We consider a Leith model of turbulence (Leith C 1967 Phys. Fluids 10 1409) in which the energy spectrum obeys a nonlinear diffusion equation. We analytically prove the existence of a self-similar solution with a power-law asymptotic on the low-wavenumber end and a sharp boundary on the high-wavenumber end, which propagates to infinite wavenumbers in a finite-time t*. We prove that this solution has a power-law asymptotic with an anomalous exponent x*, which is less than the Kolmogorov value, x* > 5/3. This is a result that was previously discovered by numerical simulations in Connaughton and Nazarenko (2004 Phys. Rev. Lett. 92 044501). We also prove the convergence to this self-similar solution of the spectrum evolving from an arbitrary finitely supported initial data as t → t*.

The notion of generalized equivalence transformations for which the equivalence transformations considered by Ovsyannikov are universal equivalence transformations is introduced for a system of differential equations. An algorithm of the group classification of the system of differential equations with the help of these generalized equivalence transformations is proposed. The efficiency and advantages of this algorithm are demonstrated by examples of equations of gas dynamics and equations of nonlinear longitudinal oscillations of a viscoelastic bar in the Kelvin model.Keywords: group classification of systems of differential equations, generalized and universal equivalence transformations, group of equivalences, equations of gas dynamics, nonlocal symmetries.

A system of equations of isentropic gas motion with n 2 is classified in terms of zero-order conservation laws with the use of the method of A-operators. New conservation laws are found to be valid only for potential isentropic motion of the Chaplygin gas. In this case, the greatest number of nontrivial conservation laws is obtained, with n scalar conservation laws being nonlocal. Additional properties of symmetry of the considered equations associated with these conservation laws are indicated.Introduction. The search for conservation laws for systems of differential equations is discussed in many publications. In particular, Shmyglevskii [1] obtained a full system of zero-order conservation laws for equations of motion of a perfect gas in a three-dimensional formulation. Ibragimov [2] found additional conservation laws for equations of motion of a polytropic gas by applying point symmetry operators to the classical conservation laws. Ibragimov [2] also derived an additional conservation law for a system of equations of potential isentropic motion of a polytropic gas.In the present work, a system of equations of isentropic gas motion with n 2 is classified in terms of zero-order conservation laws with the use of the method of A-operators [3]. It is demonstrated that the condition of an isentropic character of the flow does not generate new conservation laws in this system (in contrast to the conventional system of equations of gas motion). A system of vortex-free isentropic gas motion and a system of equations of potential isentropic gas motion are classified in terms of the zero-order conservation laws by the method of A-operators. For the latter system of equations, the greatest number of nontrivial conservation laws is observed in the case of the Chaplygin gas, with n scalar conservation laws being nonlocal. A group classification is performed for the system of equations of potential isentropic gas motion, which allows the set of nontrivial zero-order conservation laws to be extended. It is found that the system for the Chaplygin gas admits the greatest Lie group of transformations. Isentropic gas motion is described by the equations

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