Well-posedness results for the Navier-Stokes equations in the rotational framework

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November 2013, 33(11&12): 5143-5151. doi: 10.3934/dcds.2013.33.5143

Well-posedness results for the Navier-Stokes equations in the rotational framework

Matthias Hieber ^1, and Sylvie Monniaux ^2,

1.	Fachbereich Mathematik, Angewandte Analysis, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289 Darmstadt
2.	LATP UMR 6632, CMI, Technopôle de Château-Gombert, 39 rue Frédéric Joliot-Curie, 13453 Marseille Cedex 13, France

Received January 2012 Revised July 2012 Published May 2013

Abstract
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Consider the Navier-Stokes equations in the rotational framework either on $\mathbb{R}^3$ or on open sets $\Omega \subset \mathbb{R}^3$ subject to Dirichlet boundary conditions. This paper discusses recent well-posedness and ill-posedness results for both situations.

Keywords: mild solutions., Coriolis force, Navier-Stokes equations, Dirichlet boundary conditions, Stokes-Coriolis semigroup.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: Matthias Hieber, Sylvie Monniaux. Well-posedness results for the Navier-Stokes equations in the rotational framework. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5143-5151. doi: 10.3934/dcds.2013.33.5143

References:

[1]	A. Babin, A. Mahalov and B. Nicolaenko, Regularity and integrability for the 3D Euler and Navier-Stokes equations for uniformly rotating fluids,, Asympt. Anal., 15 (1997), 103. Google Scholar
[2]	A. Babin, 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. doi: 10.1512/iumj.2001.50.2155. Google Scholar
[3]	I. Bejenaru and T. Tao, Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation,, J. Funct. Anal., 223 (2006), 228. doi: 10.1016/j.jfa.2005.08.004. Google Scholar
[4]	J. Bourgain and N. Pavlović, Ill-posedness of the Navier-Stokes equations in a critical space in 3D,, J. Funct. Anal., 255 (2008), 2233. doi: 10.1016/j.jfa.2008.07.008. Google Scholar
[5]	M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations,, Handbook of Mathematical Fluid Dynamics, 3 (2003). Google Scholar
[6]	C. Cao and E. Titi, Global wellposedness of the three dimensional viscous primitive equations of large scale ocean and atmosphere dynamics,, Annals of Math., 166 (2007), 245. doi: 10.4007/annals.2007.166.245. Google Scholar
[7]	J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, "Mathematical Geophysics,", Oxford University Press, (2006). Google Scholar
[8]	J. A. Goldstein, "Semigroups of Operators and Applications,", Oxford University Press, (1985). Google Scholar
[9]	Y. Giga, K. Inui and S. Matsui, On the Cauchy problem for the Navier-Stokes equations with nondecaying initial data,, Quaderni di Math., 4 (1999), 28. Google Scholar
[10]	Y. Giga, K. Inui, A. Mahalov and S. Matsui, Navier-Stokes equations in a rotating frame in $R^3$ with initial data nondecreasing at infinity,, Hokkaido Math. J., 35 (2006), 321. Google Scholar
[11]	Y. Giga, K. Inui, A. Mahalov and J. Saal, Uniform global solvability of the Navier-Stokes equations for nondecaying data,, Indiana Univ. Math. J., 57 (2008), 2775. doi: 10.1512/iumj.2008.57.3795. Google Scholar
[12]	Y. Giga, K. Inui, A. Mahalov, S. Matsui and J. Saal, Rotating NS-equations in $\mathbbR^3_+$ with initial data nondecreasing at infinity: The Ekman boundary layer problem,, Arch. Rat. Mech. Anal., (2007). Google Scholar
[13]	M. Hieber and S. Monniaux, Global solutions of the Navier-Stokes-Coriolis system in domains,, preprint, (2012). Google Scholar
[14]	M. Hieber and O. Sawada, The Navier-Stokes equations in $\mathbbR^n$ with linearly growing initial data,, Arch. Rational Mech. Anal., 175 (2005), 269. doi: 10.1007/s00205-004-0347-0. Google Scholar
[15]	M. Hieber and Y. Shibata, The Fujita-Kato approach to the Navier-Stokes equations in the rotational framework,, Math. Z., 265 (2010), 481. doi: 10.1007/s00209-009-0525-8. Google Scholar
[16]	T. Iwabuchi and R. Takada, Global well-posedness and ill-posedness for the Navier-Stokes equations with the Coriolis force in function spaces of Besov type,, preprint, (2011). doi: 10.1016/j.jmaa.2011.02.010. Google Scholar
[17]	T. Iwabuchi and R. Takada, Dispersive effects of the Coriolis force and the local well-posedness for the Navier-Stokes equations in the rotational framework,, preprint, (2011). Google Scholar
[18]	T. Kato, Strong $L^p$-solutions of Navier-Stokes equations in $\mathbbR^n$ with applications to weak solutions,, Math. Z., 187 (1984), 471. doi: 10.1007/BF01174182. Google Scholar
[19]	H. Koch and D. Tataru, Wellposedness for the Navier-Stokes equations,, Advances Math., 157 (2001), 22. doi: 10.1006/aima.2000.1937. Google Scholar
[20]	P. Konieczny and T. Yoneda, On the dispersive effect of the Coriolis force for stationary Navier-Stokes equations,, J. Diff. Equ., 250 (2011), 3859. doi: 10.1016/j.jde.2011.01.003. Google Scholar
[21]	A. Majda, "Introduction to PDEs and Waves for the Atmosphere and Ocean,", Courant Lecture Notes in Math., (2003). Google Scholar
[22]	S. Monniaux, Navier-Stokes equations in arbitrary domains: The Fujita-Kato scheme,, Math. Res. Lett., 13 (2006), 455. Google Scholar
[23]	T. Yoneda, Long-time solvability of the Navier-Stokes equations in a rotating frame with spatially almost periodic large data,, Arch. Rational Mech. Anal., 200 (2011), 225. doi: 10.1007/s00205-010-0360-4. Google Scholar

