Averaging of one-parameter semigroups and passage to the limit in the space of pseudomeasures

TL;DR

This paper investigates the limits of sequences of one-parameter semigroups approximating initial-boundary value problems with singularities, using pseudomeasures and measure theory to describe convergence and limit points.

Contribution

It introduces a novel framework linking semigroup limits to pseudomeasures and measures in a Banach space setting, providing a new approach to passage to the limit in singular problems.

Findings

Characterization of limit points via measures on pseudomeasure space

Establishment of a one-to-one correspondence between semigroups and pseudomeasures

Convergence of semigroup sequences equivalent to convergence of pseudomeasures

Abstract

The sequence of one-parameter semigroups arising as the approximation of initial-boundary value problem with singularities is the object of investigation of this paper. The set of limit points of the sequence of approximating semigroups is studied. The set of limit points of the map with values in a linear topological space is presented as the set of mean values of this map by measures on the domain of definition of the map. One to one correspondence betwin the semigroups generated by any approximating initial-boundary value problems and the pseudomeasures on the space of maps of time semiaxe into the coordinate space is studied. The linear space of pseudomesures endowed with the structure of Banach space and with the structure of the linear topological space such that the convergence of semigroup sequence is equivalent to the convergence of the sequence of corresponding pseudomeasures. The desctiption of a limit point of the sequence of approximating semigroups is obtained by a measure on the topological vector space of corresponding pseudomeasures. The trajectories the limit one-parameter family of transformations of the space of initial data is described by the mean value of the random pseudomeasure.