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Stability and instability of a one-dimensional MHD model

Nicola De Nitti, Jie Guo, Quansen Jiu

Abstract

We consider a one-dimensional magnetohydrodynamics model introduced by Dai \textit{et al.}~(2023), in a parameter regime where, in the absence of a magnetic field, the system reduces to the De Gregorio model for the Euler equations. We analyze stability and instability near the first excited state on the torus, thus generalizing the recent results obtained by Guo and Jiu~(2025) for the De Gregorio model. Specifically, we establish global well-posedness of the linearized system, local well-posedness for the nonlinear system, and demonstrate both linear and nonlinear instability for a broad class of initial data in the weighted Sobolev space introduced by Lai \textit{et al.}~(2020). We identify the principal linearized operator, which is structurally equivalent to that of the De Gregorio model, as the primary mechanism of instability. Moreover, we prove global well-posedness and stability of both linear and nonlinear systems for initial data in a particular subspace of the aforementioned weighted Sobolev space.

Stability and instability of a one-dimensional MHD model

Abstract

We consider a one-dimensional magnetohydrodynamics model introduced by Dai \textit{et al.}~(2023), in a parameter regime where, in the absence of a magnetic field, the system reduces to the De Gregorio model for the Euler equations. We analyze stability and instability near the first excited state on the torus, thus generalizing the recent results obtained by Guo and Jiu~(2025) for the De Gregorio model. Specifically, we establish global well-posedness of the linearized system, local well-posedness for the nonlinear system, and demonstrate both linear and nonlinear instability for a broad class of initial data in the weighted Sobolev space introduced by Lai \textit{et al.}~(2020). We identify the principal linearized operator, which is structurally equivalent to that of the De Gregorio model, as the primary mechanism of instability. Moreover, we prove global well-posedness and stability of both linear and nonlinear systems for initial data in a particular subspace of the aforementioned weighted Sobolev space.
Paper Structure (18 sections, 13 theorems, 282 equations)

This paper contains 18 sections, 13 theorems, 282 equations.

Table of Contents

  1. Introduction
  2. One-dimensional models in fluid dynamics and magnetohydrodynamics
  3. Singularity formation vs global well-posedness for the Okamoto--Sakajo--Wunsch model
  4. Stability for the one-dimensional MHD model
  5. Summary of the main results and outline of the paper
  6. Perturbations around the first excited state
  7. Equations of the perturbations
  8. Function spaces and notation
  9. Main results
  10. Global existence and uniqueness for the linearized problem
  11. Instability for the linearized problem
  12. Linear estimate on eta-
  13. Linear estimate on eta+
  14. Proof of the instability result for the linearized problem
  15. (Local) existence and uniqueness for the nonlinear problem
  16. ...and 3 more sections

Key Result

Theorem 3.1

Let $\rho_2^{1/2} \partial_{\theta} \eta^{\pm}_{0} \in H^{m}(\mathbb T)$ for some integer $m > 3$. Then, for any $T > 0$, there exists a unique classical odd solution $\eta^{\pm}$ to eq:linearized with initial condition $\eta^{\pm} (0, \cdot) = \eta^{\pm}_{0} (\cdot)$, satisfying

Theorems & Definitions (25)

  • Theorem 3.1: Global existence of solutions to the linearized problem
  • Theorem 3.2: Global well-posedness of the linearized problem
  • Theorem 3.3: Instability for the linearized problem
  • Remark 3.4: The constant $\lambda_1$ in \ref{['the:instability_linearized']}
  • Remark 3.5: Initial data satisfying the assumptions of \ref{['the:instability_linearized']}
  • Theorem 3.6: Local well-posedness of the nonlinear problem
  • Theorem 3.7: Instability for the nonlinear problem
  • Remark 3.8: Lack of stability
  • Remark 3.9: Initial data satisfying the assumptions of \ref{['the:instability_nonlinear']}
  • Theorem 3.10: Global well-posedness and stability for the nonlinear problem
  • ...and 15 more