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

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.

26 pages

References in corpus (2)

• Local and Global Well-posedness of Strong Solutions to the 3D Primitive Equations with Vertical Eddy Diffusivity
• Global Well-posedness of the 3D Primitive Equations with Only Horizontal Viscosity and Diffusion

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• The Primitive Equations in the scaling invariant space $L^{\infty }(L^1)$
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