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 $$\{ M_t \}_{t \ge 0}$$ { M t } t ≥ 0 of Markov operators on the set of probability measures $$\mathfrak {P}[\mathcal {D}]$$ P [ D ] on the data space $$\mathcal {D}$$ D containing the initial data $$[\varrho _0, \mathbf{m}_0]$$ [ ϱ 0 , m 0 ] and the boundary data $$\mathbf{d}_B$$ d B . $$\{ M_t \}_{t \ge 0}$$ { M t } t ≥ 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 + s ( ν ) = M t ∘ M s ( ν ) for any t ≥ 0 , a.a. s ≥ 0 , and any ν ∈ P [ D ] . $$\{ M_t \}_{t \ge 0}$$ { M t } t ≥ 0 is deterministic when restricted to deterministic data, specifically $$\begin{aligned} M_t( \delta _{[\varrho _0, \mathbf{m}_0, \mathbf{d}_B]}) = \delta _{[\varrho (t, \cdot ), \mathbf{m}(t, \cdot ), \mathbf{d}_B]},\ t \ge 0, \end{aligned}$$ M t ( δ [ ϱ 0 , m 0 , d B ] ) = δ [ ϱ ( t , · ) , m ( t , · ) , d B ] , t ≥ 0 , where $$[\varrho , \mathbf{m}]$$ [ ϱ , m ] is a finite energy weak solution of the Navier–Stokes system corresponding to the data $$[\varrho _0, \mathbf{m}_0, \mathbf{d}_B] \in \mathcal {D}$$ [ ϱ 0 , m 0 , d B ] ∈ D . $$M_t: \mathfrak {P}[\mathcal {D}] \rightarrow \mathfrak {P}[\mathcal {D}]$$ M t : P [ D ] → P [ D ] is continuous in a suitable Bregman–Wasserstein metric at measures supported by the data giving rise to regular solutions.

中文翻译：

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Barotropic Navier-Stokes 系统的统计解

我们在具有非均匀边界条件的正压 Navier-Stokes 系统的弱解框架中引入了统计解的新概念。统计解是一组 $$\{ M_t \}_{t \ge 0}$$ { M t } t ≥ 0 的概率测度集上的马尔可夫算子 $$\mathfrak {P}[\mathcal {D} ]$$ P [ D ] 数据空间 $$\mathcal {D}$$ D 包含初始数据 $$[\varrho _0, \mathbf{m}_0]$$ [ ϱ 0 , m 0 ] 和边界数据 $$\mathbf{d}_B$$ d B 。$$\{ M_t \}_{t \ge 0}$$ { M t } t ≥ 0 具有半群性质， $$\begin{aligned} M_{t + s}(\nu ) = M_t \circ M_s (\nu ) \ \text{ for } \text{ any }\ t \ge 0, \ \text{ aa }\ s \ge 0, \ \text{ and } \text{ any }\ \nu \in \ mathfrak {P}[\mathcal {D}]。\end{aligned}$$ M t + s ( ν ) = M t ∘ M s ( ν ) 对于任何 t ≥ 0 ，aa s ≥ 0 ，以及任何 ν ∈ P [ D ] 。$$\{ M_t \}_{t \ge 0}$$ { M t } t ≥ 0 当限制为确定性数据时是确定性的，特别是 $$\begin{aligned} M_t( \delta _{[\varrho _0, \mathbf{m}_0, \mathbf{d}_B]}) = \delta _{[\varrho (t, \cdot ), \mathbf{m}(t, \cdot ), \mathbf{d}_B] },\ t \ge 0, \end{aligned}$$ M t ( δ [ ϱ 0 , m 0 , d B ] ) = δ [ ϱ ( t , · ) , m ( t , · ) , d B ] , t ≥ 0 , 其中 $$[\varrho , \mathbf{m}]$$ [ ϱ , m ] 是 Navier–Stokes 系统的有限能量弱解，对应于数据 ​​$$[\varrho _0, \mathbf {m}_0, \mathbf{d}_B] \in \mathcal {D}$$ [ ϱ 0 , m 0 , d B ] ∈ D 。$$M_t: \mathfrak {P}[\mathcal {D}] \rightarrow \mathfrak {P}[\mathcal {D}]$$ M t : P [ D ] → P [ D ] 在合适的 Bregman 中是连续的–Wasserstein 度量在数据支持的度量中产生常规解决方案。