A simple proof of the generalized Leibniz rule on bounded Euclidean domains

TL;DR

This paper provides a straightforward proof of the generalized Leibniz rule for spectral and restricted Laplacians on bounded Euclidean domains, relevant for equations in fluid dynamics and porous media.

Contribution

It offers a simple proof of the generalized Leibniz rule for spectral Laplacian operators on bounded domains, facilitating analysis in related PDEs.

Findings

Proof simplifies existing derivations

Applicable to spectral Laplacian and restricted Laplacian

Supports future research on porous medium equations

Abstract

This note is devoted to a simple proof of the generalized Leibniz rule in bounded domains. The operators under consideration are the so-called spectral Laplacian and the restricted Laplacian. Equations involving such operators have been lately considered by Constantin and Ignatova in the framework of the SQG equation \cite{CI} in bounded domains and by two of the authors \cite{HungJuan} in the framework of the porous medium with nonlocal pressure in bounded domains. We will use the estimates in this work in a forthcoming paper on the study of Porous Medium Equations with pressure given by Riesz-type potentials .