Mathematical analysis of fluid flows: the challenge of randomness

ongoing

The main goal of the present project is to make substantial contributions to the understanding of fundamental problems in the mathematical theory of fluid flows. This theory is formulated in terms of systems of nonlinear partial differential equations (PDEs). Major attention has been paid to the iconic example, the Navier-Stokes system for incompressible fluids, and the corresponding Millennium Problem. Despite joint efforts and a substantial progress for various models in fluid dynamics, fundamental questions concerning existence and uniqueness of solutions as well as long time behavior remain unsolved. This project is based on the conviction that a probabilistic description is indispensable in modeling of fluid flows to capture the chaotic behavior of deterministic systems after blow-up, and to describe model uncertainties due to high sensitivity to input data or parameter reduction. For a set of selected models, we investigate different aspects of the underlying deterministic and stochastic PDE dynamics. In particular, we are concerned with the question of solvability and well-posedness or alternatively ill-posedness. For some models including the incompressible stochastic Navier-Stokes system we investigate non-uniqueness in law. For the compressible counterpart we aim to prove existence of a unique ergodic invariant measure. The guiding theme of this research program is a core question in the field, namely, how to select physically relevant solutions to PDEs in fluid dynamics. The project lies at the challenging frontiers of PDE theory and probability theory and it will tackle several long standing open problems. The results will have an impact in the deterministic PDE theory, stochastic partial differential equations and from a wider perspective also in mathematical physics.