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Uniform gradient bounds for the primitive equations of the ocean. (English) Zbl 1224.35346

Summary: In this paper we consider the 3D primitive equations of the ocean in the case of the Dirichlet boundary conditions on the side and bottom boundaries. We provide an explicit upper bound for the \(H^1\) norm of the solution. We prove that, after a finite time, this norm is less than a constant which depends only on the viscosity \(\nu \), the force \(f\) and the domain \(\Omega \). This improves our previous result from [Nonlinearity 20, No. 12, 2739–2753 (2007; Zbl 1136.35069)], where we established the global existence of strong solutions with an argument which does not give such explicit rates.

MSC:

35Q35 PDEs in connection with fluid mechanics
35K51 Initial-boundary value problems for second-order parabolic systems
76U05 General theory of rotating fluids
86A05 Hydrology, hydrography, oceanography

Citations:

Zbl 1136.35069
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