- The student is aware of some important examples of nonlinear wave equations arising in mathematical physics.
- The student understands how to prove the existence and uniqueness of solutions to the initial value problem for general linear wave equations.
- The student is familiar with the local existence and uniqueness theory for semilinear wave equations.
- The student is aware of some global existence and uniqueness results for semilinear and quasilinear wave equations.
- The student understands how to work with energy estimates and the vector field method.
- The students is comfortable working with Sobolev spaces, Sobolev inequalities, weak solutions etc. .
Nonlinear wave equations form an important class of second-order partial differential equations and are abundant in science and engineering. In the context of Einstein's equations of general relativity, they play a fundamental role in determining the dynamical and geometric properties of space and time. They are also important for understanding nonlinear physical phenomena that arise in electromagnetism, fluid dynamics and acoustics. In this course, we will formulate the initial value problem for general nonlinear wave equations and we will derive existence and uniqueness of solutions. We will introduce the basic tools for understanding both local and global aspects of solutions, which include energy estimates and the vector field method. |
Analysis 1 and 2, basic knowledge of functional analysis.
|Written exam (oral if the number participants is low) and assignments.|