Compressible Euler equations with transport noise

We consider the isentropic compressible Euler equations in multi-dimensions with stochastic perturbation of transport type. On the one hand, this is motivated by the physical modelling in turbulence theory. On the other hand, it has been shown recently that this type of noise can have regularising effects.

We aim at a rather complete picture concerning the well-posedness (and ill-posedness) of the underlying system of stochastic PDEs. As a first step, we plan to prove existence of dissipative measure-valued martingale solutions (where the probability space is not a priori given but part of the problem). Eventually, we will analysis some of their properties such as weak-strong uniqueness, their long-time behaviour and the existence of Markov selections.

A second working package is concerned with the existence of weak solutions (where the nonlinearities are described by functions rather than measures). Based on the method of convex integration we hope to prove the existence of infinitely many analytically weak solutions for any initial datum. These solutions will be strong in the probabilistic sense, i.e., they exist on a given stochastic basis.

Finally, we will analysis whether a transport noise can delay the otherwise inevitable blow-up of solutions to the compressible Euler equations.