Sobolev and Gevrey regularity results for the primitive equations in three space dimensions

Abstract

The aim of this article is to present a qualitative study of the Primitive Equations in a threedimensional domain, with periodical boundary conditions. We start by recalling some already existing results regarding the existence locally in time of weak solutions and existence and uniqueness of strong solutions, and we prove the existence of very regular solutions, up to C 1 -regularity. In the second part of the article we prove that the solution of the Primitive Equations belongs to a certain Gevrey class of functions.

Stability of the Slow Manifold in the Primitive Equations

“…Both these works followed [6] and [13]. The result (2.20) follows from [25] and using (2.19) for s = 2.…”

“…For the present article, we need uniform bounds on the norms, which have been proved recently [23] following the global regularity results of [6,13,14]. Since our result also assumes strong rotation, however, one could have used an earlier work [2] which proved global regularity under a sufficiently strong rotation and then used [25] to obtain Gevrey regularity.…”

“…As with the Navier-Stokes equations [9], in the absence of boundaries and with Gevrey forcing, one can prove that the strong solution of the primitive equations also has Gevrey regularity [25]. For the present article, we need uniform bounds on the norms, which have been proved recently [23] following the global regularity results of [6,13,14].…”

Slow Manifolds and Invariant Sets of the Primitive Equations

“…In analogy with (4), one can then prove that for t sufficiently large, E 1 (U(t)) is bounded independently of the initial conditions. Furthermore, assuming that the forcing (but not necessarily the initial conditions) is sufficiently smooth, one can prove (Petcu and Wirosoetisno 2005) that all derivatives of the solution are similarly bounded: with…”

“…Assuming that the forcing S is Gevrey, one can prove following Foias and Temam (1989) that the solution (y, b) will also be Gevrey (Petcu and Wirosoetisno 2005), in the sense that E 1 s (U(t)) is bounded uniformly for all t $ 0, and independently of the initial conditions 2 But their bound on E 1 (U(t)) , ' as t / '; all that is needed for regularity is that E 1 (U(t)) does not blow up at finite t.…”