Study of Homogenization and Optimal Control Problems for Stokes’ system in Rough Domains

Abstract

The mathematical theory of partial differential equations (PDEs) represents a long-established classical domain, holding relevance across diverse scientific and engineering disciplines. Over the previous century, as functional analysis and operator theory advanced, PDEs underwent thorough analysis. One of the more recent areas of study is the theory of homogenization (limiting or asymptotic analysis), which illuminates multi-scale phenomena present in various physical and engineering scenarios. This developing field is applicable in various domains, encompassing composite materials, porous media, rapidly oscillating boundaries, thin structures, and more. Consequently, it has attracted significant attention as both a theoretical pursuit and an area of practical utility over the last few decades. This thesis investigates homogenization and optimal control problems (OCPs) associated with the generalized stationary Stokes equations, featuring a second-order elliptic linear differential operator in divergence form instead of the classical Laplacian operator. We formulate and analyze the homogenization problems and OCPs over rough (oscillating) domains, specifically domains characterized by rapidly oscillating boundaries (comb-shaped) and domains with perforations. Furthermore, our primary focus is on analyzing the limiting analysis of the distributive OCPs. The present thesis comprises six chapters. Chapter 1 briefly introduces homogenization and OCPs, along with relevant literature, preliminaries, and a summary of the thesis. Chapter 6 encompasses the conclusion and outlines future plans. Our primary contribution lies within Chapters 2-5. In Chapter 2, we study the homogenization of the generalized stationary Stokes equations involving the unidirectional oscillating coefficient matrix posed in a two-dimensional domain with highly oscillating boundaries. We subject a segment of the oscillating boundary with the Robin boundary condition having non-negative real parameters, while its remaining portion is subject to Neumann boundary data. We derive the homogenized problem, which depends on these non-negative real parameters. Finally, we show the convergence of state and pressure within an appropriate space to those of the limit system in a fixed domain and observe a corrector-type result under the special case of stationary Stokes equations with Neumann boundary conditions throughout the highly oscillating boundaries. Chapters 3 and 4 introduce distributive OCPs governed by the stationary Stokes equations in the same two-dimensional rough domain featuring rapidly oscillating boundaries. Specifically, in Chapter 3, we address minimizing the L2−cost with distributive controls applied in the oscillating part of the domain constrained by the stationary Stokes equations. Furthermore, these controls are periodic along the direction of the periodicity of the domain. By utilizing the unfolding operator technique, we characterize the optimal controls. Ultimately, we establish the convergence results for the optimal control, state, and pressure in an appropriate space to those of the limit system in a f ixed domain. Whereas Chapter 4 considers the homogenization of a distributive OCP subjected to the more generalized stationary Stokes equation involving unidirectional oscillating coefficients. The cost functional considered is of the Dirichlet type involving a unidirectional oscillating coefficient matrix. We characterize the optimal control and study the homogenization of this OCP with the aid of the unfolding operator. Due to oscillating matrices in the governing Stokes equations and the cost functional, one obtains the limit OCP involving a perturbed tensor in the convergence analysis. Next, in Chapter 5, we study the asymptotic analysis of the OCP constrained by the generalized stationary Stokes equations over the n-dimensional (n ≥ 2) perforated domain. We implement distributive controls in the interior region of the domain. The considered Stokes operator involves an n-directional oscillating coefficient matrix for the state equations. We provide a characterization of the optimal control and by employing the method of periodic unfolding, we establish the convergence of the solutions of the considered OCP to those of the limit OCP governed by stationary Stokes equations over a non-perforated domain. Additionally, we demonstrate the convergence of the cost functional, a result not observed in Chapters 3 and 4.