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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

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 & 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,, \emph{Math. Ann.}, 338 (2007), 435.  doi: 10.1007/s00208-007-0082-6.  Google Scholar

[2]

F. Crispo and P. Maremonti, An interpolation inequality in exterior domains,, \emph{Rend. Sem. Mat. Univ. Padova}, 112 (2004), 11.   Google Scholar

[3]

D. Fang, B. Han and T. Zhang, Global wellposedness result for density-dependent incompressible viscous fluid in $\mathbbR^2$ with linearly growing initial velocity,, \emph{Math. Meth. Appl. Sciences}, 36 (2013), 921.  doi: 10.1002/mma.2649.  Google Scholar

[4]

G. P. Galdi, An Introduction to the Navier-Stokes Initial-boundary Value Problem,, Adv. Math. Fluid Mech., (2000).   Google Scholar

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G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-state Problems,, 2$^{nd}$ edition. Springer Monographs in Mathematics, (2011).  doi: 10.1007/978-0-387-09620-9.  Google Scholar

[6]

J. G. Heywood, The Navier-Stokes equations: on the existence, regularity and decay of solutions,, \emph{Indiana Univ. Math. J.}, 29 (1980), 639.  doi: 10.1512/iumj.1980.29.29048.  Google Scholar

[7]

M. Hieber and O. Sawada, The Navier-Stokes equations in $\mathbbR^n$ with linearly growing initial data,, \emph{Arch. Rational Mech. Anal.}, 175 (2005), 269.  doi: 10.1007/s00205-004-0347-0.  Google Scholar

[8]

M. Hieber, A. Rhandi and O. Sawada, The Navier-Stokes flow for globally Lipschitz continuous initial data,, in \emph{RIMS K\^oky\^uroku Bessatsu}, (2007), 159.   Google Scholar

[9]

T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle,, \emph{Arch. Rational Mech. Anal.}, 150 (1999), 307.  doi: 10.1007/s002050050190.  Google Scholar

[10]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, Gordon and Breach Science Publishers, (1963).   Google Scholar

[11]

J. Leray, Sur le mouvement d'un liquide visquex emplissant l'espace,, \emph{Acta Math.}, 63 (1996), 193.  doi: 10.1007/BF02547354.  Google Scholar

[12]

J. L. Lions, Espaces intermédiaires entre espaces Hilbertiens et applications,, \emph{Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine}, 2 (1958), 419.   Google Scholar

[13]

J. Nečas, M. Růžička and V. Šverák, On self-similiar solutions of the Navier-Stokes equations,, \emph{Acta Math.}, 176 (1996), 283.   Google Scholar

[14]

O. Sawada, The Navier-Stokes flow with linearly growing initial velocity in the whole space,, \emph{Bol. Soc. Parana. Mat.}, 22 (2004), 75.  doi: 10.5269/bspm.v22i2.7484.  Google Scholar

[15]

T.-P. Tsai, On Leray's self-similiar solutions of the Navier-Stokes equations satisfying local energy estimates,, \emph{Arch. Ration. Mech. Anal.}, 143 (1998), 29.  doi: 10.1007/s002050050099.  Google Scholar

show all references

References:
[1]

D. Chae, Nonexistence of asymptocially self-similar singularities in the Euler and Navier-Stokes equations,, \emph{Math. Ann.}, 338 (2007), 435.  doi: 10.1007/s00208-007-0082-6.  Google Scholar

[2]

F. Crispo and P. Maremonti, An interpolation inequality in exterior domains,, \emph{Rend. Sem. Mat. Univ. Padova}, 112 (2004), 11.   Google Scholar

[3]

D. Fang, B. Han and T. Zhang, Global wellposedness result for density-dependent incompressible viscous fluid in $\mathbbR^2$ with linearly growing initial velocity,, \emph{Math. Meth. Appl. Sciences}, 36 (2013), 921.  doi: 10.1002/mma.2649.  Google Scholar

[4]

G. P. Galdi, An Introduction to the Navier-Stokes Initial-boundary Value Problem,, Adv. Math. Fluid Mech., (2000).   Google Scholar

[5]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-state Problems,, 2$^{nd}$ edition. Springer Monographs in Mathematics, (2011).  doi: 10.1007/978-0-387-09620-9.  Google Scholar

[6]

J. G. Heywood, The Navier-Stokes equations: on the existence, regularity and decay of solutions,, \emph{Indiana Univ. Math. J.}, 29 (1980), 639.  doi: 10.1512/iumj.1980.29.29048.  Google Scholar

[7]

M. Hieber and O. Sawada, The Navier-Stokes equations in $\mathbbR^n$ with linearly growing initial data,, \emph{Arch. Rational Mech. Anal.}, 175 (2005), 269.  doi: 10.1007/s00205-004-0347-0.  Google Scholar

[8]

M. Hieber, A. Rhandi and O. Sawada, The Navier-Stokes flow for globally Lipschitz continuous initial data,, in \emph{RIMS K\^oky\^uroku Bessatsu}, (2007), 159.   Google Scholar

[9]

T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle,, \emph{Arch. Rational Mech. Anal.}, 150 (1999), 307.  doi: 10.1007/s002050050190.  Google Scholar

[10]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, Gordon and Breach Science Publishers, (1963).   Google Scholar

[11]

J. Leray, Sur le mouvement d'un liquide visquex emplissant l'espace,, \emph{Acta Math.}, 63 (1996), 193.  doi: 10.1007/BF02547354.  Google Scholar

[12]

J. L. Lions, Espaces intermédiaires entre espaces Hilbertiens et applications,, \emph{Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine}, 2 (1958), 419.   Google Scholar

[13]

J. Nečas, M. Růžička and V. Šverák, On self-similiar solutions of the Navier-Stokes equations,, \emph{Acta Math.}, 176 (1996), 283.   Google Scholar

[14]

O. Sawada, The Navier-Stokes flow with linearly growing initial velocity in the whole space,, \emph{Bol. Soc. Parana. Mat.}, 22 (2004), 75.  doi: 10.5269/bspm.v22i2.7484.  Google Scholar

[15]

T.-P. Tsai, On Leray's self-similiar solutions of the Navier-Stokes equations satisfying local energy estimates,, \emph{Arch. Ration. Mech. Anal.}, 143 (1998), 29.  doi: 10.1007/s002050050099.  Google Scholar

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