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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 |
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: , ().
|
| [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. |
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. |
| [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: , ().
|
| [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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