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Applications of group theoretical methods in hydrodynamics. Transl. from the Russian by S. S. Kutateldaze, V. N. Dyatlov and G. V. Dyatlov. (English) Zbl 0912.35001
Mathematics and its Applications (Dordrecht). 450. Dordrecht: Kluwer Academic Publishers. xii, 396 p. (1998).
This book has two aims: first, to present a group-theoretic classification of the equations of hydrodynamics in the presence of planar and rotational symmetry and, secondly, to give the construction of exact solutions and their physical interpretation.
The volume begins with a preface explaining its scape and organization. The first chapter is devoted to group-theoretic properties of equations for a homogeneous or inhomogeneous fluid in Lagrangian coordinates. The second chapter is devoted to searching exact solutions to the Euler equation. In the next chapters, one introduces the notion of invariant manifold for an arbitrary system of evolution equations with \(n\) independent variables, and gives the method of defining equations. Chapter 5 concentrates on stationary vortex structures in an ideal fluid in the presence of extra symmetries. The sixth chapter contains the results of studying the group-theoretic properties of the equations of motion for a viscous heat conducting liquid. The volume ends by a chapter on exact solutions to the equations of dynamics for a viscous fluid.
The references run to 174 items and go up to 1996. The textbook is recommended to a wide audience, including graduate students in mathematics, physics and engineering.
Reviewer: V.A.Sava (Iaşi)

MSC:
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
58J70 Invariance and symmetry properties for PDEs on manifolds
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
35Q35 PDEs in connection with fluid mechanics
35Q30 Navier-Stokes equations
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