Nonlinear Analysis and Partial Differential Equations

In the first part of the course, we give an introduction to stability theory for nonlinear partial differential equations (PDE). Determining the stability of solutions is of central importance when analyzing PDE arising in applications, as it is typically the stable solutions that are observed in practice. This analysis is particularly challenging when the models are nonlinear, and often relies on the presence of symmetries and corresponding conservation laws, which are a common phenomenon in physical models. We will draw inspiration from linearization and energy methods for stability theory in finite-dimensional dynamical systems and show how these concepts can be adapted to the infinite-dimensional setting when studying nonlinear Hamiltonian PDE. We will apply the theory to several equations modeling wave propagation, such as the Klein-Gordon and Nonlinear Schrödinger equations.

In the second part of this course, we will treat the Stokes and Navier-Stokes equations and construct weak and strong solutions to the latter. This will rely on spectral methods, embedding theorems for Sobolev spaces, interpolation methods, and compactness arguments. Thus, we will prove existence and uniqueness of global weak and strong solutions to the (time-dependent) Stokes equations, existence of global weak solutions to the Navier-Stokes equations, global uniqueness for the 2D Navier-Stokes equations, and existence and uniqueness of strong solutions to the Navier-Stokes equations for short times or small initial data. A discussion on the third Millennium Prize Problem of the Clay Mathematics Institute rounds off the course.