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References:

[1]	A. Babin, A. Mahalov and B. Nicolaenko, Regularity and integrability for the 3D Euler and Navier-Stokes equations for uniformly rotating fluids,, Asympt. Anal., 15 (1997), 103. Google Scholar
[2]	A. Babin, 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. doi: 10.1512/iumj.2001.50.2155. Google Scholar
[3]	I. Bejenaru and T. Tao, Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation,, J. Funct. Anal., 223 (2006), 228. doi: 10.1016/j.jfa.2005.08.004. Google Scholar
[4]	J. Bourgain and N. Pavlović, Ill-posedness of the Navier-Stokes equations in a critical space in 3D,, J. Funct. Anal., 255 (2008), 2233. doi: 10.1016/j.jfa.2008.07.008. Google Scholar
[5]	M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations,, Handbook of Mathematical Fluid Dynamics, 3 (2003). Google Scholar
[6]	C. Cao and E. Titi, Global wellposedness of the three dimensional viscous primitive equations of large scale ocean and atmosphere dynamics,, Annals of Math., 166 (2007), 245. doi: 10.4007/annals.2007.166.245. Google Scholar
[7]	J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, "Mathematical Geophysics,", Oxford University Press, (2006). Google Scholar
[8]	J. A. Goldstein, "Semigroups of Operators and Applications,", Oxford University Press, (1985). Google Scholar
[9]	Y. Giga, K. Inui and S. Matsui, On the Cauchy problem for the Navier-Stokes equations with nondecaying initial data,, Quaderni di Math., 4 (1999), 28. Google Scholar
[10]	Y. Giga, K. Inui, A. Mahalov and S. Matsui, Navier-Stokes equations in a rotating frame in $R^3$ with initial data nondecreasing at infinity,, Hokkaido Math. J., 35 (2006), 321. Google Scholar
[11]	Y. Giga, K. Inui, A. Mahalov and J. Saal, Uniform global solvability of the Navier-Stokes equations for nondecaying data,, Indiana Univ. Math. J., 57 (2008), 2775. doi: 10.1512/iumj.2008.57.3795. Google Scholar
[12]	Y. Giga, K. Inui, A. Mahalov, S. Matsui and J. Saal, Rotating NS-equations in $\mathbbR^3_+$ with initial data nondecreasing at infinity: The Ekman boundary layer problem,, Arch. Rat. Mech. Anal., (2007). Google Scholar
[13]	M. Hieber and S. Monniaux, Global solutions of the Navier-Stokes-Coriolis system in domains,, preprint, (2012). Google Scholar
[14]	M. Hieber and O. Sawada, The Navier-Stokes equations in $\mathbbR^n$ with linearly growing initial data,, Arch. Rational Mech. Anal., 175 (2005), 269. doi: 10.1007/s00205-004-0347-0. Google Scholar
[15]	M. Hieber and Y. Shibata, The Fujita-Kato approach to the Navier-Stokes equations in the rotational framework,, Math. Z., 265 (2010), 481. doi: 10.1007/s00209-009-0525-8. Google Scholar
[16]	T. Iwabuchi and R. Takada, Global well-posedness and ill-posedness for the Navier-Stokes equations with the Coriolis force in function spaces of Besov type,, preprint, (2011). doi: 10.1016/j.jmaa.2011.02.010. Google Scholar
[17]	T. Iwabuchi and R. Takada, Dispersive effects of the Coriolis force and the local well-posedness for the Navier-Stokes equations in the rotational framework,, preprint, (2011). Google Scholar
[18]	T. Kato, Strong $L^p$-solutions of Navier-Stokes equations in $\mathbbR^n$ with applications to weak solutions,, Math. Z., 187 (1984), 471. doi: 10.1007/BF01174182. Google Scholar
[19]	H. Koch and D. Tataru, Wellposedness for the Navier-Stokes equations,, Advances Math., 157 (2001), 22. doi: 10.1006/aima.2000.1937. Google Scholar
[20]	P. Konieczny and T. Yoneda, On the dispersive effect of the Coriolis force for stationary Navier-Stokes equations,, J. Diff. Equ., 250 (2011), 3859. doi: 10.1016/j.jde.2011.01.003. Google Scholar
[21]	A. Majda, "Introduction to PDEs and Waves for the Atmosphere and Ocean,", Courant Lecture Notes in Math., (2003). Google Scholar
[22]	S. Monniaux, Navier-Stokes equations in arbitrary domains: The Fujita-Kato scheme,, Math. Res. Lett., 13 (2006), 455. Google Scholar
[23]	T. Yoneda, Long-time solvability of the Navier-Stokes equations in a rotating frame with spatially almost periodic large data,, Arch. Rational Mech. Anal., 200 (2011), 225. doi: 10.1007/s00205-010-0360-4. Google Scholar

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