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Strong solutions to the Navier-Stokes equations around a rotating obstacle. (English) Zbl 1081.35076

Summary: We study the existence of strong solutions to the three-dimensional Navier-Stokes initial-boundary value problem in the domain \(\Omega\), exterior to a rigid body that rotates with constant angular velocity \(\omega\). We show that when the initial data \(u_0\), are prescribed in an appropriate functional class, a strong solution exists at least in some finite time interval. Moreover, the solution exists for all times, provided \(u_0\), in suitable norm, and the magnitude of \(\omega\) do not exceed a certain constant depending only on the kinematic viscosity and on the regularity of \(\Omega\). In this latter case, we also show that the velocity field converges to the velocity field of the corresponding steady-state solution.

MSC:

35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76U05 General theory of rotating fluids
35B45 A priori estimates in context of PDEs
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[1] Berker, R.: Contrainte sur un Paroi en Contact avec un Fluide Visqueux Classique, un Fluide de Stokes, un Fluide de Coleman-Noll, C.R. Acad. Sci. Paris 285, 5144–5147 (1964)
[2] Borchers, W.: Zur Stabilität und Faktorisierungsmethode für die Navier-Stokes Gleichungen inkompressibler viskoser Flüssigkeiten. Habilitationsschrift, University of Paderborn (1992)
[3] Chen, Z. M., Miyakawa, T.: Decay properties of weak solutions to a perturbed Navier-Stokes system in \(\mathbb{R}\)n. Adv. Math. Sci. Appl. 7, 741–770 (1997) · Zbl 0893.35092
[4] Fujita, H., Kato, T.: On the Navier-Stokes initial value problem. I. Arch. Rational Mech. Anal. 16, 269–315 (1964) · Zbl 0126.42301 · doi:10.1007/BF00276188
[5] Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Nonlinear Steady Problems. Springer Tracts in Natural Philosophy, Vol. 38, Springer-Verlag, New York, Revised Edition (1998) · Zbl 0911.35085
[6] Galdi, G.P.: On the Motion of a Rigid Body in a Viscous Liquid: A Mathematical Analysis with Applications. Handbook of Mathematical Fluid Mechanics, Elsevier Science, 653–791 (2002) · Zbl 1230.76016
[7] Galdi, G.P.: Steady Flow of a Navier-Stokes Exterior Fluid Around a Rotating Obstacle. Journal of Elasticity 71, 1–31 (2003) · Zbl 1156.76367 · doi:10.1023/B:ELAS.0000005543.00407.5e
[8] Galdi, G.P., Silvestre, A.L.: Strong Solutions to the Problem of Motion of a Rigid Body in a Navier-Stokes Liquid under the Action of Prescribed Forces and Torques. InNonlinear Problems in Mathematical Physics and Related Topics (In Honor of Professor O.A. Ladyzhenskaya), Int. Math. Ser. (N.Y.), 1 Kluwer/Plenum, N.Y., 121–144 (2002) · Zbl 1046.35084
[9] Heywood, J.G.: The Navier-Stokes Equations: on the Existence, Regularity and Decay of Solutions. Indiana Univ. Math. Journal 29, 639–681 (1980) · Zbl 0494.35077 · doi:10.1512/iumj.1980.29.29048
[10] Hishida, T.: An Existence Theorem for the Navier-Stokes Flow in the Exterior of a Rotating Obstacle. Arch. Rational. Mech. Anal. 150, 307–348 (1999) · Zbl 0949.35106 · doi:10.1007/s002050050190
[11] Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York (1969) · Zbl 0184.52603
[12] Maremonti, P.: Asymptotic Stability Theorems for Viscous Fluid Motions in Exterior Domains. Rend. Sem. Matem. Univ. Padova 71, 35–72 (1984) · Zbl 0548.76047
[13] Masuda, K.: On the Stability of Incompressible Viscous Fluid Motions Past Objects. J. Math. Soc. Japan 27, 294–327 (1975) · Zbl 0303.76011 · doi:10.2969/jmsj/02720294
[14] Prodi, G.: Teoremi di Tipo Locale per il Sistema di Navier-Stokes e Stabilità delle Soluzioni Stazionarie. Rend. Sem. Mat. Univ. Padova 32, 374–397 (1962) · Zbl 0108.28602
[15] Silvestre, A.L.: On the Existence of Steady Flows of a Navier-Stokes Liquid Around a Moving Rigid Body. Math. Meth. Appl. Sci. 27, 1399–1409 (2004) · Zbl 1061.35078 · doi:10.1002/mma.509
[16] Temam, R.: Navier-Stokes Equations. North-Holland Pub. Co. (2001) · Zbl 0981.35001
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