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May  2020, 40(5): 2987-3011. doi: 10.3934/dcds.2020158

## 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

* Corresponding author: Keyan Wang

Published  March 2020

Fund Project: The first author is supported by NSFC grant No. 11971290.

In this paper, we prove that for the ill-prepared initial data of the form
 $$$\nonumber u_0^\epsilon(x) = (v_0^h(x_\epsilon), \epsilon^{-1}v_0^3(x_\epsilon))^T,\quad x_\epsilon = (x_h, \epsilon x_3)^T,$$$
the Cauchy problem of the incompressible Navier-Stokes equations on
 $\mathbb{R}^3$
is locally well-posed for all
 $\epsilon > 0$
, provided that the initial velocity profile
 $v_0$
is analytic in
 $x_3$
but independent of
 $\epsilon$
.
Citation: Keyan Wang, Yao Xiao. Local well-posedness for Navier-Stokes equations with a class of ill-prepared initial data. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2987-3011. doi: 10.3934/dcds.2020158
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Advances in Differential Equations, 9 (2004), 353-386.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Advances in Differential Equations, 9 (2004), 353-386.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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