# American Institute of Mathematical Sciences

December  2016, 9(4): 767-776. doi: 10.3934/krm.2016015

## Global existence for the 2D Navier-Stokes flow in the exterior of a moving or rotating obstacle

 1 College of Applied Sciences, Beijing University of Technology, Beijing 100124, China 2 College of Applied Sciences, Beijing University of Technology, PingLeYuan100, Chaoyang District, Beijing 100124 3 Department of Applied Mathematics, Beijing University of Technology, Beijing 100124 4 School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, China

Received  March 2016 Revised  May 2016 Published  September 2016

We consider the global existence of the two-dimensional Navier-Stokes flow in the exterior of a moving or rotating obstacle. Bogovski$\check{i}$ operator on a subset of $\mathbb{R}^2$ is used in this paper. One important thing is to show that the solution of the equations does not blow up in finite time in the sense of some $L^2$ norm. We also obtain the global existence for the 2D Navier-Stokes equations with linearly growing initial velocity.
Citation: Shuguang Shao, Shu Wang, Wen-Qing Xu, Bin Han. Global existence for the 2D Navier-Stokes flow in the exterior of a moving or rotating obstacle. Kinetic & Related Models, 2016, 9 (4) : 767-776. doi: 10.3934/krm.2016015
##### References:
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Zhang, Global existencefor the two dimensional incompressible viscous fluids with linearly growing velocity,, Mathematical Methods in the Applied Sciences, 36 (2013), 921. doi: 10.1002/mma.2649. Google Scholar [17] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations,, Springer Tracts Natur. Philos., (1994). doi: 10.1007/978-1-4612-5364-8. Google Scholar [18] 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. doi: 10.1515/CRELLE.2006.051. Google Scholar [19] Y. Giga and T. Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem,, Arch. Rat. Mech. Anal., 89 (1985), 267. doi: 10.1007/BF00276875. Google Scholar [20] Y. Giga, S. Matsui and O. Sawada, Global existence of two-dimensional Navier-Stokes flow with nondecaying initial velocity,, J. Math. Fluid Mech., 3 (2001), 302. doi: 10.1007/PL00000973. Google Scholar [21] T. 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Okamoto, Exact solutions of the Navier-Stokes equations via Leray's scheme,, Japan J. Indust. Appl. Math., 14 (1997), 169. doi: 10.1007/BF03167263. Google Scholar [28] H. Sohr, The Navier-Stokes Equations,, Birkhäuser Advanced Texts, (2001). Google Scholar [29] V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations,, J. Sov. Math., 8 (1977), 467. Google Scholar [30] L. Tartar, An Introduction to Sobolev Space and Interpolation Spaces,, Lecture Notes of the Unione Matematica Italiana, (2007). Google Scholar

