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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Random data Cauchy theory for the incompressible three dimensional Navier-Stokes equations


Authors: Ting Zhang and Daoyuan Fang
Journal: Proc. Amer. Math. Soc. 139 (2011), 2827-2837
MSC (2010): Primary 35Q30; Secondary 76D05, 35A01, 35A02
Published electronically: January 6, 2011
MathSciNet review: 2801624
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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 $ \Vert u_0\Vert _{L^2}$ is small, we show that the probability for the global existence and uniqueness of the solution is large.


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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

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10762-7
PII: S 0002-9939(2011)10762-7
Keywords: Navier–Stokes equations, existence and uniqueness
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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.