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

Author: 

Chen, G
Secchi, P
Wang, T

Publication Date: 

May 2019

Journal: 

ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS

Last Updated: 

2019-11-10T15:36:17.067+00:00

Issue: 

2

Volume: 

232

DOI: 

10.1007/s00205-018-1330-5

page: 

591-695

abstract: 

© 2018, Springer-Verlag GmbH Germany, part of Springer Nature. 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 Lopatinskiĭ 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 the 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.

Symplectic id: 

935956

Download URL: 

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article