Mathematical Analysis Seminars @ DISIM

Title: Well-posedness and invariant measures for the 1D stochastic Navier-Stokes-Korteweg equations

Abstract: We investigate the well-posedness and the long-time behaviour of a stochastic model describing a one-dimensional compressible viscous fluid with capillarity. In particular, we consider the stochastically forced Navier-Stokes-Korteweg equations with density-dependent viscosity and general capillarity coefficient satisfying typical power-law relations. Depending on the ranges of the viscosity and capillarity exponents we establish global existence and uniqueness of strong pathwise solutions, as well as global existence of weak martingale solutions. The analysis is based on BD entropy methods together with multi-layer approximation systems and stochastic compactness arguments.

In the second part of the talk, we focus on the long-time behaviour and we prove the existence of invariant measures for the Markov semigroup associated with strong pathwise solutions. The analysis of ergodic properties for compressible fluid equations presents several structural obstacles which are related to the absence of classical regularity frameworks typically used in the incompressible setting. This talk provides an overview of these challenges and discusses recent strategies to overcome them. This is based on joint works with Donatella Donatelli and Stefano Spirito (University of L’Aquila).