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Mathematical Problems of Statistical Hydromechanics
TLDR
In this article, the Navier-Stokes System with White Noise in a bounded domain is shown to have unique solvability in large-scale 3D Navier Stokes Equations for a Dense Set of Data.Abstract:
Table Contents- 1: Functional-Analytic Expansions of Solution of Evolution Equations- 2: Elements of Measure Theory- 3: Moment Theory for Small Reynolds Numbers- 4: Space-Time Statistical Solutions of the Navier-Stokes Equations for Arbitrary Reynolds Numbers- 5: The Hopf Equation- 6: Moment Theory for Arbitrary Reynolds Numbers- 7: Homogeneous Space-Time Statistical Solutions of Navier-Stokes Equations- 8: Individual Solutions with Unbounded Energy for Navier-Stokes Equations and Other Problems- 9: Analytic First Integrals and Asymptotic Behaviour as t ? ? of Fourier Coefficients of Solutions of Two-Dimensional Navier Stokes Equations- 10: Navier-Stokes System With White Noise In A Bounded Domain- 11: The Direct and Inverse Kolmogorov Equations Corresponding to a Stochastic Navier-Stokes System- 12: Homogeneous In x Solutions of the Stochastic Navier-Stokes System With White Noise- Appendix 1: Unique Solvability "In Large" of the Three-Dimensional Navier-Stokes System and Moment Equations for a Dense Set of Data- Appendix 2: Periodic Approximations of Homogeneous Measures- Comments- Referencesread more
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Stochastic Equations in Infinite Dimensions
Giuseppe Da Prato,Jerzy Zabczyk +1 more
TL;DR: In this paper, the existence and uniqueness of nonlinear equations with additive and multiplicative noise was investigated. But the authors focused on the uniqueness of solutions and not on the properties of solutions.
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Martingale and stationary solutions for stochastic Navier-Stokes equations
Franco Flandoli,Dariusz Gatarek +1 more
TL;DR: In this paper, the authors prove the existence of martingale solutions and stationary solutions of stochastic Navier-Stokes equations under very general hypotheses on the diffusion term.
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Ergodicity of the 2-D Navier-Stokes equation under random perturbations
Franco Flandoli,Bohdan Maslowski +1 more
TL;DR: In this paper, a 2-dimensional Navier-Stokes equation perturbed by a sufficiently distributed white noise is considered and the existence of invariant measures is known from previous works.
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Stochastic Navier--Stokes Equations for Turbulent Flows
R. Mikulevicius,Boris Rozovskii +1 more
TL;DR: The existence and uniqueness of a strong local solution to the stochastic Navier--Stokes equation in W_{p}^{1}(\boldsymbol{R}^{d}),d >1,p > d is proved.
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Dynamically orthogonal field equations for continuous stochastic dynamical systems
TL;DR: In this article, a decomposition of the solution field into a mean and stochastic dynamical component is derived from the original SPDE, using nothing more than a dynamically orthogonal condition on the representation of a solution.