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We introduce some new classes of time dependent functions whose defining properties take into account of oscillations around singularities. We study properties of solutions to the heat equation with coefficients in these classes which are much more singular than those allowed under the current theory. In the case of $L^2$ potentials and $L^2$ solutions, we give a characterization of potentials which allow the Schr\"odinger heat equation to have a positive solution. This provides a new result on the long running problem of identifying potentials permitting a positive solution to the Schr\"odinger equation. We also establish a nearly necessary and sufficient condition on certain sign changing potentials such that the corresponding heat kernel has Gaussian upper and lower bound. Some applications to the Navier-Stokes equations are given. In particular, we derive a new type of a priori estimate for solutions of Navier-Stokes equations. The point is that the gap between this estimate and a sufficient condition for all time smoothness of the solution is logarithmic.

WHEN DOES A SCHRÖDINGER HEAT EQUATION PERMIT POSITIVE SOLUTIONS