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Global regularity to the Navier-Stokes equations for a class of large initial data

    Bin Han Affiliation
    ; Yukang Chen Affiliation

Abstract

In [5], Chemin, Gallagher and Paicu proved the global regularity of solutions to the classical Navier-Stokes equations with a class of large initial data on T2 × R. This data varies slowly in vertical variable and has a norm which blows up as the small parameter ( represented by ǫ in the paper) tends to zero. However, to the best of our knowledge, the result is still unclear for the whole spaces R3. In this paper, we consider the generalized Navier-Stokes equations on Rn(n ≥ 3):


∂tu + u · ∇u + Dsu + ∇P = 0, div u = 0.


For some suitable number s, we prove that the Cauchy problem with initial data of the form u0ǫ(x) = (v0h(xǫ), ǫ−1v0n(xǫ))T , xǫ = (xh, ǫxn)T , is globally well-posed for all small ǫ > 0, provided that the initial velocity profile v0 is analytic in xn and certain norm of v0 is sufficiently small but independent of ǫ. In particular, our result is true for the n-dimensional classical Navier-Stokes equations with n ≥ 4 and the fractional Navier-Stokes equations with 1 ≤ s < 2 in 3D.

Keyword : large data, global solution, slowly varying

How to Cite
Han, B., & Chen, Y. (2018). Global regularity to the Navier-Stokes equations for a class of large initial data. Mathematical Modelling and Analysis, 23(2), 262-286. https://doi.org/10.3846/mma.2018.017
Published
Apr 18, 2018
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References

[1] J.M. Bony. Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Annales Scinentifiques de l’Ecole Normale Supérieure, 14(2):209–246, 1981. https://doi.org/10.24033/asens.1404

[2] J. Bourgain and N. Pavlovic. Ill-posedness of the Navier-Stokes equations in a critical space in 3D. Journal of Functional Analysis, 255(9):2233–2247, 2008. https://doi.org/10.1016/j.jfa.2008.07.008

[3] M. Cannone, Y. Meyer and F. Planchon. Solutions autosimilaries des équations de Navier-Stokes. Séminaire Équations aux Dérivées Partielles de l’École Polytechnique, 255:2233–2247, 1993.

[4] J. Chemin and I. Gallagher. Large, global solutions to the Navier-Stokes euqations, slowly varying in one direction. Transactions of the American Mathematical Society, 362:983–1012, 2011.

[5] J. Chemin, I. Gallagher and M. Paicu. Global regularity for some classes of large solutions to the Navier-Stokes equations. Annals of Mathematics, 173(2):983–1012, 2011. https://doi.org/10.4007/annals.2011.173.2.9

[6] J.Y. Chemin. Theorémés d’unicité pour le systéme de Navier-Stokes tridimensionnel. Journal d Analyse Mathematique, 77(1):27–50, 1999. https://doi.org/10.1007/BF02791256

[7] J.Y. Chemin. Le systeme de Navier-Stokes incompressible soixante dix ansapres Jean Leray. Séminaire et congrés, 9:99–123, 2004.

[8] D. Fang and B. Han. Global solution for the generalized anisotropic Navier-Stokes equations with large data. Mathematical Modelling and Analysis, 20(2):205–231, 2015. https://doi.org/10.3846/13926292.2015.1020894

[9] H. Fujita and T. Kato. On the Navier-Stokes initial value problem I. Archive for Rational Mechanics and Analysis, 16(4):269–315, 1964. https://doi.org/10.1007/BF00276188

[10] G. Gui, J. Huang and P. Zhang. Large global solutions to 3-D inhomogeneous Navier-Stokes equations slowly varying in one variable. Journal of Functional Analysis, 261(11):3181–3210, 2011. https://doi.org/10.1016/j.jfa.2011.07.026

[11] Y. Thomas Hou, Z. Lei and C. Li. Global regularity of the 3D axi-symmetric Navier-Stokes equations with anisotropic data. Comm. Partial Differential Equations, 33(9):1622–1637, 2008. https://doi.org/10.1080/03605300802108057

[12] D. Iftimie. The resolution of the Navier-Stokes equations in anisotropic spaces. Revista Matematica Ibero-Americana, 15(1):1–36, 1999. https://doi.org/10.4171/RMI/248

[13] 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(3):1083–1156, 2007. https://doi.org/10.1512/iumj.2007.56.2834

[14] T. Kato. Strong Lp solutions of the Navier-Stokes equations in Rm with applications to weak solutions. Mathematische Zeitschrift, 187(4):471–480, 1984. https://doi.org/10.1007/BF01174182

[15] H. Koch and D. Tataru. Well-posedness for the Navier-Stokes equations. Advances in Mathematics, 157(1):22–35, 2001.https://doi.org/10.1006/aima.2000.1937.

[16] Z. Lei and F. Lin. Global mild solutions of Navier-Stokes equations. Comm.Pure Appl. Math., 64(9):1297–1304, 2011. https://doi.org/10.1002/cpa.20361

[17] Z. Lei, F. Lin and Y. Zhou. Structure of Helicity and global solutions of incompressible Navier-Stokes equation. Archive for Rational Mechanics and Analysis, 218(3):1417–1430, 2015. https://doi.org/10.1007/s00205-015-0884-8

[18] J. Leray. Essai sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Mathematica, 63:193–248, 1934. https://doi.org/10.1007/BF02547354

[19] A. Mahalov and B. Nicolaenko. Global solvability of three dimensional Navier-Stokes equations with uniformly high initial vorticity. Uspekhi Mat. Nauk, 58:79–110, 2003.

[20] M. Paicu and Z. Zhang. Global regularity for the Navier-Stokes equations with large, slowly varying initial data in the vertical direction. Analysis of Partial Differential equation, 44:95–113, 2011.

[21] M. Paicu and Z. Zhang. Global well-posedness for 3D Navier-Stokes equations with ill-prepared initial data. J. Inst. Math. Jussieu, 13(2):395–411, 2014. https://doi.org/10.1017/S1474748013000212

[22] G. Raugel and G.R. Sell. Navier-Stokes equations on thin 3D domains. I: Global attractors and global regularity of solutions. Journal of the American Mathematical Society, 6(3):503–568, 1993. https://doi.org/10.2307/2152776

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