An alternative approach to solenoidal Lipschitz truncation

TL;DR

This paper introduces a novel method for solenoidal Lipschitz truncation that effectively modifies divergence-free functions to be Lipschitz continuous while maintaining divergence-free property, with improved bounds.

Contribution

It presents an alternative approach to Lipschitz truncation for divergence-free functions, offering stricter bounds compared to previous methods.

Findings

Provides a new Lipschitz truncation technique for divergence-free functions.

Achieves tighter bounds on the $W^{1,p}$ distance between original and truncated functions.

Maintains divergence-free property after truncation.

Abstract

In this work, a new approach to obtain a solenoidal Lipschitz truncation is presented. More precisely, the goal of the truncation is to modify a function $u \in W^{1,p}(\mathbb{R}^3,\mathbb{R}^3)$ that satisfies the additional constraint $\mathrm{div}~ u=0$, such that its modification $\tilde{u}$ is in $W^{1,\infty}(\mathbb{R}^3,\mathbb{R}^3)$ and still is divergence-free. We give an alternative approach to Lipschitz truncation compared to previous works by Breit, Diening & Fuchs (2012) and Breit, Diening & Schwarzacher (2013). The ansatz pursued here allows a rather strict bound on the $W^{1,p}$ distance of $u$ and $\tilde{u}$.