for incompressible flows.

Large bulk viscosity limit for compressible MHD equations in critical Besov spaces

TL;DR

This work addresses the global behavior of the compressible MHD system with large bulk viscosity in scaling-critical Besov spaces, proving global strong solutions for large initial data and quantitative convergence to incompressible MHD as . The authors employ a stability-first strategy around an incompressible reference flow, decomposing the velocity via Leray projections and deriving coupled Besov-energy estimates that can be closed in the large- regime. They extend the analysis to -based Besov spaces, preserving global well-posedness and convergence with explicit rates, and apply the results to construct compressible MHD solutions that exhibit magnetic reconnection, transferring the reconnection mechanism from the incompressible setting. The findings provide a rigorous framework for singular limits in MHD and demonstrate that reconnection phenomena persist in the compressible regime under large bulk viscosity, with potential implications for modeling highly viscous plasmas.

Abstract

We study the large bulk viscosity limit for the compressible magnetohydrodynamics (MHD) equations in two and three dimensions. For arbitrarily large initial data in critical Besov spaces, we prove the global well-posedness of strong solutions and establish their convergence, with explicit quantitative rates, to solutions of the incompressible MHD system, as the bulk viscosity parameter tends to infinity. As an application of this singular-limit analysis, we construct global smooth solutions to the compressible MHD equations whose magnetic field undergoes reconnection, thereby extending to the compressible regime the reconnection scenarios previously identified for incompressible flows.
Paper Structure (7 sections, 23 theorems, 263 equations)

This paper contains 7 sections, 23 theorems, 263 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Besov spaces
  4. Preliminaries on Stokes and MHD systems
  5. Proof of Theorem \ref{['thm:main']}
  6. Lp Besov framework
  7. Applications to Magnetic Reconnection

Key Result

Theorem A

Let $d\geq2$ and $c_0>0$. Assume that the initial conditions satisfy and that the initial data $V_0:={\mathcal{P}} v_0$ and $B_0:={\mathcal{P}} b_0$ generates a unique global solution $V,B\in C_b(\mathbb{R}_+,\dot B_{2,1}^{d/2-1}(\mathbb{R}^d))$ of MHDincomp. Denote by There exists a (large) universal constant $C$ such that if $\nu\geq \mu$ and then MHDcomp admit a unique global-in-time solutio

Theorems & Definitions (32)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3: Bernstein inequalities
  • Proposition 2.4: Embeddings
  • Proposition 2.5: Interpolation
  • Definition 2.6
  • Lemma 2.7
  • ...and 22 more