Classical solutions for fractional porous medium flow

TL;DR

This paper studies the fractional porous medium flow, establishing local existence, uniqueness, and blow-up criteria for smooth solutions using commutator estimates involving fractional Laplacians.

Contribution

It provides new mathematical results on the fractional porous medium flow, including existence, uniqueness, and blow-up conditions, with a novel proof technique based on commutator estimates.

Findings

Proved local in time existence of smooth solutions.

Established a blow-up criterion for solutions.

Developed a commutator estimate involving fractional Laplacians.

Abstract

We consider the fractional porous medium flow introduced by Caffarelli-Vazquez and obtain local in time existence, uniqueness, and blow-up criterion for smooth solutions. The proof is based on establishing a commutator estimate involving fractional Laplacian operators.