On the fractional version of Leibniz rule

TL;DR

This paper proves a new inequality related to fractional Leibniz rule derivatives, enabling the analysis of fractional PDEs, and applies it to establish existence and uniqueness of solutions for fractional 2D Stokes equations.

Contribution

It introduces a novel inequality for fractional derivatives and demonstrates its application in proving well-posedness of fractional PDEs, specifically the 2D Stokes equations.

Findings

Established a new inequality for fractional derivatives.

Applied the inequality to prove existence and uniqueness of solutions.

Extended the Faedo-Galerkin method to fractional PDEs.

Abstract

This manuscript is dedicated to prove a new inequality that involves an important case of Leibniz rule regarding Riemann-Liouville and Caputo fractional derivatives of order $\alpha\in(0,1)$. In the context of partial differential equations, the aforesaid inequality allows us to address the Faedo-Galerkin method to study several kinds of partial differential equations with fractional derivative in the time variable; particularly, we apply these ideas to prove the existence and uniqueness of solution to the fractional version of the 2D Stokes equations in bounded domains.