On the existence of solutions for a class of systems of integro-differential equations with the logarithmic Laplacian and drift

TL;DR

This paper proves the existence of solutions for a complex system of integro-differential equations involving the logarithmic Laplacian and drift terms, using fixed point methods and addressing technical challenges.

Contribution

It extends the analysis of integro-differential systems with the logarithmic Laplacian to systems with transport, demonstrating solution existence under Fredholm conditions.

Findings

Solutions exist for the considered system.

The linear operators satisfy the Fredholm property.

The fixed point technique is effective for such systems.

Abstract

In this article, we consider a system of integro-differential equations in L^2(R, R^N), which contains the logarithmic Laplacian in the presence of transport terms. The linear operators associated with the system satisfy the Fredholm property. By virtue of a fixed point technique, we demonstrate the existence of solutions. We emphasize that the discussion is more complicated than that of the scalar situation as there are more cumbersome technicalities to overcome.