March  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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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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. Google Scholar

[5]

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

[6]

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

[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. Google Scholar

[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. Google Scholar

[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, (). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

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

[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. Google Scholar

[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. Google Scholar

[20]

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

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

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

[6]

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

[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. Google Scholar

[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. Google Scholar

[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, (). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

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

[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. Google Scholar

[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. Google Scholar

[20]

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

[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. Google Scholar

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