Nonlinear stability of compressible vortex sheets in three-dimensional elastodynamics
Robin Ming Chen, Feimin Huang, Dehua Wang, Difan Yuan
TL;DR
The paper establishes local nonlinear stability of 3D compressible elastic vortex sheets under small perturbations in a subsonic regime. It combines an in-depth spectral analysis of the Lopatinskii determinant with an advanced paralinearization and an upper triangularization that isolates outgoing modes, enabling energy estimates despite derivative losses. A key novelty is the elasticity-driven ellipticity enhancement, realized through a geometric condition on the deformation gradient, which allows elimination of the front and recovery of regularity for the free boundary. The authors refine Coulombel’s diagonalization to handle higher-order degeneracies (double roots and double poles) by constructing invertible symbol mappings, securing localized Garding-type control. The nonlinear theory is completed via a Nash–Moser scheme, aided by robust L^2 energy estimates, tangential-derivative tame estimates, and careful handling of divergences and vorticities, yielding local-in-time existence for small data in weighted Sobolev spaces. The results provide a rigorous framework for the stability of elastic vortex sheets in 3D and highlight how elasticity can stabilize interfaces that are unstable in purely compressible Euler settings.
Abstract
We investigate the nonlinear stability of compressible vortex sheet solutions for three-dimensional (3D) isentropic elastic flows. Building upon previous results on the weakly linear stability of elastic vortex sheets [19], we perform a detailed study of the roots of the Lopatinskii determinant and identify a geometric stability condition associated with the deformation gradient. We employ an upper triangularization technique that isolates the outgoing modes into a closed system, where they appear only at the leading order. This enables us to derive energy estimates despite derivative loss. The major novelty of our approach includes the following two key aspects: (1) For the 3D compressible Euler vortex sheets, the front symbol exhibits degenerate ellipticity in certain frequency directions, which makes it challenging to ensure the front's regularity using standard energy estimates. Our analysis reveals that the non-parallel structure of the deformation gradient tensor plays a crucial role in recovering ellipticity in the front symbol, thereby enhancing the regularity of the free interface. (2) Another significant challenge in 3D arises from the strong degeneracy caused by the collision of repeated roots and poles. Unlike in 2D, where such interactions are absent, we encounter a co-dimension one set in frequency space where a double root coincides with a double pole. To resolve this, we refine Coulombel's diagonalization framework [21] and construct a suitable transformation that reduces the degeneracy order of the Lopatinskii matrix, enabling the use of localized Garding-type estimates to control the characteristic components. Finally, we employ a Nash-Moser iteration scheme to establish the local existence and nonlinear stability of vortex sheets under small initial perturbations, showing stability within a subsonic regime.