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 and Related Models, 2016, 9 (4) : 767-776. doi: 10.3934/krm.2016015
References:
[1]

H. Amann, On the strong solvability of the Navier-Stokes equations, J. Math. Fluid Mech., 2 (2000), 16-98. doi: 10.1007/s000210050018.

[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), 1133-1176. doi: 10.1016/S0893-9659(99)00208-6.

[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), 1-35. doi: 10.1512/iumj.2001.50.2155.

[4]

H. Brezis and L. Nirenberg, Degree theory and BMO. I. Compact manifolds without boundaries, Selecta Math. (N.S.), 1 (1995), 197-263. doi: 10.1007/BF01671566.

[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), 1037-1040.

[6]

W. Borchers, Zur Stabilität und Faktorisierungsmethode für die Navier-Stokes-Gleichungen inkompressibler viskoser Flüssigkeiten, Habilitationschrift Universität Paderborn, 1992.

[7]

M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, in Handbook of Mathematical Fluid Dynamics, 3, North-Holland, Amsterdam, 2004, 161-244.

[8]

D. C. Chang, The dual of Hardy spaces on a domain in $\mathbb{R}^{N}$, Forum Math., 6 (1994), 65-81. doi: 10.1515/form.1994.6.65.

[9]

D. C. Chang, G. Dafni and C. Sadosky, A div-curl lemma in BMO on a domain, Progr. Math., 238 (2005), 55-65. doi: 10.1007/0-8176-4416-4_5.

[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 $\mathbb{R}^{N}$, Trans. Amer. Math. Soc., 351 (1999), 1605-1661. doi: 10.1090/S0002-9947-99-02111-X.

[11]

D. C. Chang, S. G. Krantz and E. M. Stein, $\mathcal H^p$ theory on a smooth domain in $\mathbb{R}^{N}$ and elliptic boundary value problems, J. Funct. Anal., 114 (1993), 286-347. doi: 10.1006/jfan.1993.1069.

[12]

R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, {Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286.

[13]

Z. Chen and T. Miyakawa, Decay properties of weak solutions to a perturbed Navier-Stokes system in $\mathbb{R}^{N}$, Adv. Math. Sci. Appl., 7 (1997), 741-770.

[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), 595-623. doi: 10.1002/mma.702.

[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), 1477-1493. doi: 10.1080/00036811.2011.608160.

[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), 921-935. doi: 10.1002/mma.2649.

[17]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer Tracts Natur. Philos., Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.

[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), 45-62. doi: 10.1515/CRELLE.2006.051.

[19]

Y. Giga and T. Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem, Arch. Rat. Mech. Anal., 89 (1985), 267-281. doi: 10.1007/BF00276875.

[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), 302-315. doi: 10.1007/PL00000973.

[21]

T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of rotating obstacle, Arch. Roational Mech. Anal., 150 (1999), 307-348. doi: 10.1007/s002050050190.

[22]

T. Hishida, The Stokes operator with rotation effect in exterior domains, Analysis, 19 (1999), 51-67. doi: 10.1524/anly.1999.19.1.51.

[23]

M. Hieber and O. Sawada, The Navier-Stokes equations in $\mathbb{R}^N2$ with linearly growing initial data, Arch. Roational Mech. Anal., 175 (2005), 269-285. doi: 10.1007/s00205-004-0347-0.

[24]

O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.

[25]

J. Leray, Sur le mouvement d'un liquide visqueux remplissant l'espace, Acta mathematica, 63 (1934), 193-248. doi: 10.1007/BF02547354.

[26]

A. Majda, Vorticity and the mathematical theory of incompressible fluid flow, Comm. Pure Appl. Math., 39 (1986), 187-220. doi: 10.1002/cpa.3160390711.

[27]

H. Okamoto, Exact solutions of the Navier-Stokes equations via Leray's scheme, Japan J. Indust. Appl. Math., 14 (1997), 169-197. doi: 10.1007/BF03167263.

[28]

H. Sohr, The Navier-Stokes Equations, Birkhäuser Advanced Texts, Basel 2001.

[29]

V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations, J. Sov. Math., 8 (1977), 467-529.

