Existence, Asymptotic Limits And Space-time Estimates For The Solutions Of Applied Partial Differential Equations

In this thesis we consider the problems of existence, asymptotic limits and space-time estimates for the solutions to three types of partial differential equations in applied sciences. The introduction to these problems is given in chapter 1.In chapter 2, we study the asymptotic limit of the solution to an elliptic equation on a coated body. This problem arises in elasticity. To make an elastic medium withstand heavier loads, we attach it along the boundary with a very strong layer that is thin com-pared to the scale of the interior medium. We characterize the behavior of the solution in the singular limit as the thickness of the layer tends to zero. In the first section, under the assumption that the layer has uniform thickness, we further investigate the results of Brezis, Caffarelli and Friedman. In the second section, we assume that the thickness of the thin layer oscillates periodically, and we generalize the works of Buttazzo and Kohn in various respects.In chapter 3, we study the asymptotic limit of the solution to a heat equation, which is from the composite material. We investigate the thermal insulating ability of an anisotropically conducting coating which is thin compared to the scale of the interior body. We assume that the whole thermal tensor of the coating is small. We study the asymptotic behavior of the solution to the heat equation, as the thickness of the coating goes to zero. We find that after this singular limit, the limiting condition on the bound-ary of the body, which is of Dirichlet, Robin or Neumann type, depends on the scaling relations between the thermal tensor and the thickness of the coating; thus the scaling relation that leads to the Neumann condition ensures good insulation. We also obtain similar results under the assumption that the thermal tensor of the coating is only small in the directions normal to the body (a case called "optimally aligned coating").In chapter 4, we consider the problems of existence and space-time estimates for the solutions to hyperbolic equations. We first consider an equation of conservation law arising in liquid crystals. By the method of vanishing viscosity and the theory of LP Young measure, we prove the global existence of a dissipative weak solution to this hyperbolic equation with general L2 initial datum. This equation models wave motions in nematic liquid crystals. In the second section, we consider a wave equation with a potential that is time-dependent and singular. The size of the potential is exactly a function of the spatial dimension rather than being small enough in the known results. Based on a weighted L2 estimate for the solution, we derive both the local regularity and the Strichartz estimates. The existence of a solution to the wave equation is also studied.