Statistical Solutions to the Barotropic Navier–Stokes System

Abstract

We introduce a new concept of statistical solution in the framework of weak solutions to the barotropic Navier--Stokes system with inhomogeneous boundary conditions. Statistical solution is a family $\{ M_t \}_{t \geq 0}$ of Markov operators on the set of probability measures $\mathfrak{P}[\mathcal{D}]$ on the data space $\mathcal{D}$ containing the initial data $[\varrho_0, \mathbf{m}_0]$ and the boundary data $\mathbf{d}_B$. (1) $\{ M_t \}_{t \geq 0}$ possesses a.a. semigroup property, $ M_{t + s}(\nu) = M_t \circ M_s(\nu)$ for any $t \geq 0$, a.a. $s \geq 0$, and any $\nu \in \mathfrak{P}[\mathcal{D}]$. (2) $\{ M_t \}_{t \geq 0}$ is deterministic when restricted to deterministic data, specifically $ M_t(\delta_{[\varrho_0, \mathbf{m}_0, \mathbf{d}_B]}) = \delta_{[\varrho(t, \cdot), \mathbf{m}(t, \cdot), \mathbf{d}_B]}, $ where $[\varrho, \mathbf{m}]$ is a finite energy weak solution of the Navier--Stokes system corresponding to the data $[\varrho_0, \mathbf{m}_0, \mathbf{d}_B] \in \mathcal{D}$. (3) $M_t: \mathfrak{P}[\mathcal{D}] \to \mathfrak{P}[\mathcal{D}]$ is continuous in a suitable Bregman--Wasserstein metric at measures supported by the data giving rise to regular solutions.