Global Strong Well-Posedness of the Three Dimensional Primitive Equations in L^p -Spaces

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∈[Xp,D(Ap)]1/p provided p∈ [ 6 / 5 , ∞). 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 H¹, hereby generalizing in particular a result by Cao and Titi (Ann Math 166:245–267, 2007) to the case of non-smooth initial data.