On a new harmonic heat flow with the reverse H\"older inequalities

Abstract

This paper first proposes a new approximate scheme to construct a harmonic heat flow $u$ between a parabolic cylinder to a sphere. Y.Chen and M.Struwe have proved an existence and discussed a partial regularity of harmonic heat flows by using Ginzburg-Landau heat flow and passing to the limit of a parameter appeared in the equation. To construct a new harmonic heat flow, we propose a Ginzburg-Landau type heat flow with a time-dependent parameter. and we next establish the existence of a harmonic heat flow into spheres with (i) a global energy inequality, (ii) a monotonicity for the scaled energy, (ii) a reverse Poincare inequality. These inequalities (i), (ii) and (iii) improves the Hausdorff dimensional estimates on it's singular set contrast to the former results. I believe that inequalities (i), (ii) and (iii) allow us to analyze how it behaves around its singularities.