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##### References:
 [1] H. Amann, On the strong solvability of the Navier-Stokes equations,, J. Math. Fluid Mech., 2 (2000), 16. doi: 10.1007/s000210050018. Google Scholar [2] A. Banin, A. Mahalov and B. Nicolaenko, Global regularity of 3D rotating Navier-Stokes equations for resonant domains,, Indiana Univ. Math. J., 48 (1999), 1133. doi: 10.1016/S0893-9659(99)00208-6. Google Scholar [3] A. Banin, A. Mahalov and B. Nicolaenko, 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity,, Indiana Univ. Math. J., 50 (2001), 1. doi: 10.1512/iumj.2001.50.2155. Google Scholar [4] H. Brezis and L. Nirenberg, Degree theory and BMO. I. Compact manifolds without boundaries,, Selecta Math. (N.S.), 1 (1995), 197. doi: 10.1007/BF01671566. Google Scholar [5] M. E. Bogovskiĭ, Solution of the first boundary value problem for an equation of continuity of an incompressible medium,, Dokl. Akad. Nauk SSSR, 248 (1979), 1037. Google Scholar [6] W. Borchers, Zur Stabilität und Faktorisierungsmethode für die Navier-Stokes-Gleichungen inkompressibler viskoser Flüssigkeiten,, Habilitationschrift Universität Paderborn, (1992). Google Scholar [7] M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations,, in Handbook of Mathematical Fluid Dynamics, (2004), 161. Google Scholar [8] D. C. Chang, The dual of Hardy spaces on a domain in $\mathbbR^n$,, Forum Math., 6 (1994), 65. doi: 10.1515/form.1994.6.65. Google Scholar [9] D. C. Chang, G. Dafni and C. Sadosky, A div-curl lemma in BMO on a domain,, Progr. Math., 238 (2005), 55. doi: 10.1007/0-8176-4416-4_5. Google Scholar [10] D. C. Chang, G. Dafni and E. M. Stein, Hardy spaces, BMO, and boundary value problems for the Laplacian on a smooth domain in $\mathbbR^n$,, Trans. Amer. Math. Soc., 351 (1999), 1605. doi: 10.1090/S0002-9947-99-02111-X. Google Scholar [11] D. C. Chang, S. G. Krantz and E. M. Stein, $\mathcal H^p$ theory on a smooth domain in $\mathbbR^n$ and elliptic boundary value problems,, J. Funct. Anal., 114 (1993), 286. doi: 10.1006/jfan.1993.1069. Google Scholar [12] R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, {Compensated compactness and Hardy spaces,, J. Math. Pures Appl., 72 (1993), 247. Google Scholar [13] Z. Chen and T. Miyakawa, Decay properties of weak solutions to a perturbed Navier-Stokes system in $\mathbbR^n$,, Adv. Math. Sci. Appl., 7 (1997), 741. Google Scholar [14] P. Cumsille and M. Tucsnak, Well-posedness for the Navier-Stokes flow in exterior of a rotating obstacle,, Math. Methods in the Applied Sciences, 29 (2006), 595. doi: 10.1002/mma.702. Google Scholar [15] D. Y. Fang, M. Hieber and T. Zhang, Density-dependent incompressible viscous fluid flow subject to linearly growing initial data,, Applicable Analysis, 91 (2012), 1477. doi: 10.1080/00036811.2011.608160. Google Scholar [16] D. Y. Fang, B. Han and T. Zhang, Global existencefor the two dimensional incompressible viscous fluids with linearly growing velocity,, Mathematical Methods in the Applied Sciences, 36 (2013), 921. doi: 10.1002/mma.2649. Google Scholar [17] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations,, Springer Tracts Natur. Philos., (1994). doi: 10.1007/978-1-4612-5364-8. Google Scholar [18] 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. doi: 10.1515/CRELLE.2006.051. Google Scholar [19] Y. Giga and T. Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem,, Arch. Rat. Mech. Anal., 89 (1985), 267. doi: 10.1007/BF00276875. Google Scholar [20] Y. Giga, S. Matsui and O. Sawada, Global existence of two-dimensional Navier-Stokes flow with nondecaying initial velocity,, J. Math. Fluid Mech., 3 (2001), 302. doi: 10.1007/PL00000973. Google Scholar [21] T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of rotating obstacle,, Arch. Roational Mech. Anal., 150 (1999), 307. doi: 10.1007/s002050050190. Google Scholar [22] T. Hishida, The Stokes operator with rotation effect in exterior domains,, Analysis, 19 (1999), 51. doi: 10.1524/anly.1999.19.1.51. Google Scholar [23] M. Hieber and O. Sawada, The Navier-Stokes equations in $\mathbbR^N$ with linearly growing initial data,, Arch. Roational Mech. Anal., 175 (2005), 269. doi: 10.1007/s00205-004-0347-0. Google Scholar [24] O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, Gordon and Breach, (1969). Google Scholar [25] J. Leray, Sur le mouvement d'un liquide visqueux remplissant l'espace,, Acta mathematica, 63 (1934), 193. doi: 10.1007/BF02547354. Google Scholar [26] A. Majda, Vorticity and the mathematical theory of incompressible fluid flow,, Comm. Pure Appl. Math., 39 (1986), 187. doi: 10.1002/cpa.3160390711. Google Scholar [27] H. Okamoto, Exact solutions of the Navier-Stokes equations via Leray's scheme,, Japan J. Indust. Appl. Math., 14 (1997), 169. doi: 10.1007/BF03167263. Google Scholar [28] H. Sohr, The Navier-Stokes Equations,, Birkhäuser Advanced Texts, (2001). Google Scholar [29] V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations,, J. Sov. Math., 8 (1977), 467. Google Scholar [30] L. Tartar, An Introduction to Sobolev Space and Interpolation Spaces,, Lecture Notes of the Unione Matematica Italiana, (2007). Google Scholar
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