Formal Asymptotics of Bubbling in the Harmonic Map Heat Flow

Abstract

The harmonic map heat flow is a model for nematic liquid crystals and also has origins in geometry. We present an analysis of the asymptotic behaviour of singularities arising in this flow for a special class of solutions which generalises a known (radially symmetric) reduction. Specifically, the rate at which blowup occurs is investigated in settings with certain symmetries using the method of matched asymptotic expansions. We identify a range of blowup scenarios in both finite and infinite time, including degenerate cases.

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

“…This suggests that the continuous problem has finite time blowup, consistent with the formal analysis of [35]. Finally, we consider an initial condition with two peaks,…”

“…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. PDE (1.1) is a special case of the harmonic map heat flow 6) where u(·, t) : D…”

“…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].…”

Stable self-similar blowup in the supercritical heat flow of harmonic maps

“…For d ≥ 7, there are no self-similar shrinkers [5] and the blowup is of a more complicated nature [1,4]. The case d = 2 is of special interest since it is energy-critical and blowup takes place via shrinking of a soliton [32,33,44]. The unique continuation beyond blowup is investigated in [2,22].…”

“…Indeed, this strategy works well under certain curvature assumptions as is demonstrated in the classical paper [27]. In the general case, however, the flow tends to form singularities (or "blow up") in finite time [1,2,7,8,10,20,21,25,[30][31][32][33]43,44]. This is a severe obstruction which can only be overcome if one is able to continue the flow past the singularity in a well-defined manner.…”