Asymptotics and Lower Bound for the Lifespan of Solutions to the Primitive Equations

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Abstract

In a previous work we obtained a large lower bound for the lifespan of the solutions to the Primitive Equations, and proved convergence to the 3D quasi-geostrophic system for general and ill-prepared blowing-up data, when the kinematic viscosity\(\nu \)is equal to the heat diffusivity\(\nu '\), turning the diffusion operator \(\varGamma \) into the classical Laplacian.

Obtaining the same results in the general case is much more difficult as it involves a homogeneous non-local non-radial diffusion operator \(\varGamma \) whose semi-group and singular integral form kernels present sign changes. Every classical result related to non-local operators, or to Navier-Stokes system then becomes more involved here and the key ingredient will be new transport-diffusion estimates obtained in a companion paper and a precise use of the quasi-geostrophic decomposition.

Keywords

Navier-Stokes system Besov spaces Littlewood-Paley decomposition Primitive equations 3D quasi-geostrophic system Strichartz estimates Non-local operators Boussinesq system 

Notes

Acknowledgements

The author wishes to thank R. Danchin, I. Gallagher, and T. Hmidi for useful discussions.

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Authors and Affiliations

  1. 1.Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050)Université Paris-Est CréteilCréteil CedexFrance

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