||THE HYDROSTATIC STOKES SEMIGROUP AND WELL-POSEDNESS OF THE PRIMITIVE EQUATIONS ON SPACES OF BOUNDED FUNCTIONS
Giga, Yoshikazu ,
Gries, Mathis ,
Hieber, Matthias ,
Hussein, AmruKashiwabara, Takahito
Hokkaido University Preprint Series in Mathematics
30 , 2018-02-08 , Department of Mathematics, Hokkaido University
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 = a1 + a2, where a1 ∈ C(G;Lp(−h,0)), a2 ∈ L∞(G;Lp(−h,0)) for p > 3, and where a1 is periodic in the horizontal variables and a2 is suﬃciently small. In particular, no diﬀerentiability condition on the data is assumed. The approach relies on L∞ H Lp z(Ω)-estimates for terms of the form t1/2∥∂zetAσPf∥L∞ H Lp z(Ω) ≤ Cetβ∥f∥L∞ H Lp z(Ω) for t > 0, where etAσ 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.