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 and Continuous Dynamical Systems, 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), 103-150.
[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), 1-35. doi: 10.1512/iumj.2001.50.2155.
[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), 228-259. doi: 10.1016/j.jfa.2005.08.004.
[4]	J. Bourgain and N. Pavlović, Ill-posedness of the Navier-Stokes equations in a critical space in 3D, J. Funct. Anal., 255 (2008), 2233-2247. doi: 10.1016/j.jfa.2008.07.008.
[5]	M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, Handbook of Mathematical Fluid Dynamics, (eds. S. Friedlander and D. Serre), Elsevier, 3 (2003).
[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), 245-267. doi: 10.4007/annals.2007.166.245.
[7]	J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, "Mathematical Geophysics," Oxford University Press, 2006.
[8]	J. A. Goldstein, "Semigroups of Operators and Applications," Oxford University Press, 1985.
[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), 28-68.
[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), 321-364.
[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), 2775-2792. 321-364. doi: 10.1512/iumj.2008.57.3795.
[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).
[13]	M. Hieber and S. Monniaux, Global solutions of the Navier-Stokes-Coriolis system in domains, preprint, 2012.
[14]	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.
[15]	M. Hieber and Y. Shibata, The Fujita-Kato approach to the Navier-Stokes equations in the rotational framework, Math. Z., 265 (2010), 481-491. doi: 10.1007/s00209-009-0525-8.
[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.
[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.
[18]	T. Kato, Strong $L^p$-solutions of Navier-Stokes equations in $\mathbb{R}^{N}$ with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182.
[19]	H. Koch and D. Tataru, Wellposedness for the Navier-Stokes equations, Advances Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937.
[20]	P. Konieczny and T. Yoneda, On the dispersive effect of the Coriolis force for stationary Navier-Stokes equations, J. Diff. Equ., 250 (2011), 3859-3873. doi: 10.1016/j.jde.2011.01.003.
[21]	A. Majda, "Introduction to PDEs and Waves for the Atmosphere and Ocean," Courant Lecture Notes in Math., 2003.
[22]	S. Monniaux, Navier-Stokes equations in arbitrary domains: The Fujita-Kato scheme, Math. Res. Lett., 13 (2006), 455-461.
[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), 225-237. doi: 10.1007/s00205-010-0360-4.

show all references

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), 103-150.
[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), 1-35. doi: 10.1512/iumj.2001.50.2155.
[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), 228-259. doi: 10.1016/j.jfa.2005.08.004.
[4]	J. Bourgain and N. Pavlović, Ill-posedness of the Navier-Stokes equations in a critical space in 3D, J. Funct. Anal., 255 (2008), 2233-2247. doi: 10.1016/j.jfa.2008.07.008.
[5]	M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, Handbook of Mathematical Fluid Dynamics, (eds. S. Friedlander and D. Serre), Elsevier, 3 (2003).
[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), 245-267. doi: 10.4007/annals.2007.166.245.
[7]	J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, "Mathematical Geophysics," Oxford University Press, 2006.
[8]	J. A. Goldstein, "Semigroups of Operators and Applications," Oxford University Press, 1985.
[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), 28-68.
[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), 321-364.
[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), 2775-2792. 321-364. doi: 10.1512/iumj.2008.57.3795.
[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).
[13]	M. Hieber and S. Monniaux, Global solutions of the Navier-Stokes-Coriolis system in domains, preprint, 2012.
[14]	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.
[15]	M. Hieber and Y. Shibata, The Fujita-Kato approach to the Navier-Stokes equations in the rotational framework, Math. Z., 265 (2010), 481-491. doi: 10.1007/s00209-009-0525-8.
[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.
[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.
[18]	T. Kato, Strong $L^p$-solutions of Navier-Stokes equations in $\mathbb{R}^{N}$ with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182.
[19]	H. Koch and D. Tataru, Wellposedness for the Navier-Stokes equations, Advances Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937.
[20]	P. Konieczny and T. Yoneda, On the dispersive effect of the Coriolis force for stationary Navier-Stokes equations, J. Diff. Equ., 250 (2011), 3859-3873. doi: 10.1016/j.jde.2011.01.003.
[21]	A. Majda, "Introduction to PDEs and Waves for the Atmosphere and Ocean," Courant Lecture Notes in Math., 2003.
[22]	S. Monniaux, Navier-Stokes equations in arbitrary domains: The Fujita-Kato scheme, Math. Res. Lett., 13 (2006), 455-461.
[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), 225-237. doi: 10.1007/s00205-010-0360-4.

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