MAXIMAL Lp-REGULARITY FOR STOCHASTIC EVOLUTION EQUATIONS

We prove maximal L-p-regularity for the stochastic evolution equation dU(t) + AU(t) dt = F(t, U(t)) dt + B(t, U(t)) dW(H)(t), t is an element of [0, T], U(0) = u(0), under the assumption that A is a sectorial operator with a bounded H-infinity-calculus of angle less than 1/2 pi on a space L-q(O, mu). The driving process W-H is a cylindrical Brownian motion in an abstract Hilbert space H. For p is an element of (2, infinity) and q is an element of [2, infinity) and initial conditions u(0) in the real interpolation space D-A(1 - 1/p, p) we prove existence of a unique strong solution with trajectories in L-p(0, T; D(A)) boolean AND C([0, T]; D-A(1 - 1/p, p)), provided the nonlinearities F : [0, T] x D(A) -> L-q (O, mu) and B : [0, T] x D(A) -> gamma(H, D(A 1/2)) are of linear growth and Lipschitz continuous in their second variables with small enough Lipschitz constants. Extensions to the case where A is an adapted operator-valued process are considered as well. Various applications to stochastic partial differential equations are worked out in detail. These include higher-order and time-dependent parabolic equations and the Navier-Stokes equation on a smooth bounded domain O subset of R-d with d >= 2. For the latter, the existence of a unique strong local solution with values in (H-1,H-q (O))(d) is shown.