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 \(\mathbf{d}_B\).
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\(\{ M_t \}_{t \ge 0}\) 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 \([\varrho _0, \mathbf{m}_0, \mathbf{d}_B] \in \mathcal {D}\).
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\(M_t: \mathfrak {P}[\mathcal {D}] \rightarrow \mathfrak {P}[\mathcal {D}]\) is continuous in a suitable Bregman–Wasserstein metric at measures supported by the data giving rise to regular solutions.
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Communicated by Eric A. Carlen.
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The work of F.F. has been partially supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissement d’Avenir” (ANR-11-IDEX-0007), and by the projects BORDS (ANR-16-CE40-0027-01) and SingFlows (ANR-18-CE40-0027), all operated by the French National Research Agency (ANR).
The work of E.F. was partially supported by the Czech Sciences Foundation (GAČR), Grant Agreement 18-05974S. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840.
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Fanelli, F., Feireisl, E. Statistical Solutions to the Barotropic Navier–Stokes System. J Stat Phys 181, 212–245 (2020). https://doi.org/10.1007/s10955-020-02577-1
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DOI: https://doi.org/10.1007/s10955-020-02577-1