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General initial data for a class of parabolic equations including the curve shortening problem
Local well-posedness for Navier-Stokes equations with a class of ill-prepared initial data
1. | School of Mathematics,, Shanghai University of Finance and Economics, Shanghai 200433, China |
2. | School of Mathematical Sciences, Fudan University, Shanghai, 200433, China |
$ \begin{equation} \nonumber u_0^\epsilon(x) = (v_0^h(x_\epsilon), \epsilon^{-1}v_0^3(x_\epsilon))^T,\quad x_\epsilon = (x_h, \epsilon x_3)^T, \end{equation} $ |
$ \mathbb{R}^3 $ |
$ \epsilon > 0 $ |
$ v_0 $ |
$ x_3 $ |
$ \epsilon $ |
References:
[1] |
H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343. Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
J.-M. Bony,
Calcul symbolique et propagation des singularités pour équations aux dérivées partielles nonlinéaires, Ann. Sci. École Norm. Sup. (4), 14 (1981), 209-246.
doi: 10.24033/asens.1404. |
[3] |
J. Bourgain and N. Pavlović,
Ill-posedness of the Navier-Stokes equations in a critical space in 3D, Journal of Functional Analysis, 255 (2008), 2233-2247.
doi: 10.1016/j.jfa.2008.07.008. |
[4] |
M. Cannone, Y. Meyer and F. Planchon, Solutions autosimilaries des équations de Navier-Stokes, Séminaire sur les Équations aux Dérivées Partielles, École Polytech., Palaiseau, (1994), 12 pp.
doi: 10.1108/09533239410052824. |
[5] |
J.-Y. Chemin,
Le syst\`{e}me de Navier-Stokes incopressible soixante dix ans apr\`{e}s Jean Leray, Actes des Journées Mathématiques á la Mémoire de Jean Leray, Sémin. Congr., Soc. Math. France, Paris, 9 (2004), 99-123.
|
[6] |
J.-Y. Chemin,
Theorémés d'unicité pour le systéme de Navier-Stokes tridimensionnel, J. Anal. Math., 77 (2009), 27-50.
doi: 10.1007/BF02791256. |
[7] |
J.-Y. Chemin and I. Gallagher,
Large, global solutions to the Navier-Stokes euqations, slowly varying in one direction, Transactions of the American Mathematical Society, 362 (2010), 2859-2873.
doi: 10.1090/S0002-9947-10-04744-6. |
[8] |
J.-Y. Chemin, I. Gallagher and M. Paicu,
Global regularity for some classes of large solutions to the Navier-Stokes equations, Annals of Mathematics, 173 (2011), 983-1012.
doi: 10.4007/annals.2011.173.2.9. |
[9] |
J.-Y. Chemin and N. Lerner,
Flot de champs de vecteurs non Lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), 314-328.
doi: 10.1006/jdeq.1995.1131. |
[10] |
Y. K. Chen and B. Han,
Global regularity to the Navier-Stokes equations for a class of large initial data, Math. Model. Anal., 23 (2018), 262-286.
doi: 10.3846/mma.2018.017. |
[11] |
R. Danchin,
Local and global well-posedness results for flows of inhomogeneous viscous fluids, Advances in Differential Equations, 9 (2004), 353-386.
|
[12] |
D. Y. Fang and B. Han,
Global solution for the generalized anisotropic Navier-Stokes equations with Large data, Math. Model. Anal., 20 (2015), 205-231.
doi: 10.3846/13926292.2015.1020894. |
[13] |
H. Fujita and T. Kato,
On the Navier-Stokes initial value problem. I, Archive for Rational Mechanics and Analysis, 16 (1964), 269-315.
doi: 10.1007/BF00276188. |
[14] |
G. L. Gui, J. C. Huang and P. Zhang,
Large global solutions to 3-D inhomogeneous Navier-Stokes equations slowly varying in one variable, Journal of Functional Analysis, 261 (2011), 3181-3210.
doi: 10.1016/j.jfa.2011.07.026. |
[15] |
B. Han and Y. K. Chen,
Global reqularity to the 3D incompressible Navier-Stokes equations with large initial data, Mathematical Modelling and Analysis, 23 (2018), 262-286.
