The stokes equation in the L^p-setting: Well-posedness and regularity properties

This article discusses the Stokes equation in various classes of domains Ω C Rⁿ within the L^p-setting for 1 ≤ p ≤ ∞ from the point of view of evolution equations. Classical as well as modern approaches to well-posedness results for strong solutions to the Stokes equation, to the Helmholtz decomposition, to the Stokes semigroup, and to mixed maximal L^q -L^p-regularity results for 1 < p; q < ∞ are presented via the theory of R-sectorial operators. Of concern are domains having compact or noncompact, smooth or nonsmooth boundaries, as well as various classes of boundary conditions including energy preserving boundary conditions. In addition, the endpoints of the L^p-scale, i.e., p=1 and p=∞ are considered and recent well-posedness results for the case p =∞ are described. Results on L^q -L^p-smoothing properties of the associated Stokes semigroups and on variants of the Stokes equation (e.g., nonconstant viscosity, Lorentz spaces, Stokes-Oseen system, flow past rotating obstacles, hydrostatic Stokes equation) complete this survey article.