33 research outputs found
Maximal -regularity for stochastic evolution equations
We prove maximal -regularity for the stochastic evolution equation
\{{aligned} dU(t) + A U(t)\, dt& = F(t,U(t))\,dt + B(t,U(t))\,dW_H(t),
\qquad t\in [0,T],
U(0) & = u_0, {aligned}. under the assumption that is a sectorial
operator with a bounded -calculus of angle less than on
a space . The driving process is a cylindrical
Brownian motion in an abstract Hilbert space . For and
and initial conditions in the real interpolation space
\XAp we prove existence of unique strong solution with trajectories in
L^p(0,T;\Dom(A))\cap C([0,T];\XAp), provided the non-linearities
F:[0,T]\times \Dom(A)\to L^q(\mathcal{O},\mu) and B:[0,T]\times \Dom(A) \to
\g(H,\Dom(A^{\frac12})) are of linear growth and Lipschitz continuous in their
second variables with small enough Lipschitz constants. Extensions to the case
where 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
\OO\subseteq \R^d with . For the latter, the existence of a unique
strong local solution with values in (H^{1,q}(\OO))^d is shown.Comment: Accepted for publication in SIAM Journal on Mathematical Analysi
The separability "theorem" in terms of distributions with discussion of electromagnetic scattering theory
cited By 18International audienceno abstrac