Nonlinear stability of relativistic vortex sheets in three-dimensional Minkowski spacetime
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...
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| Format: | Journal article |
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Springer Berlin Heidelberg
2018
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| _version_ | 1826290875479621632 |
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| author | Chen, G Secchi, P Wang, T |
| author_facet | Chen, G Secchi, P Wang, T |
| author_sort | Chen, G |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-07T02:50:53Z |
| format | Journal article |
| id | oxford-uuid:ada40d37-3a30-4292-a236-ece31f93ea53 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T02:50:53Z |
| publishDate | 2018 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | oxford-uuid:ada40d37-3a30-4292-a236-ece31f93ea532022-03-27T03:37:06ZNonlinear stability of relativistic vortex sheets in three-dimensional Minkowski spacetimeJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:ada40d37-3a30-4292-a236-ece31f93ea53Symplectic Elements at OxfordSpringer Berlin Heidelberg2018Chen, GSecchi, PWang, TWe 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. |
| spellingShingle | Chen, G Secchi, P Wang, T Nonlinear stability of relativistic vortex sheets in three-dimensional Minkowski spacetime |
| title | Nonlinear stability of relativistic vortex sheets in three-dimensional Minkowski spacetime |
| title_full | Nonlinear stability of relativistic vortex sheets in three-dimensional Minkowski spacetime |
| title_fullStr | Nonlinear stability of relativistic vortex sheets in three-dimensional Minkowski spacetime |
| title_full_unstemmed | Nonlinear stability of relativistic vortex sheets in three-dimensional Minkowski spacetime |
| title_short | Nonlinear stability of relativistic vortex sheets in three-dimensional Minkowski spacetime |
| title_sort | nonlinear stability of relativistic vortex sheets in three dimensional minkowski spacetime |
| work_keys_str_mv | AT cheng nonlinearstabilityofrelativisticvortexsheetsinthreedimensionalminkowskispacetime AT secchip nonlinearstabilityofrelativisticvortexsheetsinthreedimensionalminkowskispacetime AT wangt nonlinearstabilityofrelativisticvortexsheetsinthreedimensionalminkowskispacetime |