Global Strong $L^p$ Well-Posedness of the 3D Primitive Equations with Heat and Salinity Diffusion

Abstract

Consider the full primitive equations, i.e. the three dimensional primitive equations coupled to the equation for temperature and salinity, and subject to outer forces. It is shown that this set of equations is globally strongly well-posed for arbitrary large initial data lying in certain interpolation spaces, which are explicitly characterized as subspaces of $H^{2/p,p}$, $1<p<\infty$, satisfying certain boundary conditions. In particular, global well-posedeness of the full primitive equations is obtained for initial data having less differentiability properties than $H^1$, hereby generalizing by result by Cao and Titi (Ann. of Math. (2) 166 (2007), no. 1, 245-267) to the case of non-smooth data. In addition, it is shown that the solutions are exponentially decaying provided the outer forces possess this property.

24 pages, 3 figures

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• The primitive equations with stochastic wind driven boundary conditions
• Global well-posedness of $z$-weak solutions to the primitive equations without vertical diffusivity
• Strong solutions to the 3D primitive equations with only horizontal dissipation: near $H^1$ initial data
• The primitive equations approximation of the anisotropic horizontally viscous Navier-Stokes equations
• Primitive Equations with Linearly Growing Initial Data