11 research outputs found
Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations
In this paper we study the following non-autonomous stochastic evolution
equation on a UMD Banach space with type 2,
{equation}\label{eq:SEab}\tag{SE} {{aligned} dU(t) & = (A(t)U(t) + F(t,U(t)))
dt + B(t,U(t)) dW_H(t), \quad t\in [0,T],
U(0) & = u_0. {aligned}. {equation}
Here are unbounded operators with domains
which may be time dependent. We assume that
satisfies the conditions of Acquistapace and Terreni. The
functions and are nonlinear functions defined on certain interpolation
spaces and is the initial value. is a cylindrical Brownian
motion on a separable Hilbert space .
Under Lipschitz and linear growth conditions we show that there exists a
unique mild solution of \eqref{eq:SEab}. Under assumptions on the interpolation
spaces we extend the factorization method of Da Prato, Kwapie\'n, and Zabczyk,
to obtain space-time regularity results for the solution of
\eqref{eq:SEab}. For Hilbert spaces we obtain a maximal regularity result.
The results improve several previous results from the literature.
The theory is applied to a second order stochastic partial differential
equation which has been studied by Sanz-Sol\'e and Vuillermot. This leads to
several improvements of their result.Comment: Accepted for publication in Journal of Evolution Equation
Maximal regularity for non-autonomous equations with measurable dependence on time
In this paper we study maximal -regularity for evolution equations with
time-dependent operators . We merely assume a measurable dependence on time.
In the first part of the paper we present a new sufficient condition for the
-boundedness of a class of vector-valued singular integrals which does not
rely on H\"ormander conditions in the time variable. This is then used to
develop an abstract operator-theoretic approach to maximal regularity.
The results are applied to the case of -th order elliptic operators
with time and space-dependent coefficients. Here the highest order coefficients
are assumed to be measurable in time and continuous in the space variables.
This results in an -theory for such equations for .
In the final section we extend a well-posedness result for quasilinear
equations to the time-dependent setting. Here we give an example of a nonlinear
parabolic PDE to which the result can be applied.Comment: Application to a quasilinear equation added. Accepted for publication
in Potential Analysi