A bilinear fractional integral operator for Euler-Riesz systems

TL;DR

This paper develops a uniform estimate for a bilinear fractional integral operator related to Euler-Riesz systems, aiding in understanding the integrability and stability of solutions in mean-field interaction models.

Contribution

It introduces a new uniform estimate for a tensor-valued bilinear fractional integral operator associated with Euler-Riesz systems, enhancing integrability analysis.

Findings

Established a uniform estimate via restricted weak-type endpoint estimates and Marcinkiewicz interpolation.

Derived a stability result for smooth periodic solutions of the reformulated system.

Provided insights into the integrability properties of solutions with finite energy.

Abstract

We establish a uniform estimate for a bilinear fractional integral operator via restricted weak-type endpoint estimates and Marcinkiewicz interpolation. This estimate is crucial in the integrability analysis of a tensor-valued bilinear fractional integral operator associated with Euler-Riesz systems modeling mean-field interactions induced by a singular kernel. The tensorial operator arises from a reformulation of the Euler-Riesz system that yields a gain in integrability for finite energy solutions through compensated integrability. Additionally, for smooth periodic solutions of the reformulated system, we derive a stability result.