2015, 14(2): 609-622. doi: 10.3934/cpaa.2015.14.609

Local and global existence results for the Navier-Stokes equations in the rotational framework

1. 

Department of Mathematics, Zhejiang University, Hangzhou 310027

2. 

Department of Mathematics, Fudan University, Shanghai, 200433, China

3. 

Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgartenstr. 7, D-64289 Darmstadt

Received  June 2014 Revised  October 2014 Published  December 2014

Consider the equations of Navier-Stokes in $R^3$ in the rotational setting, i.e. with Coriolis force. It is shown that this set of equations admits a unique, global mild solution provided only the horizontal components of the initial data are small with respect to the norm the Fourier-Besov space $\dot{FB}_{p,r}^{2-3/p}(R^3)$, where $p \in [2,\infty]$ and $r \in [1,\infty)$.
Citation: Daoyuan Fang, Bin Han, Matthias Hieber. Local and global existence results for the Navier-Stokes equations in the rotational framework. Communications on Pure & Applied Analysis, 2015, 14 (2) : 609-622. doi: 10.3934/cpaa.2015.14.609
References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Springer Grundlehren der mathematischen Wissenschaften, (2011). doi: 10.1007/978-3-642-16830-7.

[2]

A. Babin, A. Mahalov and B. Nicolaenko, Regularity and integrability of 3D Euler and Navier-Stokes equations for rotating fluids,, \emph{Asymptot. Anal.}, 15 (1997), 103.

[3]

A. Babin, A. Mahalov and B. Nicolaenko, Global regularity of 3D rotating Navier-Stokes equations for resonant domains,, \emph{Indiana Univ. Math. J.}, 48 (1999), 1133. doi: 10.1016/S0893-9659(99)00208-6.

[4]

J. M. Bony, Calcul symbolique et propagation des singularités pour équations aux dérivées partielles nonlinéaires,, \emph{Ann. Sci. L'$\acuteE$cole Normale Sup$\acutee$rieure}, 14 (1981), 209.

[5]

M. Cannone and Y. Meyer, Littlewood-Paley decompositions and Navier-Stokes Equations,, \emph{Methods and Application in Analysis}, 2 (1997), 307.

[6]

J. Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Mathematical Geophysics,, Oxford Lecture Series in Mathematics and its Applications, (2006).

[7]

M. Cannone and G. Karch, Smooth or singular solutions to the Navier-Stokes system,, \emph{J. Diff. Equ.}, (2004), 247. doi: 10.1016/j.jde.2003.10.003.

[8]

Q. Chen, C. Miao and Z. Zhang, Global well-posedness for the 3D rotating Navier-Stokes equations with highly oscillating initial data,, \emph{Pacific Journal of Mathematics}, (2013), 263. doi: 10.2140/pjm.2013.262.263.

[9]

D. Fang, B. Han and M. Hieber, Global existence results for the Navier-Stokes equations in the rotational framework in Fourier-Besov spaces,, in W. Arendt, ().

[10]

D. Fang, S. Wang and T. Zhang, Wellposedness for anisotropic rotating fuid equations,, \emph{Appl. Math. J. Chinese Univ.}, 27 (2012), 9. doi: 10.1007/s11766-012-2534-3.

[11]

Y. Giga, K. Inui, A. Mahalov and J. Saal, Global solvabiliy of the Navier-Stokes equations in spaces based on sum-closed frequency sets,, \emph{Adv. Diff. Equ.}, (2007), 721.

[12]

Y. Giga, K. Inui, A. Mahalov and J. Saal, Uniform global solvability of the rotating Navier-Stokes equations for nondecaying initial data,, \emph{Indiana Univ. Math. J.}, (2008), 2775. doi: 10.1512/iumj.2008.57.3795.

[13]

G. Gui and P. Zhang, Stability to the global solutions of 3-D Navier-Stokes equations,, \emph{Adv. Math.}, (2010), 1248. doi: 10.1016/j.aim.2010.03.022.

[14]

M. Hieber and Y. Shibata, The Fujita-Kato approach to the Navier-Stokes equations in the rotational framework,, \emph{Math. Z.}, (2010), 481. doi: 10.1007/s00209-009-0525-8.

[15]

Y. Koh, S. Lee and R. Takada, Dispersive estimates for the Navier-Stokes equations in the rotational framework,, \emph{Adv. Diff. Equations}, 19 (2014), 857.

[16]

P. Konieczny and T. Yoneda, On dispersive effect of the Coriolis force for the stationary Navier-Stokes equations,, \emph{J. Diff. Equ.}, (2011), 3859. doi: 10.1016/j.jde.2011.01.003.

[17]

A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean,, Courant Lecture Notes in Math., (2003).

[18]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces,, \emph{Comm. Math. Phys.}, (2011), 713. doi: 10.1007/s00220-011-1350-6.

