Global Strong Well-posedness of the Three Dimensional Primitive equations in $L^p$-spaces

TL;DR

This paper proves the global existence and uniqueness of strong solutions to the three-dimensional primitive equations in $L^p$-spaces for a broad range of initial data, extending previous results to less smooth initial conditions.

Contribution

The paper introduces an $L^p$-approach and hydrostatic Stokes operator to establish global well-posedness for initial data with minimal regularity, generalizing prior work.

Findings

Global strong solutions exist for all initial data in a specified $L^p$-space range.

The approach allows for initial data with less differentiability than previously required.

Extension of well-posedness results to non-smooth initial conditions.

Abstract

In this article, an $L^p$-approach to the primitive equations is developed. In particular, it is shown that the three dimensional primitive equations admit a unique, global strong solution for all initial data $a \in [X_p,D(A_p)]_{1/p}$ provided $p \in [6/5,\infty)$. To this end, the hydrostatic Stokes operator $A_p$ defined on $X_p$, the subspace of $L^p$ associated with the hydrostatic Helmholtz projection, is introduced and investigated. Choosing $p$ large, one obtains global well-posedness of the primitive equations for strong solutions for initial data $a$ having less differentiability properties than $H^1$, hereby generalizing in particular a result by Cao and Titi (Ann. Math. 166 (2007), pp. 245-267) to the case of non-smooth initial data.