Existence and analyticity of Lei-Lin solution to the Navier-Stokes equations

Bae, Hantaek

doi:10.1090/S0002-9939-2015-12266-6

Existence and analyticity of Lei-Lin solution to the Navier-Stokes equations

Author: Hantaek Bae
Journal: Proc. Amer. Math. Soc. 143 (2015), 2887-2892
MSC (2000): Primary 35Q30, 76D03
DOI: https://doi.org/10.1090/S0002-9939-2015-12266-6
Published electronically: March 4, 2015
MathSciNet review: 3336613
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we prove the recent work of Lei-Lin in a slightly different setting, which enables us to prove analyticity of the solution.

[1] Hantaek Bae, Animikh Biswas, and Eitan Tadmor, Analyticity and decay estimates of the Navier-Stokes equations in critical Besov spaces, Arch. Ration. Mech. Anal. 205 (2012), no. 3, 963–991. MR 2960037, https://doi.org/10.1007/s00205-012-0532-5
[2] Animikh Biswas and David Swanson, Gevrey regularity of solutions to the 3-D Navier-Stokes equations with weighted 𝑙_{𝑝} initial data, Indiana Univ. Math. J. 56 (2007), no. 3, 1157–1188. MR 2333469, https://doi.org/10.1512/iumj.2007.56.2891
[3] Marco Cannone, Ondelettes, paraproduits et Navier-Stokes, Diderot Editeur, Paris, 1995 (French). With a preface by Yves Meyer. MR 1688096
[4] M. Cannone and F. Planchon, Self-similar solutions for Navier-Stokes equations in 𝑅³, Comm. Partial Differential Equations 21 (1996), no. 1-2, 179–193. MR 1373769, https://doi.org/10.1080/03605309608821179
[5] Jean-Yves Chemin, Théorèmes d’unicité pour le système de Navier-Stokes tridimensionnel, J. Anal. Math. 77 (1999), 27–50 (French). MR 1753481, https://doi.org/10.1007/BF02791256
[6]
L. Escauriaza, G. Serigin, V. Sverak,
$L_{3,\infty }$ solutions of Navier-Stokes equations and backward uniquness,
Uspekhi Mat. Nauk, 58 (2003), 3-44.
[7] C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal. 87 (1989), no. 2, 359–369. MR 1026858, https://doi.org/10.1016/0022-1236(89)90015-3
[8] Hiroshi Fujita and Tosio Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal. 16 (1964), 269–315. MR 166499, https://doi.org/10.1007/BF00276188
[9] Giulia Furioli, Pierre G. Lemarié-Rieusset, and Elide Terraneo, Unicité dans 𝐿³(ℝ³) et d’autres espaces fonctionnels limites pour Navier-Stokes, Rev. Mat. Iberoamericana 16 (2000), no. 3, 605–667 (French, with English and French summaries). MR 1813331, https://doi.org/10.4171/RMI/286
[10] Pierre Germain, Nataša Pavlović, and Gigliola Staffilani, Regularity of solutions to the Navier-Stokes equations evolving from small data in 𝐵𝑀𝑂⁻¹, Int. Math. Res. Not. IMRN 21 (2007), Art. ID rnm087, 35. MR 2352218, https://doi.org/10.1093/imrn/rnm087
[11] Zoran Grujić and Igor Kukavica, Space analyticity for the Navier-Stokes and related equations with initial data in 𝐿^{𝑝}, J. Funct. Anal. 152 (1998), no. 2, 447–466. MR 1607936, https://doi.org/10.1006/jfan.1997.3167
[12] Rafaela Guberović, Smoothness of Koch-Tataru solutions to the Navier-Stokes equations revisited, Discrete Contin. Dyn. Syst. 27 (2010), no. 1, 231–236. MR 2600769, https://doi.org/10.3934/dcds.2010.27.231
[13] Tosio Kato, Strong 𝐿^{𝑝}-solutions of the Navier-Stokes equation in 𝑅^{𝑚}, with applications to weak solutions, Math. Z. 187 (1984), no. 4, 471–480. MR 760047, https://doi.org/10.1007/BF01174182
[14] Herbert Koch and Daniel Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math. 157 (2001), no. 1, 22–35. MR 1808843, https://doi.org/10.1006/aima.2000.1937
[15] Zhen Lei and Fanghua Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math. 64 (2011), no. 9, 1297–1304. MR 2839302, https://doi.org/10.1002/cpa.20361
[16] P. G. Lemarié-Rieusset, Recent developments in the Navier-Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics, vol. 431, Chapman & Hall/CRC, Boca Raton, FL, 2002. MR 1938147
[17] Yves Le Jan and Alain Sol Sznitman, Cascades aléatoires et équations de Navier-Stokes, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 7, 823–826 (French, with English and French summaries). MR 1446587, https://doi.org/10.1016/S0764-4442(97)86952-2
[18] Hideyuki Miura and Okihiro Sawada, On the regularizing rate estimates of Koch-Tataru’s solution to the Navier-Stokes equations, Asymptot. Anal. 49 (2006), no. 1-2, 1–15. MR 2260554
[19] Marcel Oliver and Edriss S. Titi, Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations in 𝑅ⁿ, J. Funct. Anal. 172 (2000), no. 1, 1–18. MR 1749867, https://doi.org/10.1006/jfan.1999.3550
[20] Fabrice Planchon, Asymptotic behavior of global solutions to the Navier-Stokes equations in 𝑅³, Rev. Mat. Iberoamericana 14 (1998), no. 1, 71–93. MR 1639283, https://doi.org/10.4171/RMI/235