The Hardy inequality and the heat equation in twisted tubes

TL;DR

This paper demonstrates that twisting a three-dimensional tube enhances the decay rate of heat distribution, utilizing Hardy inequalities and advanced analytical methods to improve understanding of heat flow in geometrically complex structures.

Contribution

It introduces a novel approach combining Hardy inequalities and self-similar analysis to show improved heat decay in twisted tubes.

Findings

Twisting the tube improves heat decay rates.

Hardy inequalities are effective in analyzing heat equations in twisted geometries.

The method can be applied to other geometrically complex domains.

Abstract

We show that a twist of a three-dimensional tube of uniform cross-section yields an improved decay rate for the heat semigroup associated with the Dirichlet Laplacian in the tube. The proof employs Hardy inequalities for the Dirichlet Laplacian in twisted tubes and the method of self-similar variables and weighted Sobolev spaces for the heat equation.