On a New Harmonic Heat Flow with the Reverse Hölder Inequalities

Abstract

This paper first proposes a new approximate scheme to construct a harmonic heat flow u between Q and $${\mathbb {S}}^D$$ $$\subset $$ $${\mathbb {R}}^{D+1}$$ with Q $$=$$ $$(0,\infty )$$ $$\times $$ $${\mathbb {B}}^d$$ and positive integers d and D: We agree that harmonic heat flow u means a solution of $$\begin{aligned} \frac{\partial u}{\partial t} \, - \, \triangle u \, - \, | \nabla u|^2 u \; = \; 0. \end{aligned}$$ Its scheme is crucially given by $$\begin{aligned} \frac{\partial u_\lambda }{\partial t} \, - \, \triangle u_\lambda \, + \, \lambda ^{1-\kappa } \bigl ( | u_\lambda |^2 \, - \, 1 \bigr ) u_\lambda \; = \; 0, \end{aligned}$$ where the unknown mapping $$u_\lambda $$ is from Q to $${\mathbb {R}}^{D+1}$$ with positive number $$\lambda $$ and $$\kappa (t)$$ $$=$$ $$\arctan (t)/\pi $$ $$(0 \le t)$$ . The benefit to introduce a time-dependent parameter $$\lambda ^{1-\kappa }$$ is readily to see $$\begin{aligned} \int _{Q} \lambda ^{1-\kappa } ( | u_\lambda |^2 \, - \, 1 )^2 \, \mathrm{d}t \mathrm{d}x \; \le \; \frac{C}{\log \lambda } \end{aligned}$$ for some positive constant C independent of $$\lambda $$ . Next, making the best of it, we prove that a passing to the limits $$\lambda \nearrow \infty $$ ( modulo sub-sequence of $$\lambda $$ ) brings the existence of a harmonic heat flow into spheres with (i) a global energy inequality, (ii) a monotonicity for the scaled energy, (iii) a reverse Poincare inequality. These inequalities (i), (ii) and (iii) improve the estimates on its singular set of a harmonic heat flow by Chen and Struwe (Math Z 201(1):83–103, 1989), i.e. I show that a singular set of the new harmonic heat flows into spheres has at most finite $$(d-\epsilon _0)$$ -dimensional Hausdorff measure with respect to the parabolic metric whereupon $$\epsilon _0$$ is a small positive number. We finally prove that if the harmonic heat flows is a constant at the boundary, then $$u_\lambda (t)$$ strongly converges to the constant $$t \nearrow \infty $$ in $$H^{1,2} ({\mathbb {B}}^d;{\mathbb {S}}^{D+1})$$ . We call this the parabolic constancy theorem. We restrict ourselves a harmonic heat flow from the unit ball into a sphere to avoid confusion of notation. But it is readily seen that our results can be extended to it between compact Riemannian manifolds using a distance function combined with Nash’s imbedding theorem.