Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

The existence and decay estimates of the solutions to $3$D stochastic Navier-Stokes equations with additive noise in an exterior domain

Pages: 4323 - 4341, Volume 34, Issue 10, October 2014      doi:10.3934/dcds.2014.34.4323

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Takeshi Taniguchi - Division of Mathematical Sciences, Graduate School of Comparative Culture, Kurume University, Miimachi, Kurume, Fukuoka 839-8502, Japan (email)

Abstract: In this paper we consider the local existence and global existence with probability $1-\sigma $ $(0<\sigma <1)$ of pathwise solutions to the three-dimensional stochastic Navier-Stokes equation perturbed by a cylindrical Wiener processe $W(t)$ in an exteriour domain: \begin{equation*} dX(t)=[-AX(t)+B\left( X(t)\right) +f_{\ast }(t)]dt+\Phi (t)dW(t), \end{equation*} where $A=-P\Delta $ is the Stokes operator, and $f_{\ast }(t)$ and $\Phi (t)$ satisfy some conditions. We also consider the decay of pathwise solutions.

Keywords:  Stochastic Navier-Stokes equations, additive noise, strong solutions, exterior domain, asymptotic behavior of solutions.
Mathematics Subject Classification:  Primary: 76D05; Secondary: 76M30, 60H15.

Received: December 2012;      Revised: March 2013;      Available Online: April 2014.