Linear Parabolic Problems in Random Moving Domains

TL;DR

This paper develops a framework for analyzing linear parabolic equations on random non-cylindrical domains, using domain mapping and transformations, with applications to heat and Stokes equations.

Contribution

It introduces a general approach for parabolic problems on random moving domains, including cases with unbounded transformations, and demonstrates well-posedness for specific PDEs.

Findings

Established well-posedness for the heat equation on a random tube domain.

Analyzed the impact of different flow transformations, including log-normal types.

Highlighted the necessity of specialized transformations like Piola for divergence-free conditions.

Abstract

We consider linear parabolic equations on a random non-cylindrical domain. Utilizing the domain mapping method, we write the problem as a partial differential equation with random coefficients on a cylindrical deterministic domain. Exploiting the deterministic results concerning equations on non-cylindrical domains, we state the necessary assumptions about the velocity filed and in addition, about the flow transformation that this field generates. In this paper we consider both cases, the uniformly bounded with respect to the sample and log-normal type transformation. In addition, we give an explicit example of a log-normal type transformation and prove that it does not satisfy the uniformly bounded condition. We define a general framework for considering linear parabolic problems on random non-cylindrical domains. As the first example, we consider the heat equation on a random tube domain and prove its well-posedness. Moreover, as the other example we consider the parabolic Stokes equation which illustrates the case when it is not enough just to study the plain-back transformation of the function, but instead to consider for example the Piola type transformation, in order to keep the divergence free property.