Non-autonomous Ornstein-Uhlenbeck equations in exterior domains

TL;DR

This paper studies non-autonomous Ornstein-Uhlenbeck equations in exterior domains, establishing evolution systems, $L^p$-estimates, and smoothing properties under certain conditions.

Contribution

It introduces a framework for analyzing non-autonomous Ornstein-Uhlenbeck operators in exterior domains with boundary conditions, including $L^p$-estimates and smoothing results.

Findings

Existence of an evolution system for the non-autonomous problem

Derivation of $L^p$-estimates for solutions and derivatives

Establishment of $L^p$-$L^q$ smoothing properties

Abstract

In this paper, we consider non-autonomous Ornstein-Uhlenbeck operators in smooth exterior domains $\Omega\subset \R^d$ subject to Dirichlet boundary conditions. Under suitable assumptions on the coefficients, the solution of the corresponding non-autonomous parabolic Cauchy problem is governed by an evolution system $\{P_\Omega(t,s)\}_{0\leq s\leq t}$ on $L^p(\Omega)$ for $1< p < \infty$. Furthermore, $L^p$-estimates for spatial derivatives and $L^p$-$L^q$ smoothing properties of $P_\Omega(t,s)$, $0\leq s\leq t$, are obtained.