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

New Results

Multiplication of Distributions and Algebras of Mnemofunctions

Antonevich A.B., Shagova T.G.


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.
Contemporary Mathematics. Fundamental Directions. 2019;65(3):339-389
pages 339-389 views

Linear Operators and Equations with Partial Integrals

Kalitvin A.S., Kalitvin V.A.


We consider linear operators and equations with partial integrals in Banach ideal spaces, spaces of vector functions, and spaces of continuous functions. We study the action, regularity, duality, algebras, Fredholm properties, invertibility, and spectral properties of such operators. We describe principal properties of linear equations with partial integrals. We show that such equations are essentially different compared to usual integral equations. We obtain conditions for the Fredholm alternative, conditions for zero spectral radius of the Volterra operator with partial integrals, and construct resolvents of invertible equations. We discuss Volterra-Fredholm equations with partial integrals and consider problems leading to linear equations with partial integrals.
Contemporary Mathematics. Fundamental Directions. 2019;65(3):390-433
pages 390-433 views

On Oscillations of Connected Pendulums with Cavities Filled with Homogeneous Fluids

Kopachevsky N.D., Voytitsky V.I.


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.
Contemporary Mathematics. Fundamental Directions. 2019;65(3):434-512
pages 434-512 views

On Inner Regularity of Solutions of Two-Dimensional Zakharov-Kuznetsov Equation

Faminskii A.V.


In this paper, we consider questions of inner regularity of weak solutions of initial-boundary value problems for the Zakharov-Kuznetsov equation with two spatial variables. The initial function is assumed to be irregular, and the main parameter governing the regularity is the decay rate of the initial function at infinity. The main results of the paper are obtained for the problem on a semistrip. In this problem, different types of initial conditions (e. g., Dirichlet or Neumann conditions) influence the inner regularity. We also give a survey of earlier results for other types of areas: a plane, a half-plane, and a strip.
Contemporary Mathematics. Fundamental Directions. 2019;65(3):513-546
pages 513-546 views

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