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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

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, 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.  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), 1-35. 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), 228-259. 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), 2233-2247. 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, (eds. S. Friedlander and D. Serre), Elsevier, 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), 245-267. 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), 28-68. 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), 321-364.  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), 2775-2792. 321-364. 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), 269-285. 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), 481-491. 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), 471-480. doi: 10.1007/BF01174182.  Google Scholar

[19]

H. Koch and D. Tataru, Wellposedness for the Navier-Stokes equations, Advances Math., 157 (2001), 22-35. 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), 3859-3873. 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), 455-461.  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), 225-237. doi: 10.1007/s00205-010-0360-4.  Google Scholar

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.  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), 1-35. 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), 228-259. 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), 2233-2247. 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, (eds. S. Friedlander and D. Serre), Elsevier, 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), 245-267. 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), 28-68. 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), 321-364.  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), 2775-2792. 321-364. 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), 269-285. 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), 481-491. 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), 471-480. doi: 10.1007/BF01174182.  Google Scholar

[19]

H. Koch and D. Tataru, Wellposedness for the Navier-Stokes equations, Advances Math., 157 (2001), 22-35. 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), 3859-3873. 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), 455-461.  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), 225-237. doi: 10.1007/s00205-010-0360-4.  Google Scholar

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