Analysis and Homogenization of Partial Differential Equations with Discontinuous Boundary Conditions

Sato, Riuji

Etd

Analysis and Homogenization of Partial Differential Equations with Discontinuous Boundary Conditions

Public Deposited

Solutions of differential equations with discontinuous boundary conditions fail to belong in classical Sobolev spaces and hence presents a fundamental challenge in determining their effective behavior. In this dissertation, we consider three different boundary conditions for problems coming from the sciences and engineering, and use different approaches to circumvent this issue. We first study a system of parabolic PDEs in moving domains modeling mass transfer in heterogeneous catalysis with a Robin boundary condition on the interface. The behavior of such systems becomes increasingly complex as the number of catalyst particles increases, which motivates the search for a homogenized model that would describe the asymptotic behavior of the solution to the problem. We transform the moving domain problem into a problem in a fixed domain by constructing a diffeomorphism out of the known solid particle velocities. We prove that solutions exist in any finite time and show that these solutions two-scale converge to solutions of a PDE/ODE system. We further prove corrector results for the solution and show strong convergence. Finally, we provide examples of solid velocities for which our result applies. We then consider the elasticity problem for a homogeneous body with periodically distributed fractures. We first extend previous results on the dual formulations for an elastic body without fractures to a model of a homogeneous elastic body with fractures. In particular, in the framework of Legendre-Fenchel duality, ii we were able to provide three equivalent formulations for the problem where the displacement, the stress, and the strain are the unknowns respectively. We also provide a characterization of the image of the convex cone of admissible displacements under the linearized strain tensor. Finally, we prove a homogenization result using Mosco convergence. Lastly, we study the solvability of the Stokes equations in a bounded domain, describing the motion of a Newtonian fluid past moving rigid particles whose velocities are assumed to be known. We prescribe a Navier slip boundary condition on the fluid-solid interface. To solve the moving domain problem, we map the equations to a fixed domain using a diffeomorphism constructed from the solid particle velocities. The resulting equations can be thought of as a perturbation of the Stokes equations in a fixed domain. This motivates the use of a contraction mapping argument to show existence of solutions. We first construct weak solutions to the nonstationary Stokes equations in the fixed domain via Rothe’s method. We then prove the higher regularity of the solution to the stationary Stokes equations in a bounded domain with slip boundary conditions and use this to show the existence of a strong solution for the nonstationary problem for any finite time interval using fixed-point methods. We leave the homogenization of this problem for future work.

Creator