Linear Parabolic Problems in Random Moving Domains
Ana Djurdjevac · SIAM/ASA Journal on Uncertainty Quantification · 2021
We consider linear parabolic equations on a random noncylindrical domain. Utilizing the domain mapping method, we write the problem as a partial differential equation with random coefficients on a cylindrical deterministic domain. We assume we are given a deterministic initial domain and a random velocity field. Exploiting the deterministic results concerning equations on noncylindrical domains, we state the necessary assumptions about the velocity field 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.