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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

Keyan Wang 1,, and  Yao Xiao 2,

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
$ \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} $
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 $
.
Keywords: Local well-posedness, Navier-Stokes equations, ill-prepared.
Mathematics Subject Classification: Primary: 76D03, 76D05; Secondary: 35Q30.
Citation: Keyan Wang, Yao Xiao. Local well-posedness for Navier-Stokes equations with a class of ill-prepared initial data. Discrete & Continuous Dynamical Systems, 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

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.  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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