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

DOI:
https://doi.org/10.1090/S0033-569X-09-01125-0

Published electronically:
January 22, 2009

MathSciNet review:
2497602

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 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

Email:
xiaoping@math.ac.cn

DOI:
https://doi.org/10.1090/S0033-569X-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