Skip to main content
Log in

Unsteady Solutions of Euler Equations Generated by Steady Solutions

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

Invariant solutions of partial differential equations are found by solving a reduced system involving one independent variable less. When the solutions are invariant with respect to the so-called projective group, the reduced system is simply the steady version of the original system. This feature enables us to generate unsteady solutions when steady solutions are known. The knowledge of an optimal system of subalgebras of the principal Lie algebra admitted by a system of differential equations provides a method of classifying H-invariant solutions as well as constructing systematically some transformations (essentially different transformations) mapping the given system to a suitable form. Here the transformations allowing to reduce the steady two-dimensional Euler equations of gas dynamics to an equivalent autonomous form are classified by means of the program SymboLie, after that an optimal system of two-dimensional subalgebras of the principal Lie algebra has been calculated. Some steady solutions of two-dimensional Euler equations are determined, and used to build unsteady solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bianchi, L.: Lezioni sulla teoria dei gruppi continui finiti di trasformazione. Enrico Spoerri Editore, Pisa (1918)

    Google Scholar 

  2. Ames, W.F.: Nonlinear Partial Differential Equations in Engineering, vol. 2. Academic Press, New York (1972)

    MATH  Google Scholar 

  3. Bluman, G.W., Cole, J.D.: Similarity Methods of Differential Equations. Springer, New York (1974)

    Google Scholar 

  4. Ovsiannikov, L.V.: Group Analysis of Differential Equations. Academic Press, New York (1982)

    MATH  Google Scholar 

  5. Ibragimov, N.H.: Transformation Groups Applied to Mathematical Physics. Reidel, Dordrecht (1985)

    MATH  Google Scholar 

  6. Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1986)

    MATH  Google Scholar 

  7. Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer, New York (1989)

    MATH  Google Scholar 

  8. Ibragimov, N.H.: Handbook of Lie Group Analysis of Differential Equations. CRC Press, Boca Raton (1994, 1995, 1996)

    MATH  Google Scholar 

  9. Olver, P.J.: Equivalence, Invariants, and Symmetry. Cambridge University Press, New York (1995)

    Book  MATH  Google Scholar 

  10. Kumei, S., Bluman, G.W.: When nonlinear differential equations are equivalent to linear differential equations. SIAM J. Appl. Math. 42, 1157–1173 (1992)

    Article  MathSciNet  Google Scholar 

  11. Donato, A., Oliveri, F.: Linearization procedure of nonlinear first order systems of PDE’s by means of canonical variables related to Lie groups of point transformations. J. Math. Anal. Appl. 188, 552–568 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  12. Donato, A., Oliveri, F.: How to build up variable transformations allowing one to map nonlinear hyperbolic equations into autonomous or linear ones. Transp. Theory Stat. Phys. 25, 303–322 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  13. Donato, A., Oliveri, F.: Reduction to autonomous form by group analysis and exact solutions of axi-symmetric MHD equations. Math. Comput. Model. 18, 83–90 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  14. Donato, A., Oliveri, F.: When nonautonomous equations are equivalent to autonomous ones. Appl. Anal. 58, 313–323 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  15. Margheriti, L., Speciale, M.P.: Unsteady solutions of PDEs generated by steady solutions. In: Manganaro, N., Monaco, R., Rionero, S. (eds.) Proceedings “Wascom 2007”, 14th Conference on Waves and Stability in Continuous Media, pp. 388–393. World Scientific, Singapore (2008)

    Chapter  Google Scholar 

  16. Currò, C., Oliveri, F.: Reduction of nonhomogenous quasilinear 2×2 systems to homogenous and autonomous form. J. Math. Phys. 49, 1–11 (2008)

    Article  Google Scholar 

  17. Shugrin, S.M.: Galilean systems of differential equations. Differ. Equ. 16, 1402–1413 (1981)

    MATH  Google Scholar 

  18. Ruggeri, T.: Galilean invariance and entropy principle for systems of balance laws. The structure of extended thermodynamics. Contin. Mech. Termodyn. 1, 3–20 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  19. Oliveri, F.: Galilean quasilinear systems of PDE’s and the substitution principle. In: Donato, A., Oliveri, F. (eds.) Nonlinear Hyperbolic Equations: Theory, and Computational Aspects. Notes on Numerical Fluid Mechanics, vol. 43, pp. 457–464. Vieweg, Wien (1993)

    Google Scholar 

  20. Donato, A.: Similarity analysis and non-linear wave propagation. Int. J. Non-Linear Mech. 22, 307–314 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ames, W.F., Donato, A.: On the evolution of weak discontinuities in a state characterized by invariant solutions. Int. J. Non-Linear Mech. 23, 167–174 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  22. Wolfram, S.: The Mathematica Book, 5th edn. Wolfram Media (2003)

  23. Oliveri, F., Speciale, M.P.: Exact solutions to the equations of perfect gases through Lie group analysis and substitution principles. Int. J. Non-Linear Mech. 34, 1077–1087 (1999)

    Article  MathSciNet  Google Scholar 

  24. Oliveri, F., Speciale, M.P.: Exact solutions to the unsteady equations of perfect gases through Lie group analysis and substitution principles. Int. J. Non-Linear Mech. 37, 257–274 (2002)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to L. Margheriti.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Margheriti, L., Speciale, M.P. Unsteady Solutions of Euler Equations Generated by Steady Solutions. Acta Appl Math 113, 289–303 (2011). https://doi.org/10.1007/s10440-010-9600-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-010-9600-8

Keywords

Mathematics Subject Classification (2000)

Navigation