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Stochastic Tamed Navier–Stokes Equations on $${\mathbb {R}}^3$$R3 : The Existence and the Uniqueness of Solutions and the Existence of an Invariant Measure
Journal of Mathematical Fluid Mechanics ( IF 1.3 ) Pub Date : 2020-04-15 , DOI: 10.1007/s00021-020-0480-z
Zdzisław Brzeźniak , Gaurav Dhariwal

Röckner and Zhang (Probab Theory Relat Fields 145, 211–267, 2009) proved the existence of a unique strong solution to a stochastic tamed 3D Navier–Stokes equation in the whole space and for the periodic boundary case using a result from Stroock and Varadhan (Multidimensional diffusion processes, Springer, Berlin, 1979). In the latter case, they also proved the existence of an invariant measure. In this paper, we improve their results (but for a slightly simplified system) using a self-contained approach. In particular, we generalise their result about an estimate on the \(L^4\)-norm of the solution from the torus to \({\mathbb {R}}^3\), see Lemma 5.1 and thus establish the existence of an invariant measure on \({\mathbb {R}}^3\) for a time-homogeneous damped tamed 3D Navier–Stokes equation, given by (6.1).

中文翻译:

$$ {\ mathbb {R}} ^ 3 $$ R3上的随机驯服的Navier-Stokes方程:解的存在性和唯一性以及不变测度的存在

Röckner和Zhang(Probab Theory Relat Fields 145,211–267,2009)证明了对于随机驯服的3D Navier–Stokes方程在整个空间以及周期边界情况下,使用Stroock和Varadhan的结果,存在唯一的强解(多维扩散过程,Springer,柏林,1979年)。在后一种情况下,他们还证明了不变测度的存在。在本文中,我们使用一种自包含的方法来改进他们的结果(但对于稍微简化的系统)。特别是,我们将它们的结果推广到关于从圆环到\({\ mathbb {R}} ^ 3 \)的解的\(L ^ 4 \)-范数上的估计,请参见引理5.1并建立存在性\({\ mathbb {R}} ^ 3 \)上的不变测度 对于时间均质阻尼驯服的3D Navier–Stokes方程,由(6.1)给出。
更新日期:2020-04-15
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