Vol 65, No 3 (2019): Proceedings of the Crimean Autumn Mathematical School-Symposium

In this paper, we discuss methods and approaches for definition of multiplication of distributions, which is not defined in general in the classical theory. We show that this problem is related to the fact that the operator of multiplication by a smooth function is nonclosable in the space of distributions. We give the general method of construction of new objects called new distributions, or mnemofunctions, that preserve essential properties of usual distributions and produce algebras as well. We describe various methods of embedding of distribution spaces into algebras of mnemofunctions. All ideas and considerations are illustrated by the simplest example of the distribution space on a circle. Some effects arising in study of equations with distributions as coefficients are demonstrated by example of a linear first-order differential equation.

We consider the problem and normal (eigen) oscillations of the system of three connected (coupled to each other) pendulums with cavities filled with one or several immiscible homogeneous fluids. We study the case of partially dissipative system when the cavity of the first pendulum is completely filled with two ideal fluids, the cavity of the second one is filled with three viscous fluids, and the cavity third one is filled with one ideal fluid. We use methods of functional analysis. We prove the theorem on correct solvability of the initial-boundary value problem on any interval of time. We study the case of eigen oscillations of conservative system where all fluids in cavities of pendulums are ideal and the friction in joints (points of suspension) is not taken into account. We consider in detail three auxiliary problems on small oscillations of single pendulums with three above variants of fluids in cavities.