Evolutionary problems in noncylindrical domains

Partial differential equations (PDEs) encode physical laws. Thus, they are often used to describe natural phenomena mathematically. We are particularly interested in the evolution of physical quantities or objects over time. Examples include heat progression in a conductor, flow of water through a pipe, propagation of waves in the ocean and formation of color patterns in developing animal embryos. Even though the heat, the water flow or the water wave evolves along time, the underlying physical environment for them (the conductor, the pipe or the seabed) are often fixed. However, depending on the situation at hand, the environment evolves as well. For instance, the seabed rises and falls during an earthquake; the tissue in an embryo grows. These considerations lead to more complicated mathematical formulations, that is, evolutionary PDEs in time-dependent (also called noncylindrical) domains. Moreover, some physical laws behind these phenomena are nonlinear and therefore give rise to nonlinear PDEs. Our understanding of these problems is at the inception, and their solutions are challenging. This project aims to lay mathematical foundations for some of these physical phenomena. Our proposed program consists of two steps. First, we aim to establish that the mathematical formulations are well-posed. In this context, the first question is whether the investigated models admit a solution. Given some mild physical conditions from the outset, the goal is to construct solutions. Furthermore, such solutions should be uniquely determined by the given initial and boundary values. In this step, we will investigate how the evolution of domains affects the well-posedness of the problem. This will illustrate under which physical framework the mathematical model is a sufficient description of the phenomenon. In the second step of the project, the constructed solutions fine properties will be studied. Experimentally, physical objects in our models should not change dramatically along any small spatial or temporal variation. However, the sudden growth or collapse of the environment might lead to a breach of this code. Therefore, we endeavor to understand which impact the evolution of domains will have on the regularity of solutions. This step not only deepens our mathematical understanding and further confirms the framework, but also could suggest undiscovered physics.