Nonlinear Stability of Relativistic Vortex Sheets in Three-Dimensional Minkowski Spacetime

TL;DR

This paper proves the nonlinear stability of relativistic vortex sheets in 3D Minkowski spacetime by establishing a new symmetrization, analyzing linear stability conditions, and applying a Nash--Moser scheme for the nonlinear problem.

Contribution

It introduces a novel symmetrization method and provides a necessary and sufficient stability condition for relativistic vortex sheets, advancing understanding of their nonlinear stability.

Findings

Derived a new symmetrization for the problem.

Established a stability condition via the Lopatinski
2f determinant.

Proved nonlinear stability under small perturbations.

Abstract

We are concerned with the nonlinear stability of vortex sheets for the relativistic Euler equations in three-dimensional Minkowski spacetime. This is a nonlinear hyperbolic problem with a characteristic free boundary. In this paper, we introduce a new symmetrization by choosing appropriate functions as primary unknowns. A necessary and sufficient condition for the weakly linear stability of relativistic vortex sheets is obtained by analyzing the roots of the Lopatinski\u{\i} determinant associated to the constant coefficient linearized problem. Under this stability condition, we show that the variable coefficient linearized problem obeys an energy estimate with a loss of derivatives. The construction of certain weight functions plays a crucial role in absorbing error terms caused by microlocalization. Based on the weakly linear stability result, we establish the existence and nonlinear stability of relativistic vortex sheets under small initial perturbations by a Nash--Moser iteration scheme.