20 17. ENTROPY, μ-INVARIANT, AND FINITE TIME SINGULARITIES
in ci,a (D, gJRn) for all compact domains D c JR.n. Note that now we have
wdog wi --+ w 00 log woo in c^0 ( D, 9JR.n).
Moreover, (17.54) implies
for i large enough, so that
Now by (17.58), we have
for some C < oo. In particular, w 00 E W^1 '^2 (Rn).
Proof of Step 2. First we show that, after passing to a subsequence, the
limit
(17.65) μ 00 = • i-too lim μ (g· i, !) 2 - < -c:
exists. By (17.23),
μ (gi, ~) ~ ~Rmin (gi) - 2C (1, 9i) - ~ log(27r) - n
n
~ TiRmin (g) - 2C (1, g) - 2 log(27r) - nis uniformly bounded from below since by Lemma 17.2 we have C ( 1, 2 ~i g) :::;
C(l,g) because Ti:::;~·
Now we integrate the equations (which follow from (17.53))2.6..-.w· 9i i = -μ (g· i' !) 2 w· i + !R-.w· 2 9i i - (~log 2 (27r) + n) w· i -2w· i logw· i
in Ui against a compactly supported test function. Taking the limit of the
integrations, we obtain
(17.66)
2 Ln (Vii.Joo, \l<p) dμJR.n = Ln (μoo +~log (27r) + n + 2logwoo) Woo<pdμJR.n
for all <p E Cf? (JR.n) since R9i --+ 0. That is, w 00 is a weak solution of (17.56).
Proof of Step 3. By (17.88) below, we have(17.67) ~XWi · ~exp (1 4Rmin (gi) - 4 n log(27r) -. 2 -n 1 ( 2μ 9i, 1)) 2 ·
Since Ti :::; ~, we have
(17.68)