[19]

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,, \emph{Journal of Functional Analysis}, 5 (2014), 1321. doi: 10.1016/j.jfa.2014.05.022.

[20]

H. Triebel, Theory of Function Spaces,, Monographs in Mathematics, (1983). doi: 10.1007/978-3-0346-0416-1.

[21]

T. Zhang, Global wellposedness problem for the 3-D incompressible anisotropic Navier-Stokes equations in an anisotropic space,, \emph{Comm. Math. Phys.}, (2009), 211. doi: 10.1007/s00220-008-0631-1.

show all references

References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Springer Grundlehren der mathematischen Wissenschaften, (2011). doi: 10.1007/978-3-642-16830-7.

[2]

A. Babin, A. Mahalov and B. Nicolaenko, Regularity and integrability of 3D Euler and Navier-Stokes equations for rotating fluids,, \emph{Asymptot. Anal.}, 15 (1997), 103.

[3]

A. Babin, A. Mahalov and B. Nicolaenko, Global regularity of 3D rotating Navier-Stokes equations for resonant domains,, \emph{Indiana Univ. Math. J.}, 48 (1999), 1133. doi: 10.1016/S0893-9659(99)00208-6.

[4]

J. M. Bony, Calcul symbolique et propagation des singularités pour équations aux dérivées partielles nonlinéaires,, \emph{Ann. Sci. L'$\acuteE$cole Normale Sup$\acutee$rieure}, 14 (1981), 209.

[5]

M. Cannone and Y. Meyer, Littlewood-Paley decompositions and Navier-Stokes Equations,, \emph{Methods and Application in Analysis}, 2 (1997), 307.

[6]

J. Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Mathematical Geophysics,, Oxford Lecture Series in Mathematics and its Applications, (2006).

[7]

M. Cannone and G. Karch, Smooth or singular solutions to the Navier-Stokes system,, \emph{J. Diff. Equ.}, (2004), 247. doi: 10.1016/j.jde.2003.10.003.

[8]

Q. Chen, C. Miao and Z. Zhang, Global well-posedness for the 3D rotating Navier-Stokes equations with highly oscillating initial data,, \emph{Pacific Journal of Mathematics}, (2013), 263. doi: 10.2140/pjm.2013.262.263.

[9]

D. Fang, B. Han and M. Hieber, Global existence results for the Navier-Stokes equations in the rotational framework in Fourier-Besov spaces,, in W. Arendt, ().

[10]

D. Fang, S. Wang and T. Zhang, Wellposedness for anisotropic rotating fuid equations,, \emph{Appl. Math. J. Chinese Univ.}, 27 (2012), 9. doi: 10.1007/s11766-012-2534-3.

[11]

Y. Giga, K. Inui, A. Mahalov and J. Saal, Global solvabiliy of the Navier-Stokes equations in spaces based on sum-closed frequency sets,, \emph{Adv. Diff. Equ.}, (2007), 721.

[12]

Y. Giga, K. Inui, A. Mahalov and J. Saal, Uniform global solvability of the rotating Navier-Stokes equations for nondecaying initial data,, \emph{Indiana Univ. Math. J.}, (2008), 2775. doi: 10.1512/iumj.2008.57.3795.

[13]

G. Gui and P. Zhang, Stability to the global solutions of 3-D Navier-Stokes equations,, \emph{Adv. Math.}, (2010), 1248. doi: 10.1016/j.aim.2010.03.022.

[14]

M. Hieber and Y. Shibata, The Fujita-Kato approach to the Navier-Stokes equations in the rotational framework,, \emph{Math. Z.}, (2010), 481. doi: 10.1007/s00209-009-0525-8.

[15]

Y. Koh, S. Lee and R. Takada, Dispersive estimates for the Navier-Stokes equations in the rotational framework,, \emph{Adv. Diff. Equations}, 19 (2014), 857.

[16]

P. Konieczny and T. Yoneda, On dispersive effect of the Coriolis force for the stationary Navier-Stokes equations,, \emph{J. Diff. Equ.}, (2011), 3859. doi: 10.1016/j.jde.2011.01.003.

[17]

A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean,, Courant Lecture Notes in Math., (2003).

[18]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces,, \emph{Comm. Math. Phys.}, (2011), 713. doi: 10.1007/s00220-011-1350-6.

[19]

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,, \emph{Journal of Functional Analysis}, 5 (2014), 1321. doi: 10.1016/j.jfa.2014.05.022.

[20]

H. Triebel, Theory of Function Spaces,, Monographs in Mathematics, (1983). doi: 10.1007/978-3-0346-0416-1.

[21]

T. Zhang, Global wellposedness problem for the 3-D incompressible anisotropic Navier-Stokes equations in an anisotropic space,, \emph{Comm. Math. Phys.}, (2009), 211. doi: 10.1007/s00220-008-0631-1.

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