论文标题
Barotropic Navier-Stokes系统的统计解决方案
Statistical solutions to the barotropic Navier-Stokes system
论文作者
论文摘要
我们在弱解决方案的框架中引入了一个新的统计解决方案概念，该框架的弱解决方案 - 具有不均匀边界条件的Stokes System。统计解决方案是Markov运算符的一组概率度量的统计解决方案$ \ {M_T \} _ {t \ geq 0} $，$ \ mathfrak {p} [\ Mathcal {d}] $ in Data Space $ \ \ \ \ \ m natercal {d} $包含初始数据$ nourcation $ [\ varr] $ and [\ varr varr] $ and \ varrhho__0，数据$ \ mathbf {d} _b $。 （1）$ \ {m_t \} _ {t \ geq 0} $具有A.A. semigroup属性，$ m_ {t + s}（ν）= m_t \ circ m_s（ν）$ for noth $ t \ geq 0 $，a.a. $ s \ geq 0 $，以及\ mathfrak {p} [\ mathcal {d}] $中的任何$ν\。 （2）$ \ {m_t \} _ {t \ geq 0} $在限于确定性数据时是确定性的，特别是$ m_t（δ_ {[\ varrho_0，\ varrho_0，\ mathbf {m mathbf {m} \ Mathbf {M}（T，\ CDOT），\ Mathbf {d} _b]}，$其中$ [\ varrho，\ Mathbf {M}] $是Navier-Stokes - Stokes系统的有限能量解决方案，该解决方案与数据$ [\ varrho_0，\ varrho_0，\ varhho_0，\ mathbff {\ navarho_0 \ Mathbf {d} _b] \ in \ Mathcal {d} $。 （3）$ m_t：\ Mathfrak {p} [\ Mathcal {d}] \ to \ Mathfrak {p} [\ Mathcal {d}] $在合适的Bregman-WassErstein Metric中是连续的，该指标是按数据支持的，可以通过数据为常规解决方案提供的措施。
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}(ν) = M_t \circ M_s(ν)$ for any $t \geq 0$, a.a. $s \geq 0$, and any $ν\in \mathfrak{P}[\mathcal{D}]$. (2) $\{ M_t \}_{t \geq 0}$ is deterministic when restricted to deterministic data, specifically $ M_t(δ_{[\varrho_0, \mathbf{m}_0, \mathbf{d}_B]}) = δ_{[\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.
