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October  2014, 34(10): 4323-4341. doi: 10.3934/dcds.2014.34.4323

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

 1 Division of Mathematical Sciences, Graduate School of Comparative Culture, Kurume University, Miimachi, Kurume, Fukuoka 839-8502, Japan

Received  December 2012 Revised  March 2013 Published  April 2014

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.
Citation: Takeshi Taniguchi. The existence and decay estimates of the solutions to $3$D stochastic Navier-Stokes equations with additive noise in an exterior domain. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4323-4341. doi: 10.3934/dcds.2014.34.4323
##### References:
 [1] M. Capinski and D. Gatarek, Stochastic equations in Hilbert space with application to Navier-Stokes equations in any dimension, J. Functional Analysis, 126 (1994), 26-35. doi: 10.1006/jfan.1994.1140. [2] T. Caraballo, J. Langa and T. Taniguchi, The exponential behavior and stabilizability of stochastic 2D-Navier-Stokes equations, J. Diff. Equations, 179 (2002), 714-737. doi: 10.1006/jdeq.2001.4037. [3] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992. doi: 10.1017/CBO9780511666223. [4] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, 1996. doi: 10.1017/CBO9780511662829. [5] F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Naver-Stokes equations, Proba. Theory and Related Fields, 102 (1995), 367-391. doi: 10.1007/BF01192467. [6] H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269-315. doi: 10.1007/BF00276188. [7] Y. Giga and T. Miyakawa, Solution in $L_r$ of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267-281. doi: 10.1007/BF00276875. [8] T. Kato and H. Fujita, On the nonstationary Navier-Stokes system, Rend. Semi. Math. Univ. Padova, 32 (1962), 243-260. [9] T. Kato, Strong $L^p-$solutions of the Navier-Stokes equation in $R^n$, with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182. [10] T. Miyakawa, On nonstationary solution of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140. [11] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983. doi: 10.1007/978-1-4612-5561-1. [12] J. Seidler and T. Sobukawa, Exponential integrability of stochastic convolutions, J. London Math. Soc., 67 (2003), 245-258. doi: 10.1112/S0024610702003745. [13] H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analysis Approach, Birkhäuser, Boston, 2001. doi: 10.1007/978-3-0348-8255-2. [14] S. Sritharan and P. Sundar, Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise, Stochastic Processes and Applications, 116 (2006), 1636-1659. doi: 10.1016/j.spa.2006.04.001. [15] T. Taniguchi, The exponential behaviour of Navier-Stokes equations with time delay external force, Discrete Continuous Dynamical Systems-A, 12 (2005), 997-1018. doi: 10.3934/dcds.2005.12.997. [16] T. Taniguchi, Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces, Stochastics and Stochastics Reports, 53 (1994), 41-52. doi: 10.1080/17442509508833982. [17] T. Taniguchi, The existence of energy solutions to 2-dimensional non-Lipschitz stochastic Navier-Stokes equations in unbounded domains, J. Diff. Equations, 251 (2011), 3329-3362. doi: 10.1016/j.jde.2011.07.029. [18] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, 2nd edition, ().  doi: 10.1007/978-1-4684-0313-8. [19] M. Wiegner, Decay estimates for strong solutions of the Navier-Stokes equations in exterior domain, Ann. Univ. Ferrara-Sez. VII (N.S.), 46 (2000), 61-79.

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##### References:
 [1] M. Capinski and D. Gatarek, Stochastic equations in Hilbert space with application to Navier-Stokes equations in any dimension, J. Functional Analysis, 126 (1994), 26-35. doi: 10.1006/jfan.1994.1140. [2] T. Caraballo, J. Langa and T. Taniguchi, The exponential behavior and stabilizability of stochastic 2D-Navier-Stokes equations, J. Diff. Equations, 179 (2002), 714-737. doi: 10.1006/jdeq.2001.4037. [3] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992. doi: 10.1017/CBO9780511666223. [4] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, 1996. doi: 10.1017/CBO9780511662829. [5] F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Naver-Stokes equations, Proba. Theory and Related Fields, 102 (1995), 367-391. doi: 10.1007/BF01192467. [6] H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269-315. doi: 10.1007/BF00276188. [7] Y. Giga and T. Miyakawa, Solution in $L_r$ of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267-281. doi: 10.1007/BF00276875. [8] T. Kato and H. Fujita, On the nonstationary Navier-Stokes system, Rend. Semi. Math. Univ. Padova, 32 (1962), 243-260. [9] T. Kato, Strong $L^p-$solutions of the Navier-Stokes equation in $R^n$, with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182. [10] T. Miyakawa, On nonstationary solution of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140. [11] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983. doi: 10.1007/978-1-4612-5561-1. [12] J. Seidler and T. Sobukawa, Exponential integrability of stochastic convolutions, J. London Math. Soc., 67 (2003), 245-258. doi: 10.1112/S0024610702003745. [13] H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analysis Approach, Birkhäuser, Boston, 2001. doi: 10.1007/978-3-0348-8255-2. [14] S. Sritharan and P. Sundar, Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise, Stochastic Processes and Applications, 116 (2006), 1636-1659. doi: 10.1016/j.spa.2006.04.001. [15] T. Taniguchi, The exponential behaviour of Navier-Stokes equations with time delay external force, Discrete Continuous Dynamical Systems-A, 12 (2005), 997-1018. doi: 10.3934/dcds.2005.12.997. [16] T. Taniguchi, Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces, Stochastics and Stochastics Reports, 53 (1994), 41-52. doi: 10.1080/17442509508833982. [17] T. Taniguchi, The existence of energy solutions to 2-dimensional non-Lipschitz stochastic Navier-Stokes equations in unbounded domains, J. Diff. Equations, 251 (2011), 3329-3362. doi: 10.1016/j.jde.2011.07.029. [18] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, 2nd edition, ().  doi: 10.1007/978-1-4684-0313-8. [19] M. Wiegner, Decay estimates for strong solutions of the Navier-Stokes equations in exterior domain, Ann. Univ. Ferrara-Sez. VII (N.S.), 46 (2000), 61-79.
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