Well-posedness of a non-local abstract Cauchy problem with a singular integral

TL;DR

This paper proves the well-posedness of a non-local abstract Cauchy problem with a singular integral, extending results to a generalized stationary Wigner equation, thus advancing understanding of its mathematical properties.

Contribution

It introduces a Banach space framework to establish well-posedness for a class of non-local evolution equations with singular kernels, including the stationary Wigner equation.

Findings

Established well-posedness under boundedness and smoothness conditions.

Extended results to a generalized stationary Wigner equation.

Improved understanding of the mathematical properties of the stationary Wigner equation.

Abstract

A non-local abstract Cauchy problem with a singular integral is studied, which is a closed system of two evolution equations for a real-valued function and a function-valued function. By proposing an appropriate Banach space, the well-posedness of the evolution system is proved under some boundedness and smoothness conditions on the coefficient functions. Furthermore, an isomorphism is established to extend the result to a partial integro-differential equation with singular convolution kernel, which is a generalized form of the stationary Wigner equation. Our investigation considerably improves the understanding of the open problem concerning the well-posedness of the stationary Wigner equation with inflow boundary conditions.