# American Institute of Mathematical Sciences

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

 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.
Citation: Weimin Peng, Yi Zhou. Global well-posedness of axisymmetric Navier-Stokes equations with one slow variable. Discrete and 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. [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. [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. [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. [5] Clay Mathematics Institute, Available from: http://www.claymath.org/millennium-problems/navier-stokes-equation. [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. [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. [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. [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. [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. [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). [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. [13] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354. [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. [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. [16] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27. Cambridge University, Cambridge, Mass. 2002. [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. [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.

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##### 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. [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. [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. [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. [5] Clay Mathematics Institute, Available from: http://www.claymath.org/millennium-problems/navier-stokes-equation. [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. [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. [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. [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. [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. [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). [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. [13] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354. [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. [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. [16] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27. Cambridge University, Cambridge, Mass. 2002. [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. [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.
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