241042 Variational structures for dissipative PDEs (S) (SoSe 2024)

This seminar is concerned with the analysis of dissipative PDEs and in particular with their long-time behavior. In physics, dissipativity generally refers to some energy associated to a system being dissipated over time. In mathematical terms, this means that the energy E acts as a Lyapunov function of the dynamic E is non-increasing as a function of time. This structure often allows one to deduce convergence to equilibrium of the PDE in question, e.g. in case the the energy dissipation is controlled by the energy itself via a Gronwall inequality of the form d/dt E(t)< - A(E(t)), an instance of an energy-energy-dissipation (EED) principle. A prime example is the Fokker-Planck equation where such EED principles are related to (logarithmic) Sobolev inequalities.
In many cases more structure is available and the dynamic can be characterised as a steepest descent (or gradient flow) of the energy in a suitable functional space. A particularly interesting structure in the case of the Fokker-Planck equation and many other PDEs is given by the geometric on the space of probability measures induced by the Wasserstein distance from optimal transport.
A refined analysis also allows one to exploit energy-energy-dissipation principles for dynamics which contain dissipative effects as well as conservative Hamiltonian parts using the concept of hypocoercivity. A major example and the main motivation for this approach is the proof of the trend to equilibrium for the inhomogeneous Boltzmann equation.

In the seminar (depending on the background and interest of the participants) we will investigate energy-energy dissipation principles, the Wasserstein geometry, the concept of gradient flows in metric spaces, the phenomenon of hypocoercivity, as well as the consequences for the trend to equilibrium.

Requirements for participation, required level

Functional analysis and a course in PDE