Physics, mathematics, and technology

2013 Issue №4

Back to the list Download the article

One method for constructing exact solutions of equations of two-dimensional hydrodynamics of an incompressible fluid



We propose a simple algebraic method for constructing exact solutions of equations of two-dimensional hydrodynamics of an incompressible fluids.The problem reduces to consecutively solution three linear partial differential equations for a nonviscous fluid and to solving three linear partial differential equations and one first-order ordinary differential equation for a viscous fluid.


1. Kato T. Global solutions of two-dimensional Navier-Stokes and Euler equations // Proc. Symp. Pure Math. 1986. Pt. 2. Vol. 45.
2. Kato T. Quasi-linear equations of evolution, with applications to partial differential equastions // Journal of Functional Analysis. 1972. Vol. 9. P. 296.
3. Kato T. Spectral theory and differential equastions // Lecture Notes in Mathematics / ed. W.N. Everitt. Berlin, 1975. Vol. 448. P. 25.
4. Constantin P., Wu J. The inviscid limit for non-smooth vortivity // Indiana University Mathematics Journal. 1996. № 1. P. 45.
5. Wu J. J. The Inviscid Limit of the Complex Ginzburg-Landau Equastion.//Differential Equestions. 1998. Vol. 142. P. 413.
6. Arnold V. I. Sur la Geometrie Differentielle des Groupes de Lie de Dimension Infinie et ses Applications a Láľhydrodinamique des Fluides Parfaits //Ann. Inst.Fourier. Grenoble, 1966. Vol. 16. P. 319.
7. Marsden J. E. Lectures on mechanics. Navier – Stokes equastions: a mathematical analysis //Lond Math. Soc. Lect. Note. Ser. 174. Cambridge Univ. Press, 1992.
8. Fridlander S., Vishik M. Lax pair formulation for the Euler equation. // Phys.Lett. 1990. A. 148(6—7). P. 313—319.
9. Fridlander S., Vishik M. An inverse scattering treatment for the flow of an ideal fluid in two dimensions // Nonlinearity. 1993. Vol. 6. P. 231.
10. Li Y., Yurov A. Lax pairs and Darboux transformations for Euler equastions.// Studies In Applied Math. 2003. Vol. 111. P. 101—103.
11. Ландау Л. Д., Лифшиц Е. М. Теоретическая физика. М., 1986. Т. 6 : Гидродинамика.
12. Итс А. Р., Рыбин А. В., Салль М. А. К вопросу о точном интегрировании нелинейного уравнения Шредингера // ТМФ. 1988. № 1. C. 20, 74.