doi: 10.3846/mma.2018.017. |
[16] |
T. Y. Hou, Z. Lei and C. M. 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. |
[17] |
D. Iftimie,
The resolution of the Navier-Stokes equations in anisotropic spaces, Revista Matematica Ibero-Americana, 15 (1999), 1-36.
doi: 10.4171/RMI/248. |
[18] |
D. Iftimie, G. Raugel and G. R. Sell,
Navier-Stokes equations in thin 3D domains with the Navier boundary conditions, Indiana University Mathematical Journal, 56 (2007), 1083-1156.
doi: 10.1512/iumj.2007.56.2834. |
[19] |
T. Kato,
Strong $L^p$ solutions of the Navier-Stokes equations in $\mathbb{R}^m$ with applications to weak solutions, Mathematische Zeitschrift, 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[20] |
H. Koch and D. Tataru,
Well-posedness for the Navier-Stokes equations, Advances in Maththematics, 157 (2001), 22-35.
doi: 10.1006/aima.2000.1937. |
[21] |
Z. Lei and F. H. Lin,
Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math, 64 (2011), 1297-1304.
doi: 10.1002/cpa.20361. |
[22] |
Z. Lei, F.-H. Lin and Y. Zhou,
Structure of Helicity and global solutions of incompressible Navier-Stokes equation, Arch. Ration. Mech. Anal., 218 (2015), 1417-1430.
doi: 10.1007/s00205-015-0884-8. |
[23] |
J. Leray, Essai sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Mathematica, 63 (1933), 193-248. Google Scholar |
[24] |
A. S. Mahalov and V. P. Nicolaenko,
Global solvability of three dimensional Navier-Stokes equations with Uniformly high initial vorticity, Uspekhi Mat. Nauk, 58 (2003), 79-110.
doi: 10.1070/RM2003v058n02ABEH000611. |
[25] |
M. Paicu and Z. F. Zhang, Global regularity for the Navier-Stokes equations with large, slowly varying iitial data in the vertical direction, Analysis of Partial Differential equation, 4 (2011), 95-113. Google Scholar |
[26] |
G. Raugel and G. R. Sell,
Navier-Stokes equations on thin $3$D domains. I. Global attractors and global regularity of solutions, Journal of the American Mathematical Society, 6 (1993), 503-568.
doi: 10.2307/2152776. |
show all references
References:
[1] |
H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343. Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
J.-M. Bony,
Calcul symbolique et propagation des singularités pour équations aux dérivées partielles nonlinéaires, Ann. Sci. École Norm. Sup. (4), 14 (1981), 209-246.
doi: 10.24033/asens.1404. |
[3] |
J. Bourgain and N. Pavlović,
Ill-posedness of the Navier-Stokes equations in a critical space in 3D, Journal of Functional Analysis, 255 (2008), 2233-2247.
doi: 10.1016/j.jfa.2008.07.008. |
[4] |
M. Cannone, Y. Meyer and F. Planchon, Solutions autosimilaries des équations de Navier-Stokes, Séminaire sur les Équations aux Dérivées Partielles, École Polytech., Palaiseau, (1994), 12 pp.
doi: 10.1108/09533239410052824. |
[5] |
J.-Y. Chemin,
Le syst\`{e}me de Navier-Stokes incopressible soixante dix ans apr\`{e}s Jean Leray, Actes des Journées Mathématiques á la Mémoire de Jean Leray, Sémin. Congr., Soc. Math. France, Paris, 9 (2004), 99-123.
|
[6] |
J.-Y. Chemin,
Theorémés d'unicité pour le systéme de Navier-Stokes tridimensionnel, J. Anal. Math., 77 (2009), 27-50.
doi: 10.1007/BF02791256. |
[7] |
J.-Y. Chemin and I. Gallagher,
Large, global solutions to the Navier-Stokes euqations, slowly varying in one direction, Transactions of the American Mathematical Society, 362 (2010), 2859-2873.
