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July  2016, 36(7): 3845-3856. doi: 10.3934/dcds.2016.36.3845

Global well-posedness of axisymmetric Navier-Stokes equations with one slow variable

Weimin Peng 1, and  Yi Zhou 2,

1. 

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
2. 

Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433

Received  March 2015 Revised  November 2015 Published  March 2016

This paper deals with the global well-posedness of axisymmetric Navier-Stokes equations with swirl. We prove that there exists a global solution of Navier-Stokes equations under some weighted energy for a class of large anisotropic initial data slowly varying in the vertical variable.
Keywords: global well-posedness, axisymmetric, swirl, large anisotropic data., Navier-Stokes equations.
Mathematics Subject Classification: Primary: 35Q30; Secondary: 76D03, 76D0.
Citation: Weimin Peng, Yi Zhou. Global well-posedness of axisymmetric Navier-Stokes equations with one slow variable. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3845-3856. doi: 10.3934/dcds.2016.36.3845
References:
[1]

M. Cannone, Chapter 3: Harmonic analysis tools for solving the incompressible navier-stokes equations, in Handbook of Mathmatical Fluid Dynamics, (Edited by S.J. Friedlander and D. Serre), Elsevier B. V., 3 (2004), 161-244.  Google Scholar

[2]

M. Cannone, Y. Meyer and F. Planchon, Solutions auto-similaires des équations de Navier-Stokes, Sémin. Équations aux Dérivées Partielles de I'École polytechnique, Expose, 8 (1994), 12pp.  Google Scholar

[3]

D. Chae and J. Lee, On the regularity of the axisymmetric solutions of the Navier-Stokes equations, Math. Z., 239 (2002), 645-671. doi: 10.1007/s002090100317.  Google Scholar

[4]

C. C. Chen, R. M. Strain, T. P. Tsai and H. T. Yau, Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations. II, Comm. Partial Differential Equations, 34 (2009), 203-232. doi: 10.1080/03605300902793956.  Google Scholar

[5]

Clay Mathematics Institute, Available, from: , ().   Google Scholar

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H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech Anal., 16 (1964), 269-315. doi: 10.1007/BF00276188.  Google Scholar

[7]

Y. Giga and T. Miyakama, Solutions in $L^r$ of the Navier-Stokes initial value problem, Arch. Ration. Mech. Anal., 89 (1985), 267-281. doi: 10.1007/BF00276875.  Google Scholar

[8]

T. Y. Hou, Z. Lei and C. Li, Global regularity of the 3D axi-symmetric Navier-Stokes equations with anisotropic data, Comm. Partial Differential Equations, 33 (2008), 1622-1637. doi: 10.1080/03605300802108057.  Google Scholar

[9]

T. Y. Hou and C. Li, Dynamic stability of the 3D axi-symmetric Navier-Stokes equations with swirl, Comm. Pure Appl. Math., 61 (2008), 661-697. doi: 10.1002/cpa.20212.  Google Scholar

[10]

S. Leonardi, J. Málek, J. Nečas and M. Pokorný, On axially symmetric flows in $ R^3$, Z. Anal. Anwendungen, 18 (1999), 639-649. doi: 10.4171/ZAA/903.  Google Scholar

[11]

O. A. Ladyzhenskaya, Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI),7 (1968), 155-177 (Russian).  Google Scholar

[12]

J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problems que pose I'hydrodynamique, Journal Math. Pures et Appliquées, 12 (1933), 1-82. Google Scholar

[13]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.  Google Scholar

[14]

H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937.  Google Scholar

[15]

T. Kato, Strong $L^p$-solutions of the Navier-Stokes equations in $\mathbbR^m$ with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182.  Google Scholar

[16]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27. Cambridge University, Cambridge, Mass. 2002.  Google Scholar

[17]

M. R. Ukhovskii and V. I. Iudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math. Mech., 32 (1968), 52-61. doi: 10.1016/0021-8928(68)90147-0.  Google Scholar

[18]

F. B. Weissler, The Navier-Stokes initial value problem in $L^p$, Arch. Ration. Mech. Anal., 74 (1980), 219-230. doi: 10.1007/BF00280539.  Google Scholar

show all references

References:
[1]

M. Cannone, Chapter 3: Harmonic analysis tools for solving the incompressible navier-stokes equations, in Handbook of Mathmatical Fluid Dynamics, (Edited by S.J. Friedlander and D. Serre), Elsevier B. V., 3 (2004), 161-244.  Google Scholar

[2]

M. Cannone, Y. Meyer and F. Planchon, Solutions auto-similaires des équations de Navier-Stokes, Sémin. Équations aux Dérivées Partielles de I'École polytechnique, Expose, 8 (1994), 12pp.  Google Scholar

[3]

D. Chae and J. Lee, On the regularity of the axisymmetric solutions of the Navier-Stokes equations, Math. Z., 239 (2002), 645-671. doi: 10.1007/s002090100317.  Google Scholar

[4]

C. C. Chen, R. M. Strain, T. P. Tsai and H. T. Yau, Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations. II, Comm. Partial Differential Equations, 34 (2009), 203-232. doi: 10.1080/03605300902793956.  Google Scholar

[5]

Clay Mathematics Institute, Available, from: , ().   Google Scholar

[6]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech Anal., 16 (1964), 269-315. doi: 10.1007/BF00276188.  Google Scholar

[7]

Y. Giga and T. Miyakama, Solutions in $L^r$ of the Navier-Stokes initial value problem, Arch. Ration. Mech. Anal., 89 (1985), 267-281. doi: 10.1007/BF00276875.  Google Scholar

[8]

T. Y. Hou, Z. Lei and C. Li, Global regularity of the 3D axi-symmetric Navier-Stokes equations with anisotropic data, Comm. Partial Differential Equations, 33 (2008), 1622-1637. doi: 10.1080/03605300802108057.  Google Scholar

[9]

T. Y. Hou and C. Li, Dynamic stability of the 3D axi-symmetric Navier-Stokes equations with swirl, Comm. Pure Appl. Math., 61 (2008), 661-697. doi: 10.1002/cpa.20212.  Google Scholar

[10]

S. Leonardi, J. Málek, J. Nečas and M. Pokorný, On axially symmetric flows in $ R^3$, Z. Anal. Anwendungen, 18 (1999), 639-649. doi: 10.4171/ZAA/903.  Google Scholar

[11]

O. A. Ladyzhenskaya, Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI),7 (1968), 155-177 (Russian).  Google Scholar

[12]

J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problems que pose I'hydrodynamique, Journal Math. Pures et Appliquées, 12 (1933), 1-82. Google Scholar

[13]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.  Google Scholar

[14]

H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937.  Google Scholar

[15]

T. Kato, Strong $L^p$-solutions of the Navier-Stokes equations in $\mathbbR^m$ with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182.  Google Scholar

[16]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27. Cambridge University, Cambridge, Mass. 2002.  Google Scholar

[17]

M. R. Ukhovskii and V. I. Iudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math. Mech., 32 (1968), 52-61. doi: 10.1016/0021-8928(68)90147-0.  Google Scholar

[18]

F. B. Weissler, The Navier-Stokes initial value problem in $L^p$, Arch. Ration. Mech. Anal., 74 (1980), 219-230. doi: 10.1007/BF00280539.  Google Scholar

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