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  Quarterly of Applied Mathematics
Quarterly of Applied Mathematics
Online ISSN 1552-4485; Print ISSN 0033-569X

Asymmetric and moving-frame approaches to Navier-Stokes equations

Author: Xiaoping Xu
Journal: Quart. Appl. Math. 67 (2009), 163-193
MSC (2000): Primary 35C05, 35Q35; Secondary 35C10, 35C15
Published electronically: January 22, 2009
MathSciNet review: 2497602
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Abstract: In this paper, we introduce a method of imposing asymmetric conditions on the velocity vector with respect to independent variables and a method of moving frame for solving the three-dimensional Navier-Stokes equations. Seven families of non-steady rotating asymmetric solutions with various parameters are obtained. In particular, one family of solutions blows up at any point on a moving plane with a line deleted, which may be used to study turbulence. Using Fourier expansion and two families of our solutions, one can obtain discontinuous solutions that may be useful in the study of shock waves. Another family of solutions are partially cylindrical invariant, contain two parameter functions of $ t$ and structurally depend on two arbitrary polynomials, which may be used to describe incompressible fluid in a nozzle. Most of our solutions are globally analytic with respect to spacial variables.

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Additional Information

Xiaoping Xu
Affiliation: Institute of Mathematics, Academy of Mathematics & System Sciences Chinese Academy of Sciences, Beijing 100190, People’s Republic of China

PII: S 0033-569X(09)01125-0
Keywords: Navier-Stokes equations, asymmetric condition, moving frame, exact solution, symmetry transformation.
Received by editor(s): September 5, 2007
Published electronically: January 22, 2009
Additional Notes: Research for this article was supported by China NSF Grant #10871193.
Article copyright: © Copyright 2009 Brown University

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