doi: 10.1090/S0002-9947-10-04744-6. |
[8] |
J.-Y. Chemin, I. Gallagher and M. Paicu,
Global regularity for some classes of large solutions to the Navier-Stokes equations, Annals of Mathematics, 173 (2011), 983-1012.
doi: 10.4007/annals.2011.173.2.9. |
[9] |
J.-Y. Chemin and N. Lerner,
Flot de champs de vecteurs non Lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), 314-328.
doi: 10.1006/jdeq.1995.1131. |
[10] |
Y. K. Chen and B. Han,
Global regularity to the Navier-Stokes equations for a class of large initial data, Math. Model. Anal., 23 (2018), 262-286.
doi: 10.3846/mma.2018.017. |
[11] |
R. Danchin,
Local and global well-posedness results for flows of inhomogeneous viscous fluids, Advances in Differential Equations, 9 (2004), 353-386.
|
[12] |
D. Y. Fang and B. Han,
Global solution for the generalized anisotropic Navier-Stokes equations with Large data, Math. Model. Anal., 20 (2015), 205-231.
doi: 10.3846/13926292.2015.1020894. |
[13] |
H. Fujita and T. Kato,
On the Navier-Stokes initial value problem. I, Archive for Rational Mechanics and Analysis, 16 (1964), 269-315.
doi: 10.1007/BF00276188. |
[14] |
G. L. Gui, J. C. Huang and P. Zhang,
Large global solutions to 3-D inhomogeneous Navier-Stokes equations slowly varying in one variable, Journal of Functional Analysis, 261 (2011), 3181-3210.
doi: 10.1016/j.jfa.2011.07.026. |
[15] |
B. Han and Y. K. Chen,
Global reqularity to the 3D incompressible Navier-Stokes equations with large initial data, Mathematical Modelling and Analysis, 23 (2018), 262-286.
doi: 10.3846/mma.2018.017. |
[16] |
T. Y. Hou, Z. Lei and C. M. 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. |
[17] |
D. Iftimie,
The resolution of the Navier-Stokes equations in anisotropic spaces, Revista Matematica Ibero-Americana, 15 (1999), 1-36.
doi: 10.4171/RMI/248. |
[18] |
D. Iftimie, G. Raugel and G. R. Sell,
Navier-Stokes equations in thin 3D domains with the Navier boundary conditions, Indiana University Mathematical Journal, 56 (2007), 1083-1156.
doi: 10.1512/iumj.2007.56.2834. |
[19] |
T. Kato,
Strong $L^p$ solutions of the Navier-Stokes equations in $\mathbb{R}^m$ with applications to weak solutions, Mathematische Zeitschrift, 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[20] |
H. Koch and D. Tataru,
Well-posedness for the Navier-Stokes equations, Advances in Maththematics, 157 (2001), 22-35.
doi: 10.1006/aima.2000.1937. |
[21] |
Z. Lei and F. H. Lin,
Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math, 64 (2011), 1297-1304.
doi: 10.1002/cpa.20361. |
[22] |
Z. Lei, F.-H. Lin and Y. Zhou,
Structure of Helicity and global solutions of incompressible Navier-Stokes equation, Arch. Ration. Mech. Anal., 218 (2015), 1417-1430.
doi: 10.1007/s00205-015-0884-8. |
[23] |
J. Leray, Essai sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Mathematica, 63 (1933), 193-248. Google Scholar |
[24] |
A. S. Mahalov and V. P. Nicolaenko,
Global solvability of three dimensional Navier-Stokes equations with Uniformly high initial vorticity, Uspekhi Mat. Nauk, 58 (2003), 79-110.
doi: 10.1070/RM2003v058n02ABEH000611. |
[25] |
M. Paicu and Z. F. Zhang, Global regularity for the Navier-Stokes equations with large, slowly varying iitial data in the vertical direction, Analysis of Partial Differential equation, 4 (2011), 95-113. Google Scholar |
[26] |
G. Raugel and G. R. Sell,
Navier-Stokes equations on thin $3$D domains. I. Global attractors and global regularity of solutions, Journal of the American Mathematical Society, 6 (1993), 503-568.
doi: 10.2307/2152776. |
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