Global existence of strong solutions for 2-dimensional Navier-Stokes equations on exterior domains with growing data at infinity

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July 2014, 13(4): 1613-1627. doi: 10.3934/cpaa.2014.13.1613

Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity

Michele Campiti ^1, , Giovanni P. Galdi ^2, and Matthias Hieber ^3,

1.	Dipartimento di Matematica e Fisica ``Ennio De Giorgi'', Università del Salento, Via per Arnesano, Lecce 73100, Italy
2.	Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA 15261
3.	Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgartenstr. 7, D-64289 Darmstadt

Received November 2013 Revised January 2014 Published February 2014

Abstract
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It is proved the existence of a unique, global strong solution to the two-dimensional Navier-Stokes initial-value problem in exterior domains in the case where the velocity field tends, at large spatial distance, to a prescribed velocity field that is allowed to grow linearly.

Keywords: exterior domains., Navier-Stokes with linearly growing data.

Mathematics Subject Classification: Primary: 76D03; Secondary: 35Q30,35B3.

Citation: Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613

References:

[1]	D. Chae, Nonexistence of asymptocially self-similar singularities in the Euler and Navier-Stokes equations, Math. Ann., 338 (2007), 435-449. doi: 10.1007/s00208-007-0082-6.
[2]	F. Crispo and P. Maremonti, An interpolation inequality in exterior domains, Rend. Sem. Mat. Univ. Padova, 112 (2004), 11-39.
[3]	D. Fang, B. Han and T. Zhang, Global wellposedness result for density-dependent incompressible viscous fluid in $\mathbb{R}^2$ with linearly growing initial velocity, Math. Meth. Appl. Sciences, 36 (2013), 921-935. doi: 10.1002/mma.2649.
[4]	G. P. Galdi, An Introduction to the Navier-Stokes Initial-boundary Value Problem, Adv. Math. Fluid Mech., Birkhäuser, Basel, 2000.
[5]	G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-state Problems, 2nd edition. Springer Monographs in Mathematics, New York, 2011. doi: 10.1007/978-0-387-09620-9.
[6]	J. G. Heywood, The Navier-Stokes equations: on the existence, regularity and decay of solutions, Indiana Univ. Math. J., 29 (1980), 639-681. doi: 10.1512/iumj.1980.29.29048.
[7]	M. Hieber and O. Sawada, The Navier-Stokes equations in $\mathbb{R}^{N}$ with linearly growing initial data, Arch. Rational Mech. Anal., 175 (2005), 269-285. doi: 10.1007/s00205-004-0347-0.
[8]	M. Hieber, A. Rhandi and O. Sawada, The Navier-Stokes flow for globally Lipschitz continuous initial data, in RIMS Kôkyûroku Bessatsu, vol. B1, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007, 159-165.
[9]	T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle, Arch. Rational Mech. Anal., 150 (1999), 307-348. doi: 10.1007/s002050050190.
[10]	O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Science Publishers, New York-London (1963).
[11]	J. Leray, Sur le mouvement d'un liquide visquex emplissant l'espace, Acta Math., 63 (1996), 193-248. doi: 10.1007/BF02547354.
[12]	J. L. Lions, Espaces intermédiaires entre espaces Hilbertiens et applications, Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine, 2 (1958), 419-432.
[13]	J. Nečas, M. Růžička and V. Šverák, On self-similiar solutions of the Navier-Stokes equations, Acta Math., 176 (1996), 283-294.
[14]	O. Sawada, The Navier-Stokes flow with linearly growing initial velocity in the whole space, Bol. Soc. Parana. Mat., 22 (2004), 75-96. doi: 10.5269/bspm.v22i2.7484.
[15]	T.-P. Tsai, On Leray's self-similiar solutions of the Navier-Stokes equations satisfying local energy estimates, Arch. Ration. Mech. Anal., 143 (1998), 29-51. Erratum 147 (1999), 363. doi: 10.1007/s002050050099.

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References:

[1]	D. Chae, Nonexistence of asymptocially self-similar singularities in the Euler and Navier-Stokes equations, Math. Ann., 338 (2007), 435-449. doi: 10.1007/s00208-007-0082-6.
[2]	F. Crispo and P. Maremonti, An interpolation inequality in exterior domains, Rend. Sem. Mat. Univ. Padova, 112 (2004), 11-39.
[3]	D. Fang, B. Han and T. Zhang, Global wellposedness result for density-dependent incompressible viscous fluid in $\mathbb{R}^2$ with linearly growing initial velocity, Math. Meth. Appl. Sciences, 36 (2013), 921-935. doi: 10.1002/mma.2649.
[4]	G. P. Galdi, An Introduction to the Navier-Stokes Initial-boundary Value Problem, Adv. Math. Fluid Mech., Birkhäuser, Basel, 2000.
[5]	G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-state Problems, 2nd edition. Springer Monographs in Mathematics, New York, 2011. doi: 10.1007/978-0-387-09620-9.
[6]	J. G. Heywood, The Navier-Stokes equations: on the existence, regularity and decay of solutions, Indiana Univ. Math. J., 29 (1980), 639-681. doi: 10.1512/iumj.1980.29.29048.
[7]	M. Hieber and O. Sawada, The Navier-Stokes equations in $\mathbb{R}^{N}$ with linearly growing initial data, Arch. Rational Mech. Anal., 175 (2005), 269-285. doi: 10.1007/s00205-004-0347-0.
[8]	M. Hieber, A. Rhandi and O. Sawada, The Navier-Stokes flow for globally Lipschitz continuous initial data, in RIMS Kôkyûroku Bessatsu, vol. B1, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007, 159-165.
[9]	T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle, Arch. Rational Mech. Anal., 150 (1999), 307-348. doi: 10.1007/s002050050190.
[10]	O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Science Publishers, New York-London (1963).
[11]	J. Leray, Sur le mouvement d'un liquide visquex emplissant l'espace, Acta Math., 63 (1996), 193-248. doi: 10.1007/BF02547354.
[12]	J. L. Lions, Espaces intermédiaires entre espaces Hilbertiens et applications, Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine, 2 (1958), 419-432.
[13]	J. Nečas, M. Růžička and V. Šverák, On self-similiar solutions of the Navier-Stokes equations, Acta Math., 176 (1996), 283-294.
[14]	O. Sawada, The Navier-Stokes flow with linearly growing initial velocity in the whole space, Bol. Soc. Parana. Mat., 22 (2004), 75-96. doi: 10.5269/bspm.v22i2.7484.
[15]	T.-P. Tsai, On Leray's self-similiar solutions of the Navier-Stokes equations satisfying local energy estimates, Arch. Ration. Mech. Anal., 143 (1998), 29-51. Erratum 147 (1999), 363. doi: 10.1007/s002050050099.

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