Global solutions for the critical, higher-degree corotational harmonic map heat flow to $\mathbb{S}^2$

Abstract

We study m-corotational solutions to the Harmonic Map Heat Flow from R 2 to S 2 . We first consider maps of zero topological degree, with initial energy below the threshold given by twice the energy of the harmonic map solutions. For m ≥ 2, we establish the smooth global existence and decay of such solutions via the concentration-compactness approach of Kenig-Merle, recovering classical results of Struwe by this alternate method. The proof relies on a profile decomposition, and the energy dissipation relation. We then consider maps of degree m and initial energy above the harmonic map threshold energy, but below three times this energy. For m ≥ 4, we establish the smooth global existence of such solutions, and their decay to a harmonic map (stability), extending results of Gustafson-Nakanishi-Tsai to higher energies. The proof rests on a stability-type argument used to rule out finitetime bubbling.

Finite Time Blowup for the Nematic Liquid Crystal Flow in Dimension Two

“…More precisely, for any given finite set of points in , they constructed solution blowing up exactly at those points simultaneously under suitable initial and boundary conditions. In another aspect, for higher-degree corotational harmonic map heat flow, global existence and blowup have been investigated in a series of works [29][30][31][32] and the references therein. For the general analysis of the bubbling phenomena and regularity results of the harmonic map heat flow, we refer the readers to the book [49].…”

Finite time blow-up for the nematic liquid crystal flow in dimension two

“…More precisely, for any given finite set of points in Ω, they constructed solution blowing up exactly at those points simultaneously under suitable initial and boundary conditions. In another aspect, for higher-degree corotational harmonic map heat flow, global existence and blow-up have been investigated in a series of works [29][30][31][32] and the references therein. For other bubbling phenomena and regularity results of the heat flow of harmonic maps, we refer the readers to the book [49] by Lin and Wang.…”