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

Abstract

Consider the 3-d primitive equations in a layer domain Ω = G × (−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 ∈ C(G; L p (−h, 0)), a 2 ∈ L ∞ (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 L p z (Ω)-estimates for terms of the form t 1/2 ∂z e tA σ Pf L ∞for t > 0, where e tA σ 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 ∞ -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.2010 Mathematics Subject Classification. Primary: 35Q35; Secondary: 47D06, 76D03, 86A05.

Analyticity of solutions to the primitive equations

“…Furthermore, choosing allows us to enlarge the space of admissible initial values as constructed in to the above more general Besov space setting since . This class of initial values is also used in our works on rough initial data lying in for the case of mixed Dirichlet and Neumann boundary conditions. There, the solutions obtained in this article serve as reference solutions belonging to initial values in and where parameters are chosen in such a way that .…”

On operator semigroups arising in the study of incompressible viscous fluid flows

“…In this subsection, we study properties of the hydrostatic Stokes semigroup and terms of the form normal∇ etAσ¯P on spaces of bounded functions. As shown in [24], these properties yield then a global, strong well-posedness result of the primitive equations in the case of mixed Dirichlet–Neumann boundary conditions. More precisely, global, strong well-posedness of the primitive equations is proved in [24] for initial data of the form a=a1+a2,1ema1∈Cfalse(G¯;Lpfalse(−h,0false)false)1emand1ema2∈Lnormal∞false(G;Lpfalse(−h,0false)false)1emfor1emp>3, where a 1 is periodic in the horizontal variables and a 2 is sufficiently small.…”

“…As shown in [24], these properties yield then a global, strong well-posedness result of the primitive equations in the case of mixed Dirichlet–Neumann boundary conditions. More precisely, global, strong well-posedness of the primitive equations is proved in [24] for initial data of the form a=a1+a2,1ema1∈Cfalse(G¯;Lpfalse(−h,0false)false)1emand1ema2∈Lnormal∞false(G;Lpfalse(−h,0false)false)1emfor1emp>3, where a 1 is periodic in the horizontal variables and a 2 is sufficiently small. The main difficulty when dealing with the primitive equations on spaces of bounded functions is that the hydrostatic Helmholtz projection P fails to be bounded with respect to the L ∞ -norm.…”

“…The approach to the primitive equations by evolution equations is based on the hydrostatic Stokes semigroup which was introduced first in [23]. Its extension to the setting of bounded functions was investigated only recently in [24]. The iteration scheme for the nonlinear problem rests on the fact that the combination of the terms normal∇, P and etAσ¯ nevertheless gives rise to bounded operators on the associated spaces which satisfy in addition typical second-order parabolic estimates.…”

The primitive equations in the scaling-invariant space $$L^{\infty }(L^1)$$

“…In this article, we choose Neumann boundary conditions for v, i.e. ∂ ∂z v = 0, on ∂ × (0, T ), w = 0, on ∂ × (0, T ), (1.2) and mixed boundary conditions are discussed in [8].…”