2006/01/24 by Qi S. Zhang, Zhang, Qi S.
Engineering · Mathematics · #35k05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations #math.AP #msc:35k05
paper · pdf · doi:10.48550/arxiv.math/0601593
arxiv created 2006/01/24 · arxiv updated 2009/12/01
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 L2 potentials and L2 solutions, we give a characterization of potentials which allow the Schrödinger 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 Schrödinger 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.