American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain
  • DCDS Home
  • This Issue
  • Next Article
    Skew-product semiflows for non-autonomous partial functional differential equations with delay
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

[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

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

[1]

Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613
[2]

Kumarasamy Sakthivel, Sivaguru S. Sritharan. Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise. Evolution Equations & Control Theory, 2012, 1 (2) : 355-392. doi: 10.3934/eect.2012.1.355
[3]

Jochen Merker. Strong solutions of doubly nonlinear Navier-Stokes equations. Conference Publications, 2011, 2011 (Special) : 1052-1060. doi: 10.3934/proc.2011.2011.1052
[4]

Changjiang Zhu, Ruizhao Zi. Asymptotic behavior of solutions to 1D compressible Navier-Stokes equations with gravity and vacuum. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1263-1283. doi: 10.3934/dcds.2011.30.1263
[5]

Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073
[6]

G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123
[7]

Hamid Bellout, Jiří Neustupa, Patrick Penel. On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1353-1373. doi: 10.3934/dcds.2010.27.1353
[8]

Tomás Caraballo, Xiaoying Han. A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1079-1101. doi: 10.3934/dcdss.2015.8.1079
[9]

Yoshihiro Shibata. On the local wellposedness of free boundary problem for the Navier-Stokes equations in an exterior domain. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1681-1721. doi: 10.3934/cpaa.2018081
[10]

Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747
[11]

Manil T. Mohan, Sivaguru S. Sritharan. $\mathbb{L}^p-$solutions of the stochastic Navier-Stokes equations subject to Lévy noise with $\mathbb{L}^m(\mathbb{R}^m)$ initial data. Evolution Equations & Control Theory, 2017, 6 (3) : 409-425. doi: 10.3934/eect.2017021
[12]

Zdeněk Skalák. On the asymptotic decay of higher-order norms of the solutions to the Navier-Stokes equations in R3. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 361-370. doi: 10.3934/dcdss.2010.3.361
[13]

Tian Ma, Shouhong Wang. Asymptotic structure for solutions of the Navier--Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 189-204. doi: 10.3934/dcds.2004.11.189
[14]

Gung-Min Gie, Makram Hamouda, Roger Temam. Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary. Networks & Heterogeneous Media, 2012, 7 (4) : 741-766. doi: 10.3934/nhm.2012.7.741
[15]

Lihuai Du, Ting Zhang. Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4745-4765. doi: 10.3934/dcds.2018209
[16]

Peter E. Kloeden, José Valero. The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 161-179. doi: 10.3934/dcds.2010.28.161
[17]

Joanna Rencławowicz, Wojciech M. Zajączkowski. Global regular solutions to the Navier-Stokes equations with large flux. Conference Publications, 2011, 2011 (Special) : 1234-1243. doi: 10.3934/proc.2011.2011.1234
[18]

Peter Anthony, Sergey Zelik. Infinite-energy solutions for the Navier-Stokes equations in a strip revisited. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1361-1393. doi: 10.3934/cpaa.2014.13.1361
[19]

Tomás Caraballo, Peter E. Kloeden, José Real. Invariant measures and Statistical solutions of the globally modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 761-781. doi: 10.3934/dcdsb.2008.10.761
[20]

Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences