SINGULARITIES AT

t

=

=

IN EQUIVARIANT HARMONIC MAP FLOW

3

(We postpone the short proof until the end of this section.) Under these con-

ditions the initial map

F0

:

D

2

---+

S2

maps the unit disc onto the unit sphere,

collapsing the boundary

8D

2

to one point (the south pole). We define

r

(1.6)

rp(r, t)

=

u( R'P(t), t).

THEOREM 1.2.

Assume

(1.4)

and

(1.5).

Then the solution

p

exists for all

t

0.

The radius R'P(t) converges to zero as t---+

oo,

with

R'P(t)

=

e-(2+o(l))Vt

(t---+

oo).

Furthermore, one has

lim

u(y,t)

=

U(y)

=

2arctany

t---CXJ

uniformly on arbitrary but bounded intervals

0 :::;

y :::; Y.

The limiting map

U(y)

=

2arctany corresponds to stereographic projection

from the plane to the sphere. It is to be expected from much more general results

on harmonic map flow (see

[S85])

that formation of a singularity should proceed

by the "bubbling off" of a sphere in the way described here.

Indeed, if

T

were finite, then for some sequence

tk /

T

and some sequence of

points

Pk

E

D

2

the maps

(1.7)

( )

-1

- def

Fk(x)

=

Ftk (Pk

+

Akx),

with

Ak

= s~p

IV'

Ftk

I

would converge to a harmonic map

P= :

JR2

---+

S

2

.

The only way in which this

can happen is for the

Pk

to converge to the origin, and for the limiting map

P

=

to

be stereographic projection onto the sphere. In corollary 6.4 we show that this is

impossible.

Thus the general theory in

[S85]

implies that the solution exists for all

t

oo,

and that its gradient remains uniformly bounded on any finite time interval 0 :::;

t :::; to.

The general theory does however not predict at what rate the gradient should

blow up. In

[BHK]

van den Berg, Hulshof and King gave a formal derivation of

what the blow-up rate for the gradient should be in all imaginable variations of

boundary and initial conditions (for the rotationally symmetric case at least). Our

main observation here is that one can rigorously prove the blow-up in the current

setting by modifying the formal solutions in

[BHK]

until they become sub- and

supersolutions for (1.2).

The plan of this paper is as follows: In §2 we recall how the formal solutions

were constructed in

[BHK].

A formal solution is strictly speaking not a solution,

but a function which satisfies an equation obtained from the original equation by

dropping terms which are "small" or otherwise unworthy of our attention. Thus, in

§2 we create a function

U(y)

+

o:(t)7,b

1

(y)

which almost satisfies the harmonic map

flow equation (2.1). After discussing a few useful differential equations in §3, we

begin the process of finding correction terms to add to

U(y)

+

o:(t)7,U1

(y) so as to

obtain a true solution of (2.1), and from there of (1.2). This analysis will suggest

a correction term v2 in §4.4, which will again not lead to an exact solution of the

equation. This time however, an appropriate choice of a free parameter in v2 turns

out to provide us with sub- and super solutions. These are exhibited in § 5. Finally

we show how the existence of these sub&super solutions implies Theorem 1.2.