Abstract
In this paper we consider the existence of a weak solution to a 3d stochastic Navier–Stokes equation perturbed by a noise g(X(t))dW, where W(t) is a cylindrical Wiener process, in an exterior domain D:
where \({A = -P_{2}\Delta}\) is the Stokes operator and g satisfies some conditions.
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References
Brzezniak Z., Li Y.: Asymptotic compactness and absorbing sets of 2D stochastic Navier–Stokes equations on some unbounded domains. Trans. Am. Math. Soc. 358, 5587–5629 (2006)
Brzezniak Z., Motyl E.: Existence of a martingale solution to the stochastic Boussinesq equations. Glob. Stoch. 1, 175–216 (2011)
Brzezniak Z., Motyl E.: The existence of martingale solutions to the stochastic Navier–Stokes equations in unbounded 2D and 3D-domains. J. Differ. Equ. 254, 1627–1685 (2013)
Brzezniak, Z., Peszat, S.: Strong local and global solutions for stochastic Navier–Stokes equations. In: Infinite Dimensional Stochastic Analysis, pp. 85–98, North Holland, Amsterdam (1999)
Capinski M., Gatarek D.: Stochastic equations in Hilbert space with application to Navier–Stokes equations in any dimension. J. Funct. Anal. 126, 26–35 (1994)
Capinski M., Peszat S.: On the existence of a solution to stochastic Navier–Stokes equations. Nonlinear Anal. 44, 141–177 (2001)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)
Da Prato, G.; Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge (1996)
Farwig R., Komo C.: Regularity of weak solutions to the Navier–Stokes equations in exterior domains. Nonlinear Diff. Equ. Appl. 17, 303–321 (2010)
Flandoli F., Gatarek D.: Martingale and stationary solutions for stochastic Naver–Stokes equations. Proba. Theory Relat. Fields 102, 365–391 (1995)
Flandoli, F.: An introduction to 3D stochastic fluid dynamics, Lecture notes, vol. 1942, pp. 51–150 (2008)
Giga Y., Sohr H.: Abstract L p estimates for the Cauchy probulem with applications to the Navier–Stokes equations in exterior Domains. J. Funct. Anal. 102, 72–94 (1991)
Fujita H., Kato T.: On the Navier–Stokes initial value problem 1. Arch. Ration. Mech. Anal. 16, 269–315 (1964)
Miyakawa T., Sohr H.: On energy inequality, smoothness and large time behavior in L 2 for weak solutions of the Navier–Stokes equations in exterior domains. Math. Z 199, 455–478 (1988)
Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, Berlin (1991)
Kato T., Fujita H.: On the nonstationary Navier–Stokes system. Rend. Semi. Math. Univ. Padova 32, 243–260 (1962)
Kato T.: Strong L p−solutions of the Navier–Stokes equation in R n, with applications to weak solutions. Math. Z. 187, 471–480 (1984)
Kim J.: On the stochastic wave equation with nonlinear damping. Appl. Math. Optim 58, 29–67 (2008)
Mikulevicius R., Rozovskii B.L.: Global L 2−solutions of stochastic Navier–Stokes equations. Ann. Prob. 33, 137–176 (2005)
Miyakawa T.: On nonstationary solution of the Navier–Stokes equations in an exterior domain. Hiroshima Math. J. 12, 115–140 (1982)
van Neerven J., Veraar M., Weis L.: Maximal L p− regularity for stochastic evolution equations. SIAM Math. Anal 44, 1372–1414 (2012)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983)
Seidler J., Sobukawa T.: Exponential integrability of stochastic convolutions. J. Lond. Math. Soc. 67, L245–258 (2003)
Sritharan S., Sundar P.: Large deviation for the two dimensional Navier–Stokes equations with multiplicative noise. Stoch. Pro. Appl. 116, 1636–1659 (2006)
Sohr, H.: The Navier–Stokes Equations: An Elementary Functional Analysis Approach. Birkhauser, Boston (2001)
Taniguchi, T.: The existence and decay estimates of the solutions to 3D stochastic Naver-Stokes equations with additive noise in an exterior domain, Discrete and Continuous Dynamical Systems-A (to appear)
Taniguchi T.: The existence of energy solutions to 2-dimensional non-Lipschitz stochastic Navier–Stokes equations in unbounded domains. J. Diff. Equ. 251, 3329–3362 (2011)
Temam, R.: Navier–Stokes Equation. North-Holland, Amsterdam (1977)
Temam, R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Springer, New York
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Taniguchi, T. Global existence of a weak solution to 3d stochastic Navier–Stokes equations in an exterior domain. Nonlinear Differ. Equ. Appl. 21, 813–840 (2014). https://doi.org/10.1007/s00030-014-0268-z
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DOI: https://doi.org/10.1007/s00030-014-0268-z