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

• Published in: Calculus of Variations and Partial Differential Equations
• Main Author: Krejčiřík, D. (David)
• Format: Article in Journal/Newspaper
• Language: English
• Published: 2013
• Subjects: convection-diffusion equations; large time behavior; singular potentials; hardy inequality; dirichlet forms; operators; laptev
• Online Access: https://doi.org/10.1007/s00526-012-0516-1; http://hdl.handle.net/11104/0221701

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 Schrodinger 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 Kovarik (Calc Var doi:10.1007/s00526-011-0437-4) about large-time asymptotics of the heat kernel of magnetic Schrodinger operators with radially symmetric field in a more general setting.