Abstract
This article develops a general approach to time periodic incompressible fluid flow problems and semilinear evolution equations. It yields, on the one hand, a unified approach to various classical problems in incompressible fluid flow and, on the other hand, gives new results for periodic solutions to the Navier–Stokes–Oseen flow, the Navier–Stokes flow past rotating obstacles, and, in the geophysical setting, for Ornstein–Uhlenbeck and various diffusion equations with rough coefficients. The method is based on a combination of interpolation and topological arguments, as well as on the smoothing properties of the linearized equation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transform and Cauchy Problems. Monographs in Mathematics, 96, 2nd edition. Birkhäuser Verlag, Basel, 2011
Bergh J., Löfström J.: Interpolation Spaces. Springer, Berlin-Heidelberg-NewYork (1976)
Borchers W., Miyakawa T.: On stability of exterior stationary Navier–Stokes flows. Acta Math. 174, 311–382 (1995)
Borchers W., Sohr H.: On the semigroup of the Stokes operator for exterior domains in L q-spaces. Math. Z. 196, 415–425 (1987)
Burton, T.: Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Academic Press, 1985
Crispo F., Maremonti P.: Navier–Stokes equations in aperture domains: Global existence with bounded flux and time-periodic solutions. Math. Methods Appl. Sci. 31, 249–277 (2008)
Duong X., Ouhabaz E.: Complex multiplicative perturbations of elliptic operators: heat kernel bounds and holomorphic functional calculus. Diff. Integral Equ. 12, 395–418 (1999)
Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, Steady State Problems. Springer Monographs in Math., 2nd edition, 2011
Galdi G.P.: Existence and uniqueness of time-periodic solutions to the Navier–Stokes equations in the whole plane. Discrete Contin. Dyn. Syst. 6, 1237–1257 (2013)
Galdi G.P.: On the time-periodic flow of a viscous liquid past a moving cylinder. Arch. Ration. Mech. Anal. 210, 451–498 (2013)
Galdi G.P., Silvestre A.L.: Existence of time-periodic solutions to the Navier–Stokes equations around a moving body. Pacific J. Math. 223, 251–267 (2006)
Galdi G.P., Silvestre A.L.: On the motion of a rigid body in a Navier–Stokes liquid under the action of a time-periodic force. Indiana Univ. Math. J. 58, 2805–2842 (2009)
Galdi G.P., Sohr H.: Existence and uniqueness of time-periodic physically reasonable Navier–Stokes flows past a body. Arch. Ration. Mech. Anal. 172, 363–406 (2004)
Geissert, M., Heck, H., Hieber, M.: On the equation div u = g and Bogovskii’s operator in Sobolev spaces of negative order. Partial differential equations and functional analysis, Oper. Theory Adv. Appl., vol. 168, pp. 113–121. Birkhäuser, Basel, 2006
Geissert M., Heck H., Hieber M.: L p -Theory of the Navier–Stokes flow in the exterior of a moving or rotating obstacle. J. Reine Angew. Math. 596, 45–62 (2006)
Geissert M., Heck H., Hieber M., Wood I.: The Ornstein–Uhlenbeck semigroup in exterior domains. Arch. Math. 85, 554–562 (2005)
Giga Y.: Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier–Stokes system. J. Differ. Equ. 61, 186–212 (1986)
Grafakos L.: Classical Fourier Analysis. Springer, Graduate Texts in Mathematics (2008)
Heywood J. G.: The Navier–Stokes equations: On the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29, 639–681 (1980)
Heck H., Kim H., Kozono H.: On the stationary Navier–Stokes flows around a rotating body. Manuscripta Math. 138, 315–345 (2012)
Hieber M., Shibata Y.: The Fujita–Kato approach to the Navier–Stokes equations in the rotational framework. Math. Z. 265, 481–491 (2010)
Hishida, T.: The nonstationary Stokes and Navier–Stokes flows through an aperture. In: Contributions to Current Challenges in Mathematical Fluid Mechanics, (Eds. Galdi G.P. et al.) Advances in Mathematical Fluid Mechanics, pp. 79–123. Birkhäuser, Basel, 2004
Hishida T., Shibata Y.: \({L_{p}-L_{q}}\) Estimate of the Stokes Operator and Navier–Stokes Flows in the Exterior of a Rotating Obstacle. Arch. Ration. Mech. Anal. 193, 339–421 (2009)
Iwashita H.: \({L_{q}-L_{r}}\) estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier–Stokes initial value problems in L q spaces. Math. Ann. 285, 265–288 (1989)
