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Ill-posedness of incompressible Kelvin-Helmholtz problem with transverse magnetic field

Binqiang Xie, Boling Guo, Bin Zhao

TL;DR

This work proves that the incompressible, inviscid Kelvin–Helmholtz problem for two immiscible MHD fluids with a transverse magnetic field is linearly and nonlinearly ill-posed. By combining Eulerian and Lagrangian formulations, linearizing around a shear state yields a boundary equation whose high-frequency tangential modes grow exponentially, indicating instability persists despite the transverse field. The nonlinear analysis follows an Ebin-type perturbation framework, decomposing perturbations into harmonic and remainder parts and showing nonlinear terms cannot cancel the linear growth, leading to Hadamard-type ill-posedness in Sobolev spaces. The findings reinforce Chandrasekhar's linear intuition in the nonlinear regime and demonstrate that a transverse magnetic field does not stabilize the Kelvin–Helmholtz instability in this setting, highlighting fundamental limitations for stability in current-vortex sheet MHD problems.

Abstract

In this paper, we prove the linear and nonlinear ill-posedness of the well-known Kelvin-Helmholtz problem of the incompressible ideal magnetohydrodynamics (MHD) equations with transverse magnetic field. Our proof rigorously verifies that "the development of the Kelvin-Helmholtz instability, in the direction of the streaming, is uninfluenced by the presence of the magnetic field in the transverse direction" which was proposed by S. Chandrasekhar' book named by Hydrodynamic and Hydromagnetic stability.

Ill-posedness of incompressible Kelvin-Helmholtz problem with transverse magnetic field

TL;DR

This work proves that the incompressible, inviscid Kelvin–Helmholtz problem for two immiscible MHD fluids with a transverse magnetic field is linearly and nonlinearly ill-posed. By combining Eulerian and Lagrangian formulations, linearizing around a shear state yields a boundary equation whose high-frequency tangential modes grow exponentially, indicating instability persists despite the transverse field. The nonlinear analysis follows an Ebin-type perturbation framework, decomposing perturbations into harmonic and remainder parts and showing nonlinear terms cannot cancel the linear growth, leading to Hadamard-type ill-posedness in Sobolev spaces. The findings reinforce Chandrasekhar's linear intuition in the nonlinear regime and demonstrate that a transverse magnetic field does not stabilize the Kelvin–Helmholtz instability in this setting, highlighting fundamental limitations for stability in current-vortex sheet MHD problems.

Abstract

In this paper, we prove the linear and nonlinear ill-posedness of the well-known Kelvin-Helmholtz problem of the incompressible ideal magnetohydrodynamics (MHD) equations with transverse magnetic field. Our proof rigorously verifies that "the development of the Kelvin-Helmholtz instability, in the direction of the streaming, is uninfluenced by the presence of the magnetic field in the transverse direction" which was proposed by S. Chandrasekhar' book named by Hydrodynamic and Hydromagnetic stability.
Paper Structure (11 sections, 9 theorems, 146 equations)

This paper contains 11 sections, 9 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Eulerian formulation
  3. Lagrangian reformulation
  4. History result
  5. Main result
  6. Main idea
  7. The Linearized Equations in both Lagrangian coordinates and Euler coordinates
  8. Discontinuous Dependence on Initial Data
  9. Estimates of nonlinear terms.
  10. Proof of main theorem.
  11. Acknowledgements

Key Result

Theorem 1.1

Suppose that the initial discontinuous velocity field and the initial discontinuous magnetic field satisfies 2.2, 2.3. Let the initial domain to be $\Omega_{0}= \mathbb{T}^{2} \times (-1,0)\cup(0,1)$. Then the Kelvin-Helmholtz problem of 1.15 is linear and nonlinear ill-posedness: for any small $\de but for $t>0$, we have as $n \rightarrow \infty$.

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 9 more