## Anisotropic $$L^2$$-estimates of weak solutions to the stationary Oseen-type equations in 3D-exterior domain for a rotating body.(English)Zbl 1186.35163

Summary: We study the Oseen problem with rotational effect in exterior three-dimensional domains. Using a variational approach we prove existence and uniqueness theorems in anisotropically weighted Sobolev spaces in the whole three-dimensional space. As the main tool we derive and apply an inequality of the Friedrichs-Poincaré type and the theory of Calderon-Zygmund kernels in weighted spaces. For the extension of results to the case of exterior domains we use a localization procedure.

### MSC:

 35Q35 PDEs in connection with fluid mechanics 35B45 A priori estimates in context of PDEs 76E07 Rotation in hydrodynamic stability 76M30 Variational methods applied to problems in fluid mechanics
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### References:

 [1] R. Farwig, A variational approach in weighted Sobolev spaces to the operator $$-\triangle+\partial/\partial x_{1}$$ in exterior domains of $$\mbi{R}^{3}$$, Math. Z., 210 (1992), 449-464. · Zbl 0727.35041 [2] R. Farwig, The stationary exterior 3-D problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces, Math. Z., 211 (1992), 409-447. · Zbl 0727.35106 [3] R. Farwig, An $$L^{q}$$-analysis of viscous fluid flow past a rotating obstacle, Tôhoku Math. J. (2), 58 (2006), 129-147. · Zbl 1136.76340 [4] R. Farwig, Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle, Banach Center Publ., 70 (2005), 73-84. · Zbl 1101.35348 [5] R. Farwig, T. Hishida and D. Müller, $$L^{q}$$-Theory of a singular “winding” integral operator arising from fluid dynamics, Pacific J. Math., 215 (2004), 297-312. · Zbl 1057.35028 [6] R. Farwig and T. Hishida, Stationary Navier-Stokes flow around a rotating obstacles, Funkcial. Ekvac., 50 (2007), 371-403. · Zbl 1180.35408 [7] R. Farwig, M. Krbec and Š. Nečasová, A weighted $$L^{q}$$-approach to Stokes flow around a rotating body, Ann. Univ. Ferrara Sez. VII Sci. Mat., 54 (2008), 61-84. · Zbl 1248.35158 [8] R. Farwig, M. Krbec and Š. Nečasová, A weighted $$L^{q}$$-approach to Oseen flow around a rotating body, accepted in Math. Methods Appl. Sci. · Zbl 1224.76034 [9] R. Finn, Estimates at infinity for stationary solution of Navier-Stokes equations, Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine (N.S.), 3, 51 (1959), 387-418. · Zbl 0106.39402 [10] R. Finn, On the exterior stationary problem for the Navier-Stokes equations, and associated perturbation problems, Arch. Rational Mech. Anal., 19 (1965), 363-406. · Zbl 0149.44606 [11] G. P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications, Handbook of Mathematical Fluid Dynamics, 1 , (eds. S. Friedlander and D. Serre), Elsevier, North-Holland, Amsterdam, 2002, pp. 653-791. · Zbl 1230.76016 [12] G. P. Galdi, Steady flow of a Navier-Stokes fluid around a rotating obstacle, Essays and papers dedicated to the memory of Clifford Ambrose Truesdell III, II , J. Elasticity, 71 (2003), 1-31. · Zbl 1156.76367 [13] G. P. Galdi and A. L. Silvestre, On the steady motion of a Navier-Stokes liquid around a rigid body, Arch. Ration. Mech. Anal., 184 (2007), 371-400. · Zbl 1111.76010 [14] G. P. Galdi and A. L. Silvestre, Further results on steady-state flow of a Navier-Stokes liquid around a rigid body, Existence of the wake, RIMS Kôkyûroku Bessatsu, B1, Res. Inst. Math. Sci., Kyoto, 2007, pp. 127-143. · Zbl 1119.76011 [15] M. Geissert, H. Heck and M. Hieber, $$L^{p}$$ theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle, J. Reine Angew. Math., 596 (2006), 45-62. · Zbl 1102.76015 [16] V. Girault and P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, Lecture Notes in Mathematics, 749 , Springer-Verlag, Berlin, Heidelberg, New York, 1979. · Zbl 0413.65081 [17] T. Hishida, The Stokes operator with rotation effect in exterior domains, Analysis (Munich), 19 (1999), 51-67. · Zbl 0938.35114 [18] T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle, Arch. Ration. Mech. Anal., 150 (1999), 307-348. · Zbl 0949.35106 [19] T. Hishida, $$L^{q}$$-estimates of weak solutions to the stationary Stokes equations around a rotating body, J. Math. Soc. Japan, 58 (2006), 743-767. · Zbl 1184.35241 [20] J.-L. Impagliazzo, Resolution des equations de Navier-Stokes compressibles a l’aide de la methode de decomposition, These de doctorat de l’Universite de Toulon et du Var, 1997. [21] T. Kobayashi and Y. Shibata, On the Oseen equation in the three-dimensional exterior domains, Math. Ann., 310 (1998), 1-45. · Zbl 0891.35114 [22] H. Kozono and H. Sohr, New a priori estimates for the Stokes equations in exterior domains, Indiana Univ. Math. J., 40 (1991), 1-27. · Zbl 0732.35068 [23] S. Kračmar, Š. Nečasová and P. Penel, Anisotropic $$L^{2}$$-estimates of weak solutions to the stationary Oseen type equations in $$\mbi{R}^{3}$$ for a rotating body, RIMS Kôkyûroku Bessatsu, B1, Res. Inst. Math. Sci., Kyoto, 2007, pp. 219-235. · Zbl 1153.35060 [24] S. Kračmar, Š. Nečasová and P. Penel, Remarks on the non-homogeneous Oseen problem arising from modeling of the fluid around a rotating body, Proceedings of the International Conference “Hyperbolic equations”, held in Lyon, France, 2006, [25] S. Kračmar, A. Novotný and M. Pokorný, Estimates of three dimensional Oseen kernels in weighted $$L^{p}$$ spaces, In: Applied Nonlinear Analysis, London-New York, Kluwer Academic, 1999, pp. 281-316. · Zbl 0961.35105 [26] S. Kračmar, A. Novotný and M. Pokorný, Estimates of Oseen kernels in weighted $$L^{p}$$ spaces, J. Math. Soc. Japan, 53 (2001), 59-111. · Zbl 0988.76021 [27] S. Kračmar and P. Penel, Variational properties of a generic model equation in exterior 3D domains, Funkcial. Ekvac., 47 (2004), 499-523. · Zbl 1114.35053 [28] S. Kračmar and P. Penel, New regularity results for a generic model equation in exterior 3D domains, Banach Center Publ., 70 (2005), 139-155. · Zbl 1101.35350 [29] Š. Nečasová, Asymptotic properties of the steady fall of a body in a viscous fluid, Math. Methods Appl. Sci., 27 (2004), 1969-1995. · Zbl 1174.76306 [30] C. W. Oseen, Neuere Methoden und Ergebnisse in der Hydrodynamik, Leipzig, Akad. Verlagsges. M.B.H., 1927. · JFM 53.0773.02 [31] H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analytic Approach, Berlin, Birkhäuser, 2001. · Zbl 0983.35004 [32] E. A. Thomann and R. B. Guenther, The fundamental solution of the linearized Navier-Stokes equations for spinning bodies in three spatial dimensions,–,time dependent case, J. Math. Fluid Mech., 8 (2006), 77-98. · Zbl 1125.35076
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