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.
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