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Valerii Los

Publications and source records attributed to Valerii Los.

8 recordsLinked to original sources
arXiv2026

Matrix parabolic problems in Sobolev spaces of generalized smoothness

We study a general linear parabolic problem for Petrovskii parabolic differential system in Sobolev anisotropic distribution spaces of generalized smoothness. Slowly varying functions are used to characterize supplementary generalized smoothness that cannot be determined by number indexes. We prove that this problem induces topological isomorphisms on appropriate pairs of such spaces. As an application, we give sufficient and necessary conditions for the problem solutions to have prescribed generalized regularity expressed in terms of these spaces. Their use allows obtaining exact conditions for indicated generalized partial derivatives of the solutions to be continuous.

math.AP↗
arXiv2022

Regularity conditions for solutions to some parabolic systems

We investigate global and local regularity of generalized solutions to parabolic initial-boundary value problem for Petrovskii system of second order differential equations. Results are formulated in terms of the belonging of right-hand sides of the problem to some generalized Sobolev spaces. We also obtain new sufficient conditions under which the generalized solution should be classical.

math.AP↗
arXiv2021

Some problems for Petrovskii parabolic systems in generalized Sobolev spaces

We consider an inhomogeneous initial-boundary value problem for a Petrovskii parabolic system of second order PDEs. We prove that this problem induces isomorphisms between appropriate anisotropic generalized Sobolev spaces. The regularity of these spaces are given by a pair of real numbers and by a function parameter. The latter allows us to characterize the regularity of solutions to the problem more finely as compared with anisotropic Sobolev spaces.

math.AP↗
arXiv2019

Parabolic problems in generalized Sobolev spaces

We consider a general inhomogeneous parabolic initial-boundary value problem for a $2b$-parabolic differential equation given in a finite multidimensional cylinder. We investigate the solvability of this problem in some generalized anisotropic Sobolev spaces. They are parametrized with a pair of positive numbers $s$ and $s/(2b)$ and with a function $φ:[1,\infty)\to(0,\infty)$ that varies slowly at infinity. The function parameter $φ$ characterizes subordinate regularity of distributions with respect to the power regularity given by the number parameters. We prove that the operator corresponding to this problem is an isomorphism on appropriate pairs of these spaces. As an application, we give a theorem on the local regularity of the generalized solution to the problem. We also obtain sharp sufficient conditions under which chosen generalized derivatives of the solution are continuous on a given set.

math.AP↗
arXiv2017

Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces

In Hörmander inner product spaces, we investigate initial-boundary value problems for an arbitrary second order parabolic partial differential equation and the Dirichlet or a general first-order boundary conditions. We prove that the operators corresponding to these problems are isomorphisms between appropriate Hörmander spaces. The regularity of the functions which form these spaces is characterized by a pair of number parameters and a function parameter varying regularly at infinity in the sense of Karamata. Owing to this function parameter, the Hörmander spaces describe the regularity of functions more finely than the anisotropic Sobolev spaces.

math.AP↗
arXiv2015

An isomorphism theorem for parabolic problems in Hörmander spaces and its applications

We investigate a general parabolic initial-boundary value problem with zero Cauchy data in some anisotropic Hörmander inner product spaces. We prove that the operators corresponding to this problem are isomorphisms between appropriate Hörmander spaces. As an application of this result, we establish a theorem on the local increase in regularity of solutions to the problem. We also obtain new sufficient conditions under which the generalized derivatives, of a given order, of the solutions should be continuous.

math.AP↗
arXiv2013

Parabolic problems and interpolation with a function parameter

We give an application of interpolation with a function parameter to parabolic differential operators. We introduce the refined anisotropic Sobolev scale that consists of some Hilbert function spaces of generalized smoothness. The latter is characterized by a real number and a function varying slowly at infinity in Karamata's sense. This scale is connected with anisotropic Sobolev spaces by means of interpolation with a function parameter. We investigate a general initial--boundary value parabolic problem in the refined Sobolev scale. We prove that the operator corresponding to this problem sets isomorphisms between appropriate spaces pertaining to this scale.

math.AP↗