Research interests: Nonlinear partial differential equations, thermistor problem, existence, regularity, and large-time behavior of solutions.
My current research is about analytic validation of various types of partial differential equation models. For it to be valid, a mathematical model must have a physically meaningful solution, and we concern ourselves with the existence of such a solution. The existence assertion usually involves three steps:
- 1. a prior estimates;
- 2. approximate scheme;
- 3. convergence.
In the first step, one assumes that the model under consideration has a “nice” solution. One proceeds to derive quantitative properties this solution enjoys. The second step is devoted to the construction of approximate solutions. These solutions must be sufficiently smooth, yet they still satisfy the same a priori estimates obtained in the first step. Various versions of regularization, discretization, or penalty are often employed. In the last step one shows that the sequence of approximate solutions has a limit and this limit is a solution to the original model. Here one often faces the question of how to improve weak convergence to strong convergence. Some types of compactness arguments must be developed.
Curriculum Vitae: Dr. Xu's CV