The existence of time-periodic stochastic motions of an incompressible fluid is obtained. Here the fluid is subject to a time-periodic body force and an additional time-periodic stochastic force that is produced by a rigid body moves periodically stochastically with the same period in the fluid.
REFERENCES
1.
2.
A. Ya.
Dorogovtsev
, “Some remarks on differential equations perturbed by periodic random processes
,” Ukrain. Mat. Zh.
14
, 119
–128
(1962
)3.
A. Ya.
Dorogovtsev
, “Periodic and Stationary Modes of Infinite-Dimensional Deterministic and Stochastic Dynamical Systems [in Russian]
” (Vyshcha Shkola, Kiev, 1992
).4.
G. P.
Galdi
, “On the motion of a rigid body in a viscous fluid: A mathematical analysis with applications
,” in Handbook of Mathematical Fluid Dynamics
, edited by S.
Friedlander
and D.
Serre
(North-Holland
, Amsterdam
, 2002
), pp. 653
–791
.5.
G. P.
Galdi
and A. L.
Silvestre
, “Existence of time-periodic solutions to the Navier-Stokes equations around a moving body
,” Pacific J. Math.
223
, 251
–267
(2006
).6.
G. P.
Galdi
and H.
Sohr
, “Existence and uniqueness of time-periodic physically reasonable Navier-Stokes flow past a body
,” Arch. Ration. Mech. Anal.
172
, 363
–406
(2004
).7.
J. G.
Heywood
, “The Navier-Stokes equations: on the existence, regularity and decay of solutions
,” Indiana Univ. Math. J.
29
, 639
–681
(1980
).8.
H. L.
Hurd
and A. G.
Miamee
, Periodically Correlated Random Sequences. Spectral Theory and Practice
, Wiley Series in Probability and Statistics
(Wiley-Interscience
, Hoboken, NJ
, 2007
).9.
R.
Khasminskii
, Stochastic Stability of Differential Equations. With contributions by G. N. Milstein and M. B. Nevelson
, Stochastic Modelling and Applied Probability, 66
(Springer
, Heidelberg
, 2012
).10.
H.
Kozono
, M.
Nako
, “Periodic solutions of the Navier-Stokes equations in unbounded domains
,” Tohoku Math. J.
48
, 33
–50
(1996
).11.
P.
Maremonti
, “Existence and stability of time-periodic solutions to the Navier-Stokes equations in the whole space
,” Nonlinearity
4
, 503
–529
(1991
).12.
P.
Maremonti
, “Some theorems of existence for solutions of the Navier-Stokes equations with slip boundary conditions in half-space
,” Ricerche Mat.
40
, 81
–135
(1991
).13.
J. C.
Mattingly
, “The stochastic Navier-Stokes equation: Energy estimates and phase space contraction
,” Ph.D. thesis (Princeton University
, 1998
).14.
H.
Morimoto
, “On existence of periodic weak solutions to the Navier-Stokes equations in regions with periodically moving boundaries
,” J. Fac. Sci. Univ. Tokyo Sect. IA Math.
18
, 499
–524
(1971
).15.
G.
Prouse
, “Soluzioni periodiche dell'equazione di Navier-Stokes
,” Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat.
35
, 443
–447
(1963
).16.
R.
Salvi
, “On the existence of periodic weak solutions on the Navier-Stokes equations in exterior regions with periodically moving boundaries
,” in Navier-Stokes Equations and Related Nonlinear Problems
, edited by A.
Sequeira
(Plenum
, New York
, 1995
), pp. 63
–73
.17.
A. L.
Silvestre
, “On the existence of steady flows of a Navier-Stokes liquid around a moving rigid body
,” Math. Methods Appl. Sci.
27
, 1399
–1409
(2004
).18.
R. L.
Stratonovich
, Topics in the theory of random noise. Vol. I: General theory of random processes. Nonlinear transformations of signals and noise, revised English edition. Translated from the Russian by Richard A. (Silverman Gordon and Breach Science Publishers, New York-London 1963
).19.
R.
Temam
, Navier-Stokes Equations: Studies in Mathematics and its Applications
(North-Holland
, Amsterdam
, 1984
), Vol. 2
.20.
M.
Yamazaki
, “The Navier-Stokes equations in the weak Ln space with time-dependent external force
,” Math. Ann.
317
, 635
–675
(2000
).© 2013 AIP Publishing LLC.
2013
AIP Publishing LLC
You do not currently have access to this content.