Cloaking via mapping for the heat equation

TL;DR

This paper investigates near-cloaking for the heat equation, demonstrating that boundary measurements become indistinguishable from a cloaked region after a certain time, using regularized transformation media theory.

Contribution

It introduces a novel approach to near-cloaking in heat propagation, analyzing long-term behavior and boundary measurements with high contrast coefficients.

Findings

Boundary measurements approximate cloaked heat problem after threshold time

Long-term solutions exhibit stability despite high contrast in parameters

Numerical examples confirm theoretical near-cloaking results

Abstract

This paper explores the concept of near-cloaking in the context of time-dependent heat propagation. We show that after the lapse of a certain threshold time instance, the boundary measurements for the homogeneous heat equation are close to the cloaked heat problem in a certain Sobolev space norm irrespective of the density-conductivity pair in the cloaked region. A regularised transformation media theory is employed to arrive at our results. Our proof relies on the study of the long time behaviour of solutions to the parabolic problems with high contrast in density and conductivity coefficients. It further relies on the study of boundary measurement estimates in the presence of small defects in the context of steady conduction problem. We then present some numerical examples to illustrate our theoretical results.