scholarly journals A Sufficient Condition for the Kolmogorov 4/5 Law for Stationary Martingale Solutions to the 3D Navier–Stokes Equations

2019 ◽  
Vol 367 (3) ◽  
pp. 1045-1075 ◽  
Author(s):  
Jacob Bedrossian ◽  
Michele Coti Zelati ◽  
Samuel Punshon-Smith ◽  
Franziska Weber
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Sufficient Condition ◽  
Navier Stokes Equations ◽  
Martingale Solutions

scholarly journals On the zeroth law of turbulence for the stochastically forced Navier-Stokes equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yat Tin Chow ◽  
Ali Pakzad
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Energy Dissipation Rate ◽  
Mean Value ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Additional Hypothesis ◽  
The Mean ◽  
Deterministic Force ◽  
Martingale Solutions

<p style='text-indent:20px;'>We consider the three-dimensional stochastically forced Navier–Stokes equations subjected to white-in-time (colored-in-space) forcing in the absence of boundaries. Upper bounds of the mean value of the time-averaged energy dissipation rate are derived directly from the equations for weak (martingale) solutions. This estimate is consistent with the Kolmogorov dissipation law. Moreover, an additional hypothesis of energy balance implies the zeroth law of turbulence in the absence of a deterministic force.</p>

scholarly journals On weak martingale solutions to a stochastic Allen-Cahn-Navier-Stokes model with inertial effects

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
T. Tachim Medjo
Keyword(s):  
Representation Theorem ◽  
Multiplicative Noise ◽  
Stokes Equations ◽  
Galerkin Approximation ◽  
Navier Stokes ◽  
Inertial Effects ◽  
Navier Stokes Equations ◽  
Martingale Solution ◽  
Skorokhod Representation Theorem ◽  
Martingale Solutions

<p style='text-indent:20px;'>We consider a stochastic Allen-Cahn-Navier-Stokes equations with inertial effects in a bounded domain <inline-formula><tex-math id="M1">\begin{document}$ D\subset\mathbb{R}^{d} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ d = 2, 3 $\end{document}</tex-math></inline-formula>, driven by a multiplicative noise. The existence of a global weak martingale solution is proved under non-Lipschitz assumptions on the coefficients. The construction of the solution is based on the Faedo-Galerkin approximation, compactness method and the Skorokhod representation theorem.</p>

scholarly journals Hyperviscous stochastic Navier–Stokes equations with white noise invariant measure

2020 ◽  
Vol 20 (06) ◽  
pp. 2040005
Author(s):  
M. Gubinelli ◽  
M. Turra
Keyword(s):  
Invariant Measure ◽  
Existence And Uniqueness ◽  
Stokes Equations ◽  
Initial Conditions ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Absolutely Continuous ◽  
Navier Stokes Equations ◽  
Whole Space ◽  
Martingale Solutions

We prove existence and uniqueness of martingale solutions to a (slightly) hyper-viscous stochastic Navier–Stokes equation in 2d with initial conditions absolutely continuous with respect to the Gibbs measure associated to the energy, getting the results both in the torus and in the whole space setting.

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