Martingale solutions to stochastic nonlocal Cahn–Hilliard–Navier–Stokes equations with multiplicative noise of jump type

2019 ◽  
Vol 398 ◽  
pp. 23-68 ◽  
Author(s):  
G. Deugoué ◽  
A. Ndongmo Ngana ◽  
T. Tachim Medjo
Keyword(s):  
Multiplicative Noise ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Martingale Solutions

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>

Navier-Stokes equations with multiplicative noise

Nonlinearity ◽  
1993 ◽  
Vol 6 (1) ◽  
pp. 71-78 ◽  
Author(s):  
M Capinski ◽  
N J Cutland
Keyword(s):  
Multiplicative Noise ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations

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>

The exponential behavior and stabilizability of the stochastic 3D Navier–Stokes equations with damping

2019 ◽  
Vol 31 (07) ◽  
pp. 1950023 ◽  
Author(s):  
Hui Liu ◽  
Lin Lin ◽  
Chengfeng Sun ◽  
Qingkun Xiao
Keyword(s):  
Weak Solutions ◽  
Multiplicative Noise ◽  
Stokes Equations ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
Exponential Behavior ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
The Mean ◽  
The Stability

The stochastic 3D Navier–Stokes equation with damping driven by a multiplicative noise is considered in this paper. The stability of weak solutions to the stochastic 3D Navier–Stokes equations with damping is proved for any [Formula: see text] with any [Formula: see text] and [Formula: see text] as [Formula: see text]. The weak solutions converge exponentially in the mean square and almost surely exponentially to the stationary solutions are proved for any [Formula: see text] with any [Formula: see text] and [Formula: see text] as [Formula: see text]. The stabilization of these equations is obtained for any [Formula: see text] with any [Formula: see text] and [Formula: see text] as [Formula: see text].

ATTRACTORS AND NEO-ATTRACTORS FOR 3D STOCHASTIC NAVIER–STOKES EQUATIONS

2005 ◽  
Vol 05 (04) ◽  
pp. 487-533 ◽  
Author(s):  
NIGEL J. CUTLAND ◽  
H. JEROME KEISLER
Keyword(s):  
Global Attractor ◽  
Ad Hoc ◽  
Multiplicative Noise ◽  
Stokes Equations ◽  
Nonstandard Analysis ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Loeb Space

In [14] nonstandard analysis was used to construct a (standard) global attractor for the 3D stochastic Navier–Stokes equations with general multiplicative noise, living on a Loeb space, using Sell's approach [26]. The attractor had somewhat ad hoc attracting and compactness properties. We strengthen this result by showing that the attractor has stronger properties making it a neo-attractor — a notion introduced here that arises naturally from the Keisler–Fajardo theory of neometric spaces [18]. To set this result in context we first survey the use of Loeb space and nonstandard techniques in the study of attractors, with special emphasis on results obtained for the Navier–Stokes equations both deterministic and stochastic, showing that such methods are well-suited to this enterprise.

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