Vol 214, No 4 (2023)

The Cauchy problems are studied for a first-order multidimensional symmetric linear hyperbolic system of equations with variable coefficients and its singular perturbations that are second-order strongly parabolic and hyperbolic systems of equations with a small parameter τ>0 multiplying the second derivatives with respect to x and t. The existence and uniqueness of weak solutions of all three systems and τ-uniform estimates for solutions of systems with perturbations are established. Estimates for the difference of solutions of the original system and the systems with perturbations are given, including ones of order O(τα/2) in the norm of C(0,T;L2(Rn)), for an initial function w0 in the Sobolev space Hα(Rn), α=1,2, or the Nikolskii space Hα2(Rn), 0<α<2, α≠1, and under appropriate assumptions on the free term of the first-order system. For α=1/2 a wide class of discontinuous functions w0 is covered. Estimates for derivatives of any order with respect to x for solutions and of order O(τα/2) for their differences are also deduced. Applications of the results to the first-order system of gas dynamic equations linearized at a constant solution and to its perturbations, namely, the linearized second-order parabolic and hyperbolic quasi-gasdynamic systems of equations, are presented.

Sufficient conditions are presented for the construction of a regularization of a distribution in the form a(σ)f, where f is a distribution and a(σ) is an infinitely differentiable function outside a closed set N which has power-like singularities of derivatives on N. Applications of such regularizations to an effective construction of solutions of the equation Pu=f, where P(σ) is a polynomial, are considered.

In Vinberg's works certain non-Abelian gradings of simple Lie algebras were introduced and investigated, namely, short SO3- and SL3-structures. We investigate a different kind of these, short SL2-structures. The main results refer to the one-to-one correspondence between such structures and certain special Jordan algebras.