78 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS
LEMMA 29 .8.
(29.33)
J[!Vvl~2 [[w 2 .p :::; C(p) for p ?_ 1, [[IVvl;2 J[c 1 .a :::; C(a) for a E (0, 1).PROOF. By the Bochner formula for 682 IVvl;2 (see (29.26)) and by applying
IV"vl2
68 2v = R + ~ - 2v to the term 2 (\7(652v ), V'v) 8 2, we compute that
652 IV'vl~2 = 2 IV^2 vl! 2 + 2 (\7(652v), \7v) 82 + 2 IV'vl~2
I
= 2 \7^2 v l2^2 \^2 ) IVvl~2
82 + -v V IV'vl 82 , \i'v 52 - 2--v 2 -- 2 IV'vl~2 + 2 (\7 R , \i'v) 82
~B.
By (29.32), (29.25), and (29.7), we have
(29.34) IV ln vi~ = IV'vl~(^2) :::; C
v
on 52 x (-oo,-1]. From this, (29.28), and l(VR,V'v) 8 2I:::; IVRl 91 :~~q
2
:::; C
(see the remark below), we obtain llB llLP :::; C(p). Thus we conclude (29.33) for
t E (-oo,-1]. D
REMARK 29 .9. Since 0 < R :::; C on (-oo, -1], the derivative estimates imply
that there exist constants Cm < oo such that IV;' R[ 9 :::; Cm on 52 x (-oo, -1].
However, this does not imply bounds for IV'$2Rl 52.
LEMMA 29.10.[[v3/2Jlc2,a :::; C(a), l[fo\72v[[ca :::; C(a), and [lv2[[c3,a :::; C(a).P W h h
I
ROOF. e ave t at n v IV"vlv ~2^1 <^1 IV"vl!2^2 1n^2 I IV"vls2 · b d d ·
112 _^2 v 312 + v v 82 v 112 IS oun e m
LP by (29.34) and (29.28) and we have that
~6 2 (v3f2) = v1/2 R _ 2 v3/2 + ~ IV'vl;2
3 5 2 vl/2 'so that 11652 ( v^312 ) II wr.p :::; C. Hence, on the time interval (-oo, -1], we have that
(29.35)
Since we have ~v^112 V^2 v + ~v-^112 \i'v ® \i'v = \7^2 (v^312 ) E W^1 ·P as well as
v-^112 \i'v ® \i'v E W^1 ·P, it follows from (29.35) that on (-oo, -1],
(29.36) [lfoV^2 v[lw1,p:::; C(p), so llfoV^2 v[lca :::; C(a).
Since 6 52(v^2 ) = 2Rv - 4v^2 + 4 IV'vl~2 ~ D , where llDllw2,p:::; C, we obtain
(29.37) D
LEMMA 29.11.(29.38)