Quasi-periodic solutions to the incompressible Euler equations in dimensions two and higher
Building on the work of Crouseilles and Faou on the 2D case, we construct C∞ quasi-periodic solutions to the incompressible Euler equations with periodic boundary conditions in dimension 3 and in any even dimension. These solutions are genuinely high-dimensional, which is particularly interesting be...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Consejo Superior de Investigaciones Científicas (CSIC) |
| Repositorio: | DIGITAL.CSIC. Repositorio Institucional del CSIC |
| OAI Identifier: | oai:digital.csic.es:10261/348131 |
| Acceso en línea: | http://hdl.handle.net/10261/348131 https://www.scopus.com/inward/record.uri?eid=2-s2.0-85146686964&doi=10.1016%2fj.jde.2023.01.013&partnerID=40&md5=0396214c5e0c04ae6edac55900be61f3 |
| Access Level: | acceso abierto |
| Palabra clave: | Compactly supported steady states Incompressible Euler equations Quasi-periodic solutions |
| Sumario: | Building on the work of Crouseilles and Faou on the 2D case, we construct C∞ quasi-periodic solutions to the incompressible Euler equations with periodic boundary conditions in dimension 3 and in any even dimension. These solutions are genuinely high-dimensional, which is particularly interesting because there are extremely few examples of high-dimensional initial data for which global solutions are known to exist. These quasi-periodic solutions can be engineered so that they are dense on tori of arbitrary dimension embedded in the space of solenoidal vector fields. Furthermore, in the two-dimensional case we show that quasi-periodic solutions are dense in the phase space of the Euler equations. More precisely, for any integer N⩾1 we prove that any Lq initial stream function can be approximated in Lq (strongly when 1⩽q<∞ and weak-⁎ when q=∞) by smooth initial data whose solutions are dense on N-dimensional tori. © 2023 The Author(s) |
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