Random data Cauchy theory for the incompressible three dimensional Navier–Stokes equations
HTML articles powered by AMS MathViewer
- by Ting Zhang and Daoyuan Fang PDF
- Proc. Amer. Math. Soc. 139 (2011), 2827-2837 Request permission
Abstract:
We study the existence and uniqueness of the strong solution for the incompressible Navier–Stokes equations with the $L^2$ initial data and the periodic space domain $\mathbb {T}^3$. After a suitable randomization, we are able to construct the local unique strong solution for a large set of initial data in $L^2$. Furthermore, if $\|u_0\|_{L^2}$ is small, we show that the probability for the global existence and uniqueness of the solution is large.References
- Herbert Amann, On the strong solvability of the Navier-Stokes equations, J. Math. Fluid Mech. 2 (2000), no. 1, 16–98. MR 1755865, DOI 10.1007/s000210050018
- Antoine Ayache and Nikolay Tzvetkov, $L^p$ properties for Gaussian random series, Trans. Amer. Math. Soc. 360 (2008), no. 8, 4425–4439. MR 2395179, DOI 10.1090/S0002-9947-08-04456-5
- Nicolas Burq and Nikolay Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math. 173 (2008), no. 3, 449–475. MR 2425133, DOI 10.1007/s00222-008-0124-z
- M. Cannone, Y. Meyer, and F. Planchon, Solutions auto-similaires des équations de Navier-Stokes, Séminaire sur les Équations aux Dérivées Partielles, 1993–1994, École Polytech., Palaiseau, 1994, pp. Exp. No. VIII, 12 (French). MR 1300903, DOI 10.1108/09533239410052824
- Reinhard Farwig and Hermann Sohr, Optimal initial value conditions for the existence of local strong solutions of the Navier-Stokes equations, Math. Ann. 345 (2009), no. 3, 631–642. MR 2534111, DOI 10.1007/s00208-009-0368-y
- Hiroshi Fujita and Tosio Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal. 16 (1964), 269–315. MR 166499, DOI 10.1007/BF00276188
- Yoshikazu Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations 62 (1986), no. 2, 186–212. MR 833416, DOI 10.1016/0022-0396(86)90096-3
- Eberhard Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951), 213–231 (German). MR 50423, DOI 10.1002/mana.3210040121
- Tosio Kato, Strong $L^{p}$-solutions of the Navier-Stokes equation in $\textbf {R}^{m}$, with applications to weak solutions, Math. Z. 187 (1984), no. 4, 471–480. MR 760047, DOI 10.1007/BF01174182
- A. Khintchine, Über dyadische Brüche, Math. Z. 18 (1923), no. 1, 109–116 (German). MR 1544623, DOI 10.1007/BF01192399
- A.A. Kiselev, O.A. Ladyzenskaya, On the existence of uniqueness of solutions of the non-stationary problems for flows of non-compressible fluids, Amer. Math. Soc. Transl. Ser. 2., 24 (1963), 79–106.
- Herbert Koch and Daniel Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math. 157 (2001), no. 1, 22–35. MR 1808843, DOI 10.1006/aima.2000.1937
- Jean Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), no. 1, 193–248 (French). MR 1555394, DOI 10.1007/BF02547354
- R.E.A.C. Paley, A. Zygmund, On some series of functions (1) (2) (3), Proc. Cambridge Philos. Soc., 26 (1930), 337–357, 458–474, 28 (1932), 190–205.
Additional Information
- Ting Zhang
- Affiliation: Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China
- Email: zhangting79@zju.edu.cn
- Daoyuan Fang
- Affiliation: Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China
- Email: dyf@zju.edu.cn
- Received by editor(s): December 2, 2009
- Received by editor(s) in revised form: March 10, 2010, and July 27, 2010
- Published electronically: January 6, 2011
- Additional Notes: The authors were supported in part by the National Natural Science Foundation of China (NSFC) (10871175, 10931007, 10901137), the Zhejiang Provincial Natural Science Foundation of China (Z6100217), and SRFDP No. 20090101120005.
- Communicated by: Matthew J. Gursky
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 2827-2837
- MSC (2010): Primary 35Q30; Secondary 76D05, 35A01, 35A02
- DOI: https://doi.org/10.1090/S0002-9939-2011-10762-7
- MathSciNet review: 2801624