Two scenarios on a potential smoothness breakdown for the three-dimensional Navier-Stokes equations
In this paper we construct two families of initial data being arbitrarily large under any scaling-invariant norm for which their corresponding weak solution to the three-dimensional Navier- Stokes equations become smooth on either [0, T1] or [T2,1), respectively, where T1 and T2 are two times prescr...
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| Tipo de documento: | artigo |
| Estado: | Versão publicada |
| Data de publicação: | 2018 |
| País: | España |
| Recursos: | Universidad de Sevilla (US) |
| Repositório: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/89776 |
| Acesso em linha: | https://hdl.handle.net/11441/89776 https://doi.org/10.3934/dcds.2020142 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Navier-Stokes equations Weak solutions Strong solutions Breakdown of smooth solutions Regularity of solutions |
| Resumo: | In this paper we construct two families of initial data being arbitrarily large under any scaling-invariant norm for which their corresponding weak solution to the three-dimensional Navier- Stokes equations become smooth on either [0, T1] or [T2,1), respectively, where T1 and T2 are two times prescribed previously. In particular, T1 can be arbitrarily large and T2 can be arbitrarily small. Therefore, possible formation of singularities would occur after a very long or short evolution time, respectively. We further prove that if a large part of the kinetic energy is consumed prior to the first (possible) blow-up time, then the global-in-time smoothness of the solutions follows for the two families of initial data. |
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