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The lifespan of strong solutions to the compressible MHD equations with entropy transport in the presence of vacuum

Yongteng Gu, Xiangdi Huang

TL;DR

This work analyzes the finite-time blow-up of strong solutions to the compressible MHD system with entropy transport and no magnetic diffusion, including interior vacuum and, in some results, a free boundary. It establishes local well-posedness in bounded domains and derives rigorous blow-up mechanisms under radial and axisymmetric symmetry, providing explicit lifespan bounds that depend on the initial energy and magnetic flux in the vacuum region. The authors develop a fractional-moment method in the vacuum to balance viscous stresses against the Lorentz force, and they extend the analysis to free-boundary problems by coupling energy conservation with boundary growth estimates. Collectively, the results generalize prior work on isentropic MHD by handling more general pressure laws and interior vacuum, and they illuminate how magnetic effects interact with vacuum regions to trigger finite-time singularities. The paper also outlines a robust local well-posedness theory via linearization, enabling the construction of strong solutions even in the presence of vacuum.

Abstract

In this paper, we investigate the finite time blow-up of strong solutions to the compressible magnetohydrodynamic (MHD) system (without magnetic diffusion) coupled with entropy transport, and derive an upper bound for the lifespan of such solutions. We first establish the local well-posedness of strong solutions for bounded domains and study the mechanism of finite-time singularity formation in the 2D radially symmetric case and 3D cylindrically symmetric case. We prove that if the initial density vanishes in an interior region containing the origin and the magnetic field is non-trivial within this vacuum region, the strong solution must blow up in finite time. These results generalize and improve the previous results of Huang-Xin-Yan [Math. Ann. 392 (2025) 2365-2394] for the compressible isentropic MHD equations. Significantly, we extend this blow-up result to the free boundary problem. Our analysis of the boundary's expansion allows us to explicitly estimate the maximum lifespan of the solution.

The lifespan of strong solutions to the compressible MHD equations with entropy transport in the presence of vacuum

TL;DR

This work analyzes the finite-time blow-up of strong solutions to the compressible MHD system with entropy transport and no magnetic diffusion, including interior vacuum and, in some results, a free boundary. It establishes local well-posedness in bounded domains and derives rigorous blow-up mechanisms under radial and axisymmetric symmetry, providing explicit lifespan bounds that depend on the initial energy and magnetic flux in the vacuum region. The authors develop a fractional-moment method in the vacuum to balance viscous stresses against the Lorentz force, and they extend the analysis to free-boundary problems by coupling energy conservation with boundary growth estimates. Collectively, the results generalize prior work on isentropic MHD by handling more general pressure laws and interior vacuum, and they illuminate how magnetic effects interact with vacuum regions to trigger finite-time singularities. The paper also outlines a robust local well-posedness theory via linearization, enabling the construction of strong solutions even in the presence of vacuum.

Abstract

In this paper, we investigate the finite time blow-up of strong solutions to the compressible magnetohydrodynamic (MHD) system (without magnetic diffusion) coupled with entropy transport, and derive an upper bound for the lifespan of such solutions. We first establish the local well-posedness of strong solutions for bounded domains and study the mechanism of finite-time singularity formation in the 2D radially symmetric case and 3D cylindrically symmetric case. We prove that if the initial density vanishes in an interior region containing the origin and the magnetic field is non-trivial within this vacuum region, the strong solution must blow up in finite time. These results generalize and improve the previous results of Huang-Xin-Yan [Math. Ann. 392 (2025) 2365-2394] for the compressible isentropic MHD equations. Significantly, we extend this blow-up result to the free boundary problem. Our analysis of the boundary's expansion allows us to explicitly estimate the maximum lifespan of the solution.

Paper Structure

This paper contains 19 sections, 8 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Local well-posedness
  4. Finite time blow-up for 2D Disk
  5. Finite time Blow-up for 3D Axisymmetric Cylinders
  6. Finite time Blow-up for free boundary problem
  7. Sketch of the proof
  8. Finite time blow-up for fixed boundary
  9. Dynamic behavior in the vacuum region
  10. Conservation law for the magnetic field
  11. Proof of Theorem \ref{['Thm 1.2']}
  12. Proof of Theorem \ref{['th1.3']}
  13. Finite time Blow-up for free boundary problem
  14. Basic energy estimates
  15. Estimates for the free boundary radius
  16. ...and 4 more sections

Key Result

Theorem 2.1

(Local existence of strong solutions-2D case) Let $\Omega \subset \mathbb{R} ^2$ be a $2D$ bounded domain with smooth boundary. Assume that the initial data $(\rho_0, u_0, P_0, B_0)$ satisfy the following conditions for some constant $q>2$, and satisfy the compatibility condition: for some $g \in L^2(\Omega)$. Then, there exists a small time $T^* > 0$ and a unique strong solution $(\rho, u, P, B

Theorems & Definitions (24)

  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.4
  • ...and 14 more