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On three dimensional steady super-Alfvénic magnetohydrodynamics shocks with aligned fields

Shangkun Weng, Wengang Yang

TL;DR

This work addresses the existence and structural stability of three-dimensional steady magnetohydrodynamic shocks with aligned magnetic and velocity fields in a cylindrical nozzle. It introduces a deformation-tensor–modified-vorticity framework, capturing a hyperbolic–elliptic mixed system via a dynamic J = curl((1−κ^2ρ)u) and conserved quantities B, S, κ along streamlines, and recasts the problem into a fixed-domain formulation. An iterative contraction scheme on a function space combines algebraic shock conditions, transport equations for B, S and κ, a deformation–curl system, and a Poisson-type equation for solvability, yielding a unique, small-perturbation solution with sharp regularity and explicit bounds. The results establish the existence and stability of a super-Alfvénic cylindrical transonic shock under 3D perturbations of the inflow and exit pressure, advancing the mathematical theory of MHD shocks with aligned fields.

Abstract

The coupled motion between the hydrodynamic flow and magnetic field introduces significant complexity into the structure of the magnetohydrodynamic (MHD) equations. A key factor contributing to this complexity is the presence of Alfvén waves, which critically influences the character of the flow and makes the problem considerably more challenging. Within the framework where the magnetic field is everywhere parallel to the flow velocity, we give an effective decomposition of the steady MHD equations in terms of the deformation tensor and the modified vorticity, where the modification in the vorticity is to record the effect of the Lorentz force on the velocity field. The existence and structural stability of the super-Alfvénic cylindrical transonic shock solutions for the steady MHD equations are established under three-dimensional perturbations of the incoming flow and the exit total pressure (kinetic plus magnetic).

On three dimensional steady super-Alfvénic magnetohydrodynamics shocks with aligned fields

TL;DR

This work addresses the existence and structural stability of three-dimensional steady magnetohydrodynamic shocks with aligned magnetic and velocity fields in a cylindrical nozzle. It introduces a deformation-tensor–modified-vorticity framework, capturing a hyperbolic–elliptic mixed system via a dynamic J = curl((1−κ^2ρ)u) and conserved quantities B, S, κ along streamlines, and recasts the problem into a fixed-domain formulation. An iterative contraction scheme on a function space combines algebraic shock conditions, transport equations for B, S and κ, a deformation–curl system, and a Poisson-type equation for solvability, yielding a unique, small-perturbation solution with sharp regularity and explicit bounds. The results establish the existence and stability of a super-Alfvénic cylindrical transonic shock under 3D perturbations of the inflow and exit pressure, advancing the mathematical theory of MHD shocks with aligned fields.

Abstract

The coupled motion between the hydrodynamic flow and magnetic field introduces significant complexity into the structure of the magnetohydrodynamic (MHD) equations. A key factor contributing to this complexity is the presence of Alfvén waves, which critically influences the character of the flow and makes the problem considerably more challenging. Within the framework where the magnetic field is everywhere parallel to the flow velocity, we give an effective decomposition of the steady MHD equations in terms of the deformation tensor and the modified vorticity, where the modification in the vorticity is to record the effect of the Lorentz force on the velocity field. The existence and structural stability of the super-Alfvénic cylindrical transonic shock solutions for the steady MHD equations are established under three-dimensional perturbations of the incoming flow and the exit total pressure (kinetic plus magnetic).
Paper Structure (6 sections, 8 theorems, 175 equations)

This paper contains 6 sections, 8 theorems, 175 equations.

Table of Contents

  1. Introduction and main results
  2. The reformulation of the shock problem
  3. The reformulation of the steady MHD equations in terms of the deformation tensor and the modified vorticity
  4. The linearization of the Rankine-Hugoniot conditions and boundary conditions
  5. Transform to a fixed boundary value problem
  6. Proof of Theorem \ref{['main']}

Key Result

Proposition 1.1

Given the incoming supersonic flow $(\bar{U}_-(r_1){\bf e}_r, \bar{\rho}_-(r_1),\bar{S}_-,\bar{\kappa})$ at $r=r_1$, where $\bar{U}_-(r_1)>0,\bar{\rho}_-(r_1)>0, \bar{S}_->0$ and $\bar{U}_-^2(r_1)>c^2(\bar{\rho}_-(r_1),\bar{S}_-)$. Then there exist two positive constants $P_1$ and $P_2$ depending on to mhd-cyl1, which satisfies the incoming supersonic flow and the exit pressure with a shock front

Theorems & Definitions (8)

  • Proposition 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Lemma 3.1
  • Proposition 3.2