Date of Award
Doctor of Philosophy (PhD)
The underlying theme of this dissertation centers on the development of novel mathematical tools used in the analysis of some important types of nonlinear partial differential equations. Namely, this dissertation examines semilinear elliptic systems and hyperbolic conservation laws and inviscid techniques for their regularization. The existence theory for a class of global, semilinear elliptic systems, including Hardy-Littlewood-Sobolev and stationary Schrödinger type systems, is developed by combining the shooting method with topological degree theory. First, a `target' map is defined which aims the shooting method, then non-degeneracy conditions are established which guarantee the continuity of the map. The continuity of the target map allows for the application of tools from degree theory to show the map is onto. Then the existence result follows from this surjectivity property of the target map along with a non-existence result for the corresponding Navier boundary value problem. Next, an extension of the Hardy-Littlewood-Pólya inequality, which may be regarded as the discrete analogue of the Hardy-Littlewood-Sobolev inequality, is established along with an accurate estimate of the best constant for this inequality. The other class of problems examines a shock-regularization method for hyperbolic conservation laws that applies a spatial averaging of the nonlinear terms in the partial differential equations. A central motivation is to promote the idea of applying a recently developed filtering technique, rather than viscous regularization, as an alternative to the simulation of shocks and turbulence for inviscid flows. On the other hand, the study presented here also generalizes and unifies past mathematical and numerical analysis of the method applied to the one-dimensional Burgers' and Euler equations. This examination primarily concerns the analysis of this technique and addresses two fundamental issues. The first is the global existence and uniqueness of classical solutions for the regularization technique under the more general setting of quasilinear, symmetric hyperbolic systems in higher dimensions. The second issue examines scalar conservation laws and describes the sufficient conditions that guarantee this inviscid regularization technique captures the unique entropy or physically relevant solution of the original, non-averaged problem as filtering vanishes.
Villavert, John, "The Analysis of Some Nonlinear Partial Differential Equations" (2013). Applied Mathematics Graduate Theses & Dissertations. 33.