American Institute of Mathematical Sciences

• Previous Article
Note on evolutionary free piston problem for Stokes equations with slip boundary conditions
• CPAA Home
• This Issue
• Next Article
Two sequences of solutions for indefinite superlinear-sublinear elliptic equations with nonlinear boundary conditions
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 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 $\mathbbR^2$ 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 $\mathbbR^n$ 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.

show all references

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 $\mathbbR^2$ 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 $\mathbbR^n$ 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.
 [1] Yinnian He, Kaitai Li. Nonlinear Galerkin approximation of the two dimensional exterior Navier-Stokes problem. Discrete and Continuous Dynamical Systems, 1996, 2 (4) : 467-482. doi: 10.3934/dcds.1996.2.467 [2] Siegfried Maier, Jürgen Saal. Stokes and Navier-Stokes equations with perfect slip on wedge type domains. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 1045-1063. doi: 10.3934/dcdss.2014.7.1045 [3] Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1355-1369. doi: 10.3934/dcdss.2013.6.1355 [4] Paolo Maremonti. A note on the Navier-Stokes IBVP with small data in $L^n$. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 255-267. doi: 10.3934/dcdss.2016.9.255 [5] Igor Kukavica, Mohammed Ziane. Regularity of the Navier-Stokes equation in a thin periodic domain with large data. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 67-86. doi: 10.3934/dcds.2006.16.67 [6] Ben-Yu Guo, Yu-Jian Jiao. Mixed generalized Laguerre-Fourier spectral method for exterior problem of Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 315-345. doi: 10.3934/dcdsb.2009.11.315 [7] 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 [8] Yoshihiro Shibata. On the local wellposedness of free boundary problem for the Navier-Stokes equations in an exterior domain. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1681-1721. doi: 10.3934/cpaa.2018081 [9] Takeshi Taniguchi. The existence and decay estimates of the solutions to $3$D stochastic Navier-Stokes equations with additive noise in an exterior domain. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4323-4341. doi: 10.3934/dcds.2014.34.4323 [10] Alessio Falocchi, Filippo Gazzola. Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1185-1200. doi: 10.3934/dcds.2021151 [11] Paolo Maremonti. On the Stokes problem in exterior domains: The maximum modulus theorem. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2135-2171. doi: 10.3934/dcds.2014.34.2135 [12] Yutaka Tsuzuki. Solvability of $p$-Laplacian parabolic logistic equations with constraints coupled with Navier-Stokes equations in 2D domains. Evolution Equations and Control Theory, 2014, 3 (1) : 191-206. doi: 10.3934/eect.2014.3.191 [13] Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure and Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35 [14] Donatella Donatelli, Eduard Feireisl, Antonín Novotný. On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions. Discrete and Continuous Dynamical Systems - B, 2010, 13 (4) : 783-798. doi: 10.3934/dcdsb.2010.13.783 [15] Reinhard Farwig, Paul Felix Riechwald. Regularity criteria for weak solutions of the Navier-Stokes system in general unbounded domains. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 157-172. doi: 10.3934/dcdss.2016.9.157 [16] Michal Beneš. Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains. Conference Publications, 2011, 2011 (Special) : 135-144. doi: 10.3934/proc.2011.2011.135 [17] Reinhard Farwig, Yasushi Taniuchi. Uniqueness of backward asymptotically almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1215-1224. doi: 10.3934/dcdss.2013.6.1215 [18] Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3343-3366. doi: 10.3934/dcds.2020408 [19] Claude W. Bardos, Trinh T. Nguyen, Toan T. Nguyen, Edriss S. Titi. The inviscid limit for the 2D Navier-Stokes equations in bounded domains. Kinetic and Related Models, 2022, 15 (3) : 317-340. doi: 10.3934/krm.2022004 [20] Joel Avrin. Global existence and regularity for the Lagrangian averaged Navier-Stokes equations with initial data in $H^{1//2}$. Communications on Pure and Applied Analysis, 2004, 3 (3) : 353-366. doi: 10.3934/cpaa.2004.3.353

2020 Impact Factor: 1.916