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

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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

Takeshi Taniguchi ^1,

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

Abstract
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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: strong solutions, Stochastic Navier-Stokes equations, exterior domain, additive noise, asymptotic behavior of solutions..

Mathematics Subject Classification: Primary: 76D05; Secondary: 76M30, 60H1.

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, Springer, New York. 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.

show all references

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, Springer, New York. 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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