[30]

L. Tartar, An Introduction to Sobolev Space and Interpolation Spaces, Lecture Notes of the Unione Matematica Italiana, Springer-Verlag Berlin Heidelberg, 2007.

show all references

References:
[1]

H. Amann, On the strong solvability of the Navier-Stokes equations, J. Math. Fluid Mech., 2 (2000), 16-98. doi: 10.1007/s000210050018.

[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), 1133-1176. doi: 10.1016/S0893-9659(99)00208-6.

[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), 1-35. doi: 10.1512/iumj.2001.50.2155.

[4]

H. Brezis and L. Nirenberg, Degree theory and BMO. I. Compact manifolds without boundaries, Selecta Math. (N.S.), 1 (1995), 197-263. doi: 10.1007/BF01671566.

[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), 1037-1040.

[6]

W. Borchers, Zur Stabilität und Faktorisierungsmethode für die Navier-Stokes-Gleichungen inkompressibler viskoser Flüssigkeiten, Habilitationschrift Universität Paderborn, 1992.

[7]

M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, in Handbook of Mathematical Fluid Dynamics, 3, North-Holland, Amsterdam, 2004, 161-244.

[8]

D. C. Chang, The dual of Hardy spaces on a domain in $\mathbb{R}^{N}$, Forum Math., 6 (1994), 65-81. doi: 10.1515/form.1994.6.65.

[9]

D. C. Chang, G. Dafni and C. Sadosky, A div-curl lemma in BMO on a domain, Progr. Math., 238 (2005), 55-65. doi: 10.1007/0-8176-4416-4_5.

[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 $\mathbb{R}^{N}$, Trans. Amer. Math. Soc., 351 (1999), 1605-1661. doi: 10.1090/S0002-9947-99-02111-X.

[11]

D. C. Chang, S. G. Krantz and E. M. Stein, $\mathcal H^p$ theory on a smooth domain in $\mathbb{R}^{N}$ and elliptic boundary value problems, J. Funct. Anal., 114 (1993), 286-347. doi: 10.1006/jfan.1993.1069.

[12]

R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, {Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286.

[13]

Z. Chen and T. Miyakawa, Decay properties of weak solutions to a perturbed Navier-Stokes system in $\mathbb{R}^{N}$, Adv. Math. Sci. Appl., 7 (1997), 741-770.

[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), 595-623. doi: 10.1002/mma.702.

[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), 1477-1493. doi: 10.1080/00036811.2011.608160.

[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), 921-935. doi: 10.1002/mma.2649.

[17]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer Tracts Natur. Philos., Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.

[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), 45-62. doi: 10.1515/CRELLE.2006.051.

[19]

Y. Giga and T. Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem, Arch. Rat. Mech. Anal., 89 (1985), 267-281. doi: 10.1007/BF00276875.

[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), 302-315. doi: 10.1007/PL00000973.

[21]

T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of rotating obstacle, Arch. Roational Mech. Anal., 150 (1999), 307-348. doi: 10.1007/s002050050190.

[22]

T. Hishida, The Stokes operator with rotation effect in exterior domains, Analysis, 19 (1999), 51-67. doi: 10.1524/anly.1999.19.1.51.

[23]

M. Hieber and O. Sawada, The Navier-Stokes equations in $\mathbb{R}^N2$ with linearly growing initial data, Arch. Roational Mech. Anal., 175 (2005), 269-285. doi: 10.1007/s00205-004-0347-0.

[24]

O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.

[25]

J. Leray, Sur le mouvement d'un liquide visqueux remplissant l'espace, Acta mathematica, 63 (1934), 193-248. doi: 10.1007/BF02547354.

[26]

A. Majda, Vorticity and the mathematical theory of incompressible fluid flow, Comm. Pure Appl. Math., 39 (1986), 187-220. doi: 10.1002/cpa.3160390711.

[27]

H. Okamoto, Exact solutions of the Navier-Stokes equations via Leray's scheme, Japan J. Indust. Appl. Math., 14 (1997), 169-197. doi: 10.1007/BF03167263.

[28]

H. Sohr, The Navier-Stokes Equations, Birkhäuser Advanced Texts, Basel 2001.

[29]

V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations, J. Sov. Math., 8 (1977), 467-529.

[30]

L. Tartar, An Introduction to Sobolev Space and Interpolation Spaces, Lecture Notes of the Unione Matematica Italiana, Springer-Verlag Berlin Heidelberg, 2007.

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