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Steady super-Alfvénic MHD shocks with aligned fields in two-dimensional almost flat nozzles

Shangkun Weng, Wengang Yang

TL;DR

This work analyzes steady, two-dimensional MHD flows with aligned magnetic and velocity fields in almost flat nozzles, focusing on the existence and stability of super-Alfvénic transonic shocks. It introduces a deformation-curl decomposition and a Lagrangian reformulation that flattens the nozzle and rigidifies the shock as a free boundary, enabling a two-stage approach: (i) determine an initial shock location via a linearized free-boundary problem and a solvability condition linking the exit pressure to the wall profile, and (ii) implement a nonlinear iteration to close the coupling between the supersonic and subsonic regions. Under a super-Alfvénic assumption ($\bar{A}_\pm^2>1$) and small perturbations $\sigma$, the authors prove the existence and stability of a transonic shock solution $U_-,U_+;\eta$ with quantitative norm estimates, and show the admissible condition determining the initial shock position via $\bar{K}_0 \int_0^1 P_{ex}(y_2)\,dy_2 = -\bar{K} f(\bar{\eta}^*) + \bar{u}_+ f(L_1)$. The deformation-curl framework further enables a natural path to three-dimensional generalizations at the initial approximation level, broadening the applicability of the method to more complex MHD configurations.

Abstract

The Lorentz force induced by the magnetic field in MHD flow introduces a fundamental difference from pure gas dynamics by facilitating the anisotropic propagation of small disturbances, thus the type of steady MHD equations depends on not only the Mach number but also the Alfvén number. In the super-Alfvénic case, we derive an admissible condition for the locations of transonic shock fronts in terms of the nozzle wall profile and the exit total pressure (the kinetic plus magnetic pressure). Starting from this initial approximation, a nonlinear existence of super-Alfvénic transonic shock solution to steady MHD equations is established. Our admissible condition is slightly different from that first introduced by Fang-Xin in [Comm. Pure Appl. Math., 74 (2021), pp. 1493-1544], and because our formulation is based on the deformation-curl decomposition of the steady MHD equations, our admissible condition has the advantage that a direct generalization to three dimensional case is available at least at the level of the initial approximation of the shock position.

Steady super-Alfvénic MHD shocks with aligned fields in two-dimensional almost flat nozzles

TL;DR

This work analyzes steady, two-dimensional MHD flows with aligned magnetic and velocity fields in almost flat nozzles, focusing on the existence and stability of super-Alfvénic transonic shocks. It introduces a deformation-curl decomposition and a Lagrangian reformulation that flattens the nozzle and rigidifies the shock as a free boundary, enabling a two-stage approach: (i) determine an initial shock location via a linearized free-boundary problem and a solvability condition linking the exit pressure to the wall profile, and (ii) implement a nonlinear iteration to close the coupling between the supersonic and subsonic regions. Under a super-Alfvénic assumption () and small perturbations , the authors prove the existence and stability of a transonic shock solution with quantitative norm estimates, and show the admissible condition determining the initial shock position via . The deformation-curl framework further enables a natural path to three-dimensional generalizations at the initial approximation level, broadening the applicability of the method to more complex MHD configurations.

Abstract

The Lorentz force induced by the magnetic field in MHD flow introduces a fundamental difference from pure gas dynamics by facilitating the anisotropic propagation of small disturbances, thus the type of steady MHD equations depends on not only the Mach number but also the Alfvén number. In the super-Alfvénic case, we derive an admissible condition for the locations of transonic shock fronts in terms of the nozzle wall profile and the exit total pressure (the kinetic plus magnetic pressure). Starting from this initial approximation, a nonlinear existence of super-Alfvénic transonic shock solution to steady MHD equations is established. Our admissible condition is slightly different from that first introduced by Fang-Xin in [Comm. Pure Appl. Math., 74 (2021), pp. 1493-1544], and because our formulation is based on the deformation-curl decomposition of the steady MHD equations, our admissible condition has the advantage that a direct generalization to three dimensional case is available at least at the level of the initial approximation of the shock position.

Paper Structure

This paper contains 10 sections, 12 theorems, 204 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Reformulation in Lagrangian Coordinates and Main Results
  3. Reformulation in Lagrangian Coordinates and Main Results
  4. The initial approximating locations of the shock front
  5. The nonlinear shock problem
  6. The supersonic solution in
  7. The shock front and subsonic solution
  8. The linearized system and iteration scheme
  9. The solvability condition and well-posedness for the iteration system
  10. Contractive mapping

Key Result

Theorem 2.2

Assume that the background magnetic field strength $\bar{\kappa}$ satisfies assump_bar_ka. Let $\bar{\eta}^*\in(L_0,L_1)$ be such that and $f'(\bar{\eta}^*)\neq 0$, where Then there exists a small constant $\sigma_*>0$, depending only on the background transonic solution, such that for any $0<\sigma<\sigma_*$, the Problem prob2 admits a transonic shock solution $(\mathcal{U}_-,\mathcal{U}_+;\eta

Figures (1)

  • Figure 1: Nozzle

Theorems & Definitions (24)

  • Remark 1.2
  • Remark 1.3
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 14 more