RMU Seminar Solution Concepts in PDEs

Let us consider the Euler equations modeling the behavior of an incompressible, homogeneous, inviscid fluid. In the two-dimensional case, the Euler equations can be written in vorticity form as a continuity equation, in which the advecting velocity depends on the vorticity through an integral operator. In my talk, I will introduce several notions of weak solutions for the two-dimensional Euler equations in vorticity form: weak solutions, renormalized solutions, and vanishing-viscosity solutions. Relying on the linear theory for continuity equations with Sobolev velocity field by DiPerna and Lions, I will show that in the subcritical case weak solutions do not exhibit anomalies. In the supercritical case, I will show by means of a duality approach that the same holds for vanishing-viscosity solutions. This has some connections with the two-dimensional theory of turbulence of Kraichnan and Batchelor. If times allows, I will also comment on a stochastic approach which provides a convergence rate in the vanishing-viscosity limit.

The standard F-KPP (Fisher- Kolmogorov-Petrovsky-Piscounov) equation is a classical reaction-diffusion equation admitting so-called travelling wave solutions. In this talk, we will explain how certain interacting particle systems may be used to describe solutions of (inhomogeneous) F-KPP equations. We then use this description to obtain estimates on the transition front of such solutions.

Moreover, we will see how the Feynman-Kac formula (a tool from stochastic analysis) may be used to obtain estimates on solutions of F-KPP equations.

The talk is based on ideas from collaborations with A. Bovier, A. Drewitz and P. Hanigk.