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

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 Schrödinger 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 gives a normwise variant of the recent pointwise results of Kovařík (Calc Var doi:10.1007/s00526-011-0437-4) about large-time asymptotics of the heat kernel of magnetic Schrödinger operators with radially symmetric field in a more general setting.