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

摘要

Consider the 3-d primitive equations in a layer domain Omega = G x (-h,0), G = (0,1)(2), subject to mixed Dirichlet and Neumann boundary conditions at z = -h and z = 0, respectively, and the periodic lateral boundary condition. It is shown that this equation is globally, strongly well-posed for arbitrary large data of the form a = a(1) + a(2), where a(1) is an element of C((G) over bar; L-p(-h,0)), a(2) is an element of L-infinity(G;L-p(-h,0)) for p > 3, and where a(1) is periodic in the horizontal variables and a(2) is sufficiently small. In particular, no differentiability condition on the data is assumed. The approach relies on L-H(infinity) L-z(p)(Omega)-estimates for terms of the form t(1/2)parallel to partial derivative(z)e(tA (sigma) over bar)Pf parallel to L-H(infinity) L-z(p)(Omega) <= Ce-t beta parallel to f parallel to L-H(infinity) L-z(p)(Omega) for t > 0, where e(tA (sigma) over bar) denotes the hydrostatic Stokes semigroup. The difficulty in proving estimates of this form is that the hydrostatic Helmholtz projection P fails to be bounded with respect to the L-infinity-norm. The global strong well-posedness result is then obtained by an iteration scheme, splitting the data into a smooth and a rough part and by combining a reference solution for smooth data with an evolution equation for the rough part. (C) 2020 Elsevier Inc. All rights reserved.