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)) dWH (t) ,t ∈ (0 ,T ), U (0) = u0, under the assumption that A is a sectorial operator with a bounded H ∞ -calculus of angle less than 1 π on a space L q (O ,μ ). The driv- ing process WH is a cylindrical Brownian motion in an abstract Hilbert space H.F orp ∈ (2, ∞ )a nd q ∈ (2, ∞) and initial conditions u0 in the real interpolation space DA(1 − 1 ,p )w e prove existence of a unique strong solution with trajectories in L p (0 ,T ;D(A)) ∩ C((0 ,T );DA(1 − 1 ,p )), provided the nonlinearities F :( 0 ,T ) × D(A) → Lq(O ,μ )a ndB :( 0 ,T ) × D(A) → γ(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. Vari- ous 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⊆ R d with d ≥ 2. For the latter, the existence of a unique strong local solution with values in (H 1,q (O)) d is shown.

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