KanielS. Shinbrot M.: A reproductive property of the Navier–Stokes equations. Arch. Rational Mech. Anal. 24, 363–369 (1989)
Kato T.: Strong L p-solutions of Navier–Stokes equations in \({{\mathbb {R}}^{n}}\) with applications to weak solutions. Math. Z. 187, 471–480 (1984)
Kobayashi T., Shibata Y.: On the Oseen equation in the three dimensional exterior domains. Math. Ann. 310, 1– (1998)
Kozono H., Nakao M.: Periodic solutions solutions to the Navier–Stokes equations in unbounded domains. Tohoku Math. J. 48, 33–50 (1996)
Kozono H., Mashiko Y., Takada R.: Existence of periodic solutions and their asymptotic stability to the Navier–Stokes equations with Coriolis force. J. Evol. Equ. 14, 565–601 (2014)
Kubo T.: Periodic solutions to the Navier–Stokes equations in a perturbed half space and an aperture domain. Math. Methods Appl. Sci. 28, 1341–1357 (2005)
Kyed M.: The existence and regularity of time-periodic solutions to the three dimensional Navier–Stokes equations in the whole space. Nonlinearity 27, 2909–2935 (2014)
Liu, J.H., N'Guerekata, G.M., Van Minh, N.: Topics on Stability and Periodicity in Abstract Differential Equations. Series on Concrete and Applicable Mathematics. World Scientific Publishing, Singapore, 2008
Maremonti P.: Existence and stability of time periodic solutions to the Navier–Stokes equations in the whole space. Nonlinearity 4, 503–529 (1991)
Maremonti P., Padula M.: Existence, uniqueness, and attainability of periodic solutions of the Navier–Stokes equations in exterior domains. J. Math. Sci. 93, 719–746 (1999)
Massera J.: The existence of periodic solutions of systems of differential equations. Duke Math. J. 17, 457–475 (1950)
Miyakawa T.: On nonstationary solutions of the Navier–Stokes equations in an exterior domain. Hiroshima Math. J. 12, 115–140 (1982)
Miyakawa T., Teramoto Y.: Existence and periodicity of weak solutions to the Navier–Stokes equations in a time dependent domain. Hiroshima Math. J. 12, 513–528 (1982)
Nguyen T.H.: Periodic motions of Stokes and Navier–Stokes flows around a rotating obstacle. Arch. Ration. Mech. Anal. 213, 689–703 (2014)
Prodi G.: Qualche risultato riguardo alle equazioni di Navier–Stokes nel caso bidimensionale. Rend. Sem. Mat. Univ. Padova. 30, 1–15 (1960)
Prouse G.: Soluzioni periodiche dell’equazione di Navier–Stokes. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 35, 443–447 (1963)
Prüss J.: Periodic solutions of semilinear evolution equations. Nonlinear Anal. 3, 601–612 (1979)
Serrin J.: A note on the existence of periodic solutions of the Navier–Stokes equations. Arch. Rational Mech. Anal. 3, 120–122 (1959)
Shibata, Y.: On the Oseen semigroup with rotating effect. Functional Analysis and Evolution Equations, pp. 595–611. Birkhäuser, Basel, 2008
Taniuchi Y.: On stability solutions of periodic solutions in unbounded domains. Hokkaido Math. J. 28, 147–173 (1999)
Taniuchi Y.: On the uniqueness of time-periodic solutions to the Navier–Stokes equations in unbounded domains. Math. Z. 261, 597–615 (2009)
Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam, New York, Oxford, 1978
Van Baalen G., Wittwer P.: Time periodic solutions of the Navier–Stokes equations with nonzero constant boundary conditions at infinity. SIAM J. Math. Anal. 43, 1787–1809 (2011)
Yamazaki M.: The Navier–Stokes equations in the weak-L n space with time-dependent external force. Math. Ann. 317, 635–675 (2000)
Yoshizawa, T.: Stability theory and the existence of periodic solutions and almost periodic solutions. Applied Mathematical Sciences. Springer, 1975
Yudovich V.: Periodic motions of a viscous incompressible fluid. Sov. Math. Dokl. 1, 168–172 (1960)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by E. G. Virga
Thieu Huy Nguyen, on leave from Hanoi University of Science and Technology as a research fellow of the Alexander von Humboldt Foundation.
Rights and permissions
About this article
Cite this article
Geissert, M., Hieber, M. & Nguyen, T.H. A General Approach to Time Periodic Incompressible Viscous Fluid Flow Problems. Arch Rational Mech Anal 220, 1095–1118 (2016). https://doi.org/10.1007/s00205-015-0949-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-015-0949-8