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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

 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.
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:
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##### 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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