A study on stochastic maximal regularity for rough time-dependent problems

Govindaraju P,Sasikala V,Mohamed Ali A

doi:10.26524/cm92

Abstract

We unify and extend the semigroup and the PDE approaches to stochastic maximal regularity of time-dependent semilinear parabolic problems with noise given by a cylindrical Brownian motion. We treat random coefficients that are only progressively measurable in the time variable. For 2m-th order systems with VMO regularity in space, we obtain Lp(Lq) estimates for all p > 2 and q ≥ 2, leading to optimal space-time regularity results. For second order systems with continuous coefficients in space, we also include a first order linear term, under a stochastic parabolicity condition, and obtain Lp(Lp) estimates together with optimal space-time regularity. For linear second order equations in divergence form with random coefficients that are merely measurable in both space and time, we obtain estimates in the tent spaces Tp,2 of Coifman-Meyer-Stein. This is done in the deterministic case under no extra assumption, and in the stochastic case under the assumption that the coefficients are divergence free.

Highlights

On X0 (typically X0 = Lr(O; CN ) where r ∈ [2, ∞)), we consider the following stochastic evolution equation: dU (t) + A(t)U (t)dt = F (t, U (t))dt + B(t)U (t) + G(t, U (t)) dWH(t), (1) U (0) = u0, Journal of Computational Mathematica where H is a Hilbert space, WH a cylindrical Brownian motion, A : R+ × Ω → L(X1, X0) and B : R+ × Ω → L(X1, γ(H, X 1 )) are progressively measurable, the functions F and G are suitable nonlinearities, and the initial value u0 : Ω → X0 is F0-measurable
We are interested in maximal Lp-regularity results for (1)
Remark 3.1 we do allow T = ∞ in the above definition, most result will be formulated for T ∈ (0, ∞) as this is often simpler and enough for applications to PDEs

Summary

Introduction

Where H is a Hilbert space, WH a cylindrical Brownian motion, A : R+ × Ω → L(X1, X0) (for some Banach space X1 such that X1 → X0, typically a Sobolev space) and B : R+ × Ω → L(X1, γ(H, X 1 )) are progressively measurable (and satisfy a suitable stochastic parabolic estimate), the functions F and G are suitable nonlinearities, and the initial value u0 : Ω → X0 is F0-measurable (see Chapter 2 for precise definitions). We are interested in maximal Lp-regularity results for (1). This means that we want to investigate well-posedness together with sharp Lp-regularity estimates given the data F, G and u0. Knowing these sharp regularity results for equations such as (1), gives a priori estimates to nonlinear equations involving suitable nonlinearities F (t, U (t))dt and G(t, U (t))dWH(t). Well-posedness of such non-linear equations follows from these a priori estimates

Preliminaries

Maximal Regularity for Stochastic Evolution Equations