THE $L^{p}$-APPROACH TO GLOBAL STRONG WELL-POSEDNESS OF THE PRIMITIVE EQUATIONS OF OCEAN DYNAMICS (Mathematical Analysis of Viscous Incompressible Fluid)

Abstract

In this short note we summarize recent results on the L^{p}-approach to the primitive equations. By this approach, one obtains global strong well-posedness results for the primitive equations for arbitrarly large data in D((-A_{p})^{1/p}) for 1<p<infty, where A_{p} denotes the hydrostatic Stokes operator on Ldisplaystyle frac{mathrm{p}{$sigma$}($Omega$), and $Omega$ subset mathbb{R}^{3} is a cylindrical domain subject to mixed, periodic Dirichlet and Neumann boundary conditions. The above space D((-A_{mathrm{p}})^{1/p}) may be identified by a Bessel potential space on $Omega$, satisfying certain boundary conditions. Furthermore -A_{p} admits a bounded H^{infty}-calculus on Ldisplaystyle frac{p}{$sigma$}($Omega$) for all pin(1, infty) with H^{infty}-angle 0 and in particular one obtains thus maximal L^{mathrm{q}}-L^{p}- regularity estimates for the linearized primitive equations.