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 $\{ M_t \}_{t \ge 0}$ 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 $\mathbf{d}_B$.

• $\{ M_t \}_{t \ge 0}$ possesses a.a. semigroup property,

  $$\begin{aligned} M_{t + s}(\nu ) = M_t \circ M_s(\nu ) \ \text{ for } \text{ any }\ t \ge 0, \ \text{ a.a. }\ s \ge 0, \ \text{ and } \text{ any }\ \nu \in \mathfrak {P}[\mathcal {D}]. \end{aligned}$$

• $\{ M_t \}_{t \ge 0}$ is deterministic when restricted to deterministic data, specifically

  $$\begin{aligned} M_t( \delta _{[\varrho _0, \mathbf{m}_0, \mathbf{d}_B]}) = \delta _{[\varrho (t, \cdot ), \mathbf{m}(t, \cdot ), \mathbf{d}_B]},\ t \ge 0, \end{aligned}$$

  where $[\varrho , \mathbf{m}]$ is a finite energy weak solution of the Navier–Stokes system corresponding to the data $[\varrho _0, \mathbf{m}_0, \mathbf{d}_B] \in \mathcal {D}$.

• $M_t: \mathfrak {P}[\mathcal {D}] \rightarrow \mathfrak {P}[\mathcal {D}]$ is continuous in a suitable Bregman–Wasserstein metric at measures supported by the data giving rise to regular solutions.