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

TL;DR

This paper proves that the full 3D primitive equations with heat and salinity diffusion are globally well-posed for large, less smooth initial data, and solutions decay exponentially if external forces do.

Contribution

It extends previous results by establishing global strong well-posedness for initial data with lower regularity in the primitive equations.

Findings

Global well-posedness for less smooth initial data.

Solutions decay exponentially under certain force conditions.

Explicit characterization of initial data spaces.

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.