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.