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 ${\mathbf{d}}_B$. (1) $\{M_t{\}}_{t\ge 0}$ possesses a.a. semigroup property, $M_{t+s}(\nu )=M_t\circ M_s(\nu )$ for any $t\ge 0$, a.a. $s\ge 0$, and any $\nu \in \mathfrak{P}[\mathcal{D}]$. (2) $\{M_t{\}}_{t\ge 0}$ 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.

Submitted

References in corpus (2)

• Convergence of time averages of weak solutions of the three-dimensional Navier-Stokes equations
• Semiflow selection for the compressible Navier-Stokes system

Cited by in corpus (4)

• Ergodic theory for energetically open compressible fluid flows
• Probabilistic Descriptions of Fluid Flow: A Survey
• Compressible fluid motion with uncertain data
• Randomness in compressible fluid flows past an obstacle