Asymmetric and movingframe approaches to NavierStokes equations
Author:
Xiaoping Xu
Journal:
Quart. Appl. Math. 67 (2009), 163193
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 threedimensional NavierStokes equations. Seven families of nonsteady 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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 A. A. Buchnev, Lie group admitted by the equations of motion of an ideal incompressible fluid, Dinamika Sploshnoi Sredi. Int. of Hydrodynamics Novosibirsk 7 (1971), 212.
 [Bv1]
 V. O. Bytev, Nonsteady motion of a rotating ring of viscous incompressible fluid with free boundary, Zhumal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki 3 (1970), 83. MR 0285187 (44:2410)
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 V. O. Bytev, Invariant solutions of the NavierStokes equations, Zhumal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki 6 (1972), 56.
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 V. I. Gryn, Exact solutions of NavierStokes equations, J. Appl. Math. Mech. 55 (1991), 301309. MR 1134603 (92m:76042)
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 N. H. Ibragimov, Lie Group Analysis of Differential Equations, Volume 2, CRC Handbook, CRC Press, 1995.
 [J]
 G. B. Jeffery, Philosophical Magazine, Ser. 6 (1915), 29.
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 L. B. Kapitanskii, Group analysis of NavierStokes equations and Euler equations with rotational symmetry and new exact solutions of these equations, Dokl. Akad. Nauk S.S.S.R. 243 (1978), 901.
 [KKR]
 H. E. Kochin, I. A. Kibel' and N. V. Roze, Theoretical Hydromechanics, Fizmatgiz, Moscow, 1963.
 [Ll]
 L. Landau, A new exact solution of NavierStokes equations, C. R. (Doklady) Acad. Sci. URSS (N. S.) 43 (1944), 286288. MR 0011205 (6:135d)
 [Lr]
 R. B. Leipnik, Exact solutions of NavierStokes equations by recursive series of diffusive equations, C. R. Math. Rep. Acad. Sci. Canada 18 (1996), 211216. MR 1425294 (97i:76039)
 [LRT]
 C. C. Lin, E. Reissner and H. S. Tsien, On twodimensional nonsteady motion of a slender body in a compressible fluid, J. Math. Phys. 27 (1948), no. 3, 220. MR 0026499 (10:162e)
 [O]
 L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982. MR 668703 (83m:58082)
 [Pa]
 A. D. Polyanin, Exact solutions of the NavierStokes equations with generalized separation of variables, Dokl. Phys. 46 (2001), 726731. MR 1875505 (2002g:76039)
 [Pv1]
 V. V. Pukhnachev, Group properties of NavierStokes equations in twodimensional case, Zhumal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki 1 (1960), 83.
 [Pv2]
 V. V. Pukhnachev, Invariant solutions of NavierStokes equations describing motions with free boundary, Dokl. Akad. Nauk S.S.S.R. 202 (1972), 302.
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 H.C. Shen, The theory of functions of a complex variable under DiracPauli representation and its application in fluid dynamics I, Appl. Math. Mech. (English Ed.) 7 (1986), 391411. MR 861144 (88d:30057)
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 [W]
 Z. Warsi, Fluid Dynamics, CRC Press LTC, 1999. MR 1658211 (99k:76001)
 [X1]
 X. Xu, Stablerange approach to the equation of nonstationary transonic gas flows, Quart. Appl. Math 65 (2008), 529547. MR 2354886
 [X2]
 X. Xu, Parameterfunction approach to classical nonsteady boundary Layer problems, arXiv:0706.1864.
 [Y]
 A. Yu. Yakimov, Exact solutions of NavierStokes equations in the presence of a vortex singularity on a ray, Dokl. Acad. Nauk SSSR 276 (1984), 7982. MR 744894 (85e:76020)
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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:
http://dx.doi.org/10.1090/S0033569X09011250
PII:
S 0033569X(09)011250
Keywords:
NavierStokes 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
