Point Singularities and Nonuniqueness for the Heat Flow for Harmonic Maps

Moving mesh for the axisymmetric harmonic map flow

“…In this paper, we build solutions to (5)- (7) with constant corotational degree. We believe these solutions to be the ones described by Bertsch et al in the introduction of [3]. This approach seems to be more accurate for the description of nematic liquid crystals.…”

“…for some θ : [0, 1] × [0, +∞) → R. The examples constructed by Bertsch et al [3] and Topping [16] to prove the non-uniqueness of weak solution to (1)- (3) were actually corotational symmetric maps. In terms of θ, equations (1)- (3) are equivalent to…”

“…Briefly, at such a point, the energy E(u(t)) has a jump which corresponds to a concentration phenomenon called bubbling of spheres. The reverse phenomenon (debubbling) has been used in [3,16] to build a weak solution to (1)-(3) in H 1 loc (D × [0, +∞)) different from the Struwe solution. The idea is to keep in mind the energy lost at a bubbling time t 1 and to release it at some time t 2 > t 1 for a debubbling (the energy has an upward jump at time t 2 ).…”

A Numerical Study of Blowup in the Harmonic Map Heat Flow Using the MMPDE Moving Mesh Method

“…It is known [12,17,18] This is different from the widely studied blowup phenomenon associated with semilinear parabolic equations where the solution itself, instead of the spatial derivative of the solution, becomes unbounded in a finite time (e.g., see Friedman and Mcleod [25]). Moreover, via formal asymptotical analysis van den Berg et al [35] show that for θ 1 > π, the blowup behavior of IBVP (1.1)-(1.3) is given by lim t↑T θ µ κ (T − t) | ln(T − t)| 2 , t = 2 arctan(µ), for all fixed µ > 0, (1.5) where κ > 0 is a constant and µ is called the kernel coordinate in the literature.…”

“…Such a weak solution is unique if the energy is non-increasing along the flow [24] and is smooth except for at most finitely many singular space-time points where non-constant harmonic maps "separate" and a downward jump in the energy and blowup in the spatial derivative of the solution occur. Finite time blowup in the harmonic map heat flow has been a topic of extensive research; e.g., see [12,18,20,21,26,35].…”