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

Abstract

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 ∈ [Xp, D(Ap)] 1/p provided p ∈ [6/5, ∞). To this end, the hydrostatic Stokes operator Ap defined on Xp, 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 [8] to the case of non-smooth initial data.

The hydrostatic Stokes semigroup and well-posedness of the primitive equations on spaces of bounded functions

“…Taking advantage of the regularization of solutions for t > 0 one passes into the setting discussed in [11] and [9], and thus we obtain the following corollary. Our main result on the hydrostatic semigroup acting on X σ reads as follows.…”

“…Our first main result concerns the global well-posedness of the primitive equations for arbitrarily large initial data in X σ , while the second result extends this situation to the case of small perturbations in L ∞ H L p z (Ω). Here, a strong solution means -as in [11] -a solution v to the primitive equations satisfying…”

“…Using L s (G ′ ) ֒→ L 1 (G ′ ) as well as the estimate on the pressure term, cf. [11,Theorem 3.1. ], in L s (Ω) for s ∈ (1, q], we obtain…”

“…Using V ∈ S(T ), the embedding L ∞ H L p z (Ω) ֒→ L p (Ω), as well as the semigroup estimates t ϑ e tA Pf D((−Ap, σ ) ϑ ) ≤ C f L p (Ω) , t 1/2 e tA P∇ · f L p (Ω) ≤ C f L p (Ω) , t > 0, ϑ ∈ [0, 1] compare [11,Lemma 4.6] and [7,Theorem 3.7], one easily obtains that V (t 0 ) ∈ D((−A p,σ ) ϑ ) for t 0 > 0, and thus v(t 0 ) ∈ D((−A p,σ ) 1/p ) as well, so v can be extended to a global solution that is strong on (t 0 , ∞).…”

On the finite element approximation for non-stationary saddle-point problems

“…We are interested in time-global error estimates for finite element approximations of the hydrostatic Stokes and the primitive equations. In contrast to the three-dimensional Navier-Stokes equations, it is known that the three-dimensional primitive equations are globally well-posed in the L p -settings (see [2] for p = 2, and [19] for p ∈ (1, ∞)). In their proofs, the analyticity of the hydrostatic Stokes semigroup plays a crucial role.…”