Geodesic
Symmetry analysis of ideal fluid equations in terms of trajectories and Weber's potential
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 2, pp. 133-144.

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The 2D perfect fluid motions equations in Lagrangian coordinates are considered. If body forces are potential one, then there is the general integral called Weber's integral and the resulting system includes initial data which in fact make the problem of group-theoretical classification actual. It is established that the basic group becomes infinite-dimensional with respect to the space variable too. The exceptional values of arbitrary initial vorticity are obtained at which we can be observed further extension of the group. Group properties of Euler equations in arbitrary Lagrangian coordinates are also considered and some exact solutions are constructed.
Keywords: symmetry analysis
Mots-clés : Euler equations, Weber's transformation, equivalence transformation, group classification. 
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Victor K. Andreev; Daria A. Krasnova. Symmetry analysis of ideal fluid equations in terms of trajectories and Weber's potential. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 2, pp. 133-144. http://geodesic.mathdoc.fr/item/JSFU_2019_12_2_a0/

[1] L.V. Ovsyannikov, Group Properties of Differential Equations, Siberian Branch of USSR Ac. Sci., Novosibirsk, 1962 (in Russian) | MR

[2] L.V. Ovsyannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982 | MR | Zbl

[3] P.J. Olver, Application of Lie Groups to Differential Equations, Springer-Verlag, New York Inc., 1986 | MR

[4] Andreev V. K., Kaptsov O. V., Pukhnachev V. V., Rodionov A. A., Applications of Group-Theoretical Methods in Hydrodynamics, Springer, Dordrecht, 2010 | MR

[5] V.K. Andreev, V.V. Bublik, V.O. Bytev, Symmetries of Non-classical Hydrodynamics Models, Nauka, Novosibirsk, 2003 (in Russian) | MR

[6] V.K. Andreev, Yu.A. Gaponenko, O.N. Goncharova, V.V. Pukhnachev, Mathematical Models of Convection, Walter de Gruyter GmbH CO KG, Berlin–Boston, 2012 | MR

[7] V.K. Andreev, Stability of Unsteady Motions of a Fluid with a Free Boundary, Nauka, Novosibirsk, 1992 (in Russian) | MR | Zbl

[8] V.I. Nalimov, “The Cauchy–Poisson Problem”, Dinamica Sploshnoy Sredy, v. 1, 1974 (in Russian) | MR

[9] H. Yosihara, “Gravity Waves on the Free Surface of an Incompressible Perfect Fluid of Finite Depth”, Publ. RIMS Kyoto Univ., 18 | MR | Zbl

[10] H. Yosihara, “Capillary-gravity Waves for an Incompressible Ideal Fluid”, J. Math. Kyoto Univ., 23 (1983), 649–694 | DOI | MR | Zbl

[11] V.I. Nalimov, “Unsteady Vortex Surface Waves”, Siberian Math. J., 37:6 (1996), 1189–1198 (in Russian) | DOI | MR | Zbl

[12] H. Lamb, Hydrodynamics, Gostekhizdat, M., 1947 (in Russian)

[13] V.K. Andreev, A.A. Rodionov, “Group-Theoretic Classification and Exact Solutions of equations for a Plane and Rotationally-Symmetric Flow of an Ideal Fluid in Lagrangian Coordinates”, Differential Equations, 24:9 (1988), 1041–1049 (in Russian) | MR | Zbl

[14] V.K. Andreev, A.A. Rodionov, “Group Analysis of Equations for Planar Flows of an Ideal Fluid in Lagrangian Coordinates”, Dokl. Akad. Nauk SSSR, 298:6 (1988) (in Russian) | MR