The improved decay rate for the heat semigroup with local magnetic field in the plane

TL;DR

This paper demonstrates that a local magnetic field in the plane enhances the decay rate of the heat semigroup, with the improvement depending on the magnetic flux's proximity to flux quanta, using Hardy inequalities and self-similar analysis.

Contribution

It establishes a general result that magnetic fields improve heat decay rates without symmetry assumptions, extending previous radially symmetric cases.

Findings

Magnetic field causes polynomial decay rate improvement proportional to flux distance from quanta.

Asymptotic analysis shows magnetic field degenerates to Aharonov-Bohm type, influencing decay.

Results hold without symmetry assumptions, generalizing prior work.

Abstract

We consider the heat equation in the presence of compactly supported magnetic field in the plane. We show that the magnetic field leads to an improvement of the decay rate of the heat semigroup by a polynomial factor with power proportional to the distance of the total magnetic flux to the discrete set of flux quanta. The proof employs Hardy-type inequalities due to Laptev and Weidl for the two-dimensional magnetic Schroedinger operator and the method of self-similar variables and weighted Sobolev spaces for the heat equation. A careful analysis of the asymptotic behaviour of the heat equation in the similarity variables shows that the magnetic field asymptotically degenerates to an Aharonov-Bohm magnetic field with the same total magnetic flux, which leads asymptotically to the gain on the polynomial decay rate in the original physical variables. Since no assumptions about the symmetry of the magnetic field are made in the present work, it confirms that the recent results of Kovarik about large-time asymptotics of the heat kernel of magnetic Schroedinger operators with radially symmetric field hold in greater generality.