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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

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 & Continuous Dynamical Systems - A, 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.  doi: 10.1006/jfan.1994.1140.  Google Scholar

[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.  doi: 10.1006/jdeq.2001.4037.  Google Scholar

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G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992).  doi: 10.1017/CBO9780511666223.  Google Scholar

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G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems,, Cambridge University Press, (1996).  doi: 10.1017/CBO9780511662829.  Google Scholar

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F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Naver-Stokes equations,, Proba. Theory and Related Fields, 102 (1995), 367.  doi: 10.1007/BF01192467.  Google Scholar

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H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I,, Arch. Rational Mech. Anal., 16 (1964), 269.  doi: 10.1007/BF00276188.  Google Scholar

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Y. Giga and T. Miyakawa, Solution in $L_r$ of the Navier-Stokes initial value problem,, Arch. Rational Mech. Anal., 89 (1985), 267.  doi: 10.1007/BF00276875.  Google Scholar

[8]

T. Kato and H. Fujita, On the nonstationary Navier-Stokes system,, Rend. Semi. Math. Univ. Padova, 32 (1962), 243.   Google Scholar

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T. Kato, Strong $L^p-$solutions of the Navier-Stokes equation in $R^n$, with applications to weak solutions,, Math. Z., 187 (1984), 471.  doi: 10.1007/BF01174182.  Google Scholar

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T. Miyakawa, On nonstationary solution of the Navier-Stokes equations in an exterior domain,, Hiroshima Math. J., 12 (1982), 115.   Google Scholar

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A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

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J. Seidler and T. Sobukawa, Exponential integrability of stochastic convolutions,, J. London Math. Soc., 67 (2003), 245.  doi: 10.1112/S0024610702003745.  Google Scholar

[13]

H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analysis Approach,, Birkhäuser, (2001).  doi: 10.1007/978-3-0348-8255-2.  Google Scholar

[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.  doi: 10.1016/j.spa.2006.04.001.  Google Scholar

[15]

T. Taniguchi, The exponential behaviour of Navier-Stokes equations with time delay external force,, Discrete Continuous Dynamical Systems-A, 12 (2005), 997.  doi: 10.3934/dcds.2005.12.997.  Google Scholar

[16]

T. Taniguchi, Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces,, Stochastics and Stochastics Reports, 53 (1994), 41.  doi: 10.1080/17442509508833982.  Google Scholar

[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.  doi: 10.1016/j.jde.2011.07.029.  Google Scholar

[18]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, 2nd edition, ().  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[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.   Google Scholar

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.  doi: 10.1006/jfan.1994.1140.  Google Scholar

[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.  doi: 10.1006/jdeq.2001.4037.  Google Scholar

[3]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992).  doi: 10.1017/CBO9780511666223.  Google Scholar

[4]

G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems,, Cambridge University Press, (1996).  doi: 10.1017/CBO9780511662829.  Google Scholar

[5]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Naver-Stokes equations,, Proba. Theory and Related Fields, 102 (1995), 367.  doi: 10.1007/BF01192467.  Google Scholar

[6]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I,, Arch. Rational Mech. Anal., 16 (1964), 269.  doi: 10.1007/BF00276188.  Google Scholar

[7]

Y. Giga and T. Miyakawa, Solution in $L_r$ of the Navier-Stokes initial value problem,, Arch. Rational Mech. Anal., 89 (1985), 267.  doi: 10.1007/BF00276875.  Google Scholar

[8]

T. Kato and H. Fujita, On the nonstationary Navier-Stokes system,, Rend. Semi. Math. Univ. Padova, 32 (1962), 243.   Google Scholar

[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.  doi: 10.1007/BF01174182.  Google Scholar

[10]

T. Miyakawa, On nonstationary solution of the Navier-Stokes equations in an exterior domain,, Hiroshima Math. J., 12 (1982), 115.   Google Scholar

[11]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[12]

J. Seidler and T. Sobukawa, Exponential integrability of stochastic convolutions,, J. London Math. Soc., 67 (2003), 245.  doi: 10.1112/S0024610702003745.  Google Scholar

[13]

H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analysis Approach,, Birkhäuser, (2001).  doi: 10.1007/978-3-0348-8255-2.  Google Scholar

[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.  doi: 10.1016/j.spa.2006.04.001.  Google Scholar

[15]

T. Taniguchi, The exponential behaviour of Navier-Stokes equations with time delay external force,, Discrete Continuous Dynamical Systems-A, 12 (2005), 997.  doi: 10.3934/dcds.2005.12.997.  Google Scholar

[16]

T. Taniguchi, Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces,, Stochastics and Stochastics Reports, 53 (1994), 41.  doi: 10.1080/17442509508833982.  Google Scholar

[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.  doi: 10.1016/j.jde.2011.07.029.  Google Scholar

[18]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, 2nd edition, ().  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[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.   Google Scholar

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