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

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.

**[BK]**M. A. Brutyan and P. L. Krapivskiĭ,*Exact solution of Navier-Stokes equations describing the evolution of a vortex structure in a generalized shear flow*, Zh. Vychisl. Mat. i Mat. Fiz.**32**(1992), no. 2, 326–329 (Russian); English transl., Comput. Math. Math. Phys.**32**(1992), no. 2, 270–272. MR**1166987****[Ba]**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,*Unsteady movements of a rotating ring of a viscous incompressible fluid with a free boundary*, Ž. Prikl. Meh. i Tehn. Fiz.**1970**(1970), no. 3, 82–88 (Russian). MR**0285187****[Bv2]**V. O. Bytev, Invariant solutions of the Navier-Stokes equations,*Zhumal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki***6**(1972), 56.**[G]**V. I. Gryn′,*Exact solutions of Navier-Stokes equations*, Prikl. Mat. Mekh.**55**(1991), no. 3, 373–381 (Russian); English transl., J. Appl. Math. Mech.**55**(1991), no. 3, 301–309 (1992). MR**1134603**, https://doi.org/10.1016/0021-8928(91)90029-T**[I]**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.**[K]**L. B. Kapitanskii, Group analysis of Navier-Stokes 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 Navier-Stokes equations*, C. R. (Doklady) Acad. Sci. URSS (N.S.)**43**(1944), 286–288. MR**0011205****[Lr]**R. B. Leipnik,*Exact solutions of Navier-Stokes equations by recursive series of diffusive quotients*, C. R. Math. Rep. Acad. Sci. Canada**18**(1996), no. 5, 211–216. MR**1425294****[LRT]**C. C. Lin, E. Reissner, and H. S. Tsien,*On two-dimensional non-steady motion of a slender body in a compressible fluid*, J. Math. Physics.**27**(1948), 220–231. MR**0026499****[O]**L. V. Ovsiannikov,*Group analysis of differential equations*, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. Translated from the Russian by Y. Chapovsky; Translation edited by William F. Ames. MR**668703****[Pa]**A. D. Polyanin,*Exact solutions of Navier-Stokes equations with generalized separation of variables*, Dokl. Akad. Nauk**380**(2001), no. 4, 491–496 (Russian); English transl., Dokl. Phys.**46**(2001), no. 10, 726–731. MR**1875505**, https://doi.org/10.1134/1.1415590**[Pv1]**V. V. Pukhnachev, Group properties of Navier-Stokes equations in two-dimensional case,*Zhumal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki***1**(1960), 83.**[Pv2]**V. V. Pukhnachev, Invariant solutions of Navier-Stokes equations describing motions with free boundary,*Dokl. Akad. Nauk S.S.S.R.***202**(1972), 302.**[S1]**Hui Chuan Shen,*The theory of functions of a complex variable under Dirac-Pauli representation and its application in fluid dynamics. I*, Appl. Math. Mech.**7**(1986), no. 4, 365–382 (Chinese, with English summary); English transl., Appl. Math. Mech. (English Ed.)**7**(1986), no. 4, 391–411. MR**861144**, https://doi.org/10.1007/BF01898228**[S2]**Hui Chuan Shen,*Exact solution of Navier-Stokes equations—the theory of functions of a complex variable under Dirac-Pauli representation and its application in fluid dynamics. II*, Appl. Math. Mech.**7**(1986), no. 6, 517–522 (Chinese, with English summary); English transl., Appl. Math. Mech. (English Ed.)**7**(1986), no. 6, 557–562. MR**866002**, https://doi.org/10.1007/BF01899554**[V]**V. G. Vyskrebtsov,*New exact solutions of Navier-Stokes equations for axisymmetric self-similar fluid flows*, Mat. Metodi Fiz.-Mekh. Polya**41**(1998), no. 3, 44–51 (Russian, with English, Russian and Ukrainian summaries); English transl., J. Math. Sci. (New York)**104**(2001), no. 5, 1456–1463. MR**1706756**, https://doi.org/10.1023/A:1011371316589**[W]**Z. U. A. Warsi,*Fluid dynamics*, 2nd ed., CRC Press, Boca Raton, FL, 1999. Theoretical and computational approaches. MR**1658211****[X1]**Xiaoping Xu,*Stable-range approach to the equation of nonstationary transonic gas flows*, Quart. Appl. Math.**65**(2007), no. 3, 529–547. MR**2354886**, https://doi.org/10.1090/S0033-569X-07-01057-9**[X2]**X. Xu, Parameter-function approach to classical non-steady boundary Layer problems,*arXiv:0706.1864.***[Y]**A. Yu. Yakimov,*Exact solutions of Navier-Stokes equations in the presence of a vortex singularity on a ray*, Dokl. Akad. Nauk SSSR**276**(1984), no. 1, 79–82 (Russian). MR**744894**

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC (2000):
35C05,
35Q35,
35C10,
35C15

Retrieve articles in all journals with MSC (2000): 35C05, 35Q35, 35C10, 35C15

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