SIAM/ASA Journal on Uncertainty Quantification, Volume 9, Issue 2, Page 848-879, January 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.

中文翻译：

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随机运动域中的线性抛物线问题

SIAM/ASA 不确定性量化杂志，第 9 卷，第 2 期，第 848-879 页，2021 年 1 月。
我们考虑随机非圆柱域上的线性抛物线方程。利用域映射方法，我们将问题写成一个在圆柱确定域上具有随机系数的偏微分方程。我们假设给定了一个确定性初始域和一个随机速度场。利用关于非圆柱域方程的确定性结果，我们陈述了关于速度场的必要假设，此外，关于该场产生的流动变换。在本文中，我们考虑了两种情况，关于样本的一致有界和对数正态类型变换。此外，我们给出了一个对数正态类型变换的显式示例，并证明它不满足一致有界条件。我们定义了一个通用框架来考虑随机非圆柱域上的线性抛物线问题。作为第一个例子，我们考虑随机管域上的热方程并证明其适定性。此外，作为另一个例子，我们考虑抛物线斯托克斯方程，它说明了仅研究函数的平背变换是不够的情况，而是考虑例如 Piola 类型变换，以便保持无发散性质。