Dispersive effects and long-time asymptotics for wave equations in exterior domains

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Dispersive effects and long-time asymptotics for wave

equations in exterior domains

David Lafontaine

To cite this version:

David Lafontaine. Dispersive effects and long-time asymptotics for wave equations in exterior domains. Analysis of PDEs [math.AP]. COMUE Université Côte d’Azur (2015 - 2019), 2018. English. �NNT : 2018AZUR4067�. �tel-02075081�

Effets dispersifs et asymptotique en

temps long d’équations d’ondes dans des

domaines extérieurs

David LAFONTAINE

Laboratoire Jean-Alexandre Dieudonné

Présentée en vue de l’obtention du grade de docteur en Mathématiques de l’Université Côte d’Azur

Dirigée par: Fabrice Planchon Soutenue le: 25 Septembre 2018

Devant le jury, composé de :

Nicolas Burq, Rapporteur, Université Paris-Sud Jean-Marc Delort, Examinateur, Université Paris 13 Thomas Duyckaerts, Examinateur, Université Paris 13 Isabelle Gallagher, Examinatrice, Ecole Normale Supérieure Oana Ivanovici, Examinatrice, CNRS et Université Côte d’Azur Fabrice Planchon, Directeur de thèse, Université Côte d’Azur Pierre Raphaël, Examinateur, Université Côte d’Azur

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C2 M2 T2mt +?2p2` +2 KMmb+`Bi bMb T2Mb2` ¨ Gm+B2- LQ`- 1m;ĕM2- *?MiH- *Q+Q-p2+ [mB DǶm`Bb iMi BKû T`i;2` +2 KQK2MiX lM ;`M/ K2`+B ¨ JB+?ďH 2i >`QmKB-/QMi H T`ûb2M+2 ¨ H bQmi2MM+2 K2 iQm+?2 #2m+QmTX 1M}M- D2 M2 TQm``Bb DKBb bb2x `2K2`+B2` K2b T`2Mib- [mB KǶQMi iQmDQm`b 7Bi +QM}M+2 2i bQmi2Mm /Mb iQmb K2b T`QD2ibX

h#H2 /2b KiBĕ`2b

AX AMi`Q/m+iBQM U6`MÏBbV Rj

RX 1biBKiBQMb /2 ai`B+?`ix ¨ HǶ2tiû`B2m` /2 THmbB2m`b Q#bi+H2b +QMp2t2b Rd

RXRX 1biBKiBQMb /2 ai`B+?`ix /Mb HǶ2bT+2 HB#`2 X X X X X X X X X X X X X X X X X X Rd RXkX 1biBKiBQMb /2 ai`B+?`ix 2i ;ûQKûi`B2 X X X X X X X X X X X X X X X X X X X X X RN RXjX 1z2i `û;mH`BbMi- 2biBKiBQMb /2 ai`B+?`ix 2i MQM@+TiBpBiû X X X X X X X X X kk RX9X 1z2i `û;mH`BbMi 2i i`D2+iQB`2b +Tiû2b X X X X X X X X X X X X X X X X X X X X kj RX8X *QKT2Mb2` H T2`i2 /m2 ¨ mM2 i`D2+iQB`2 +TiBp2 ?vT2`#QHB[m2 X X X X X X X k8 RXeX _ûbmHii , 1biBKiBQMb /2 ai`B+?`ix 2M /2?Q`b /2 /2mt Q#bi+H2b bi`B+i2K2Mi

+QMp2t2b X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X ke RXdX _ûbmHii , GǶ2tiû`B2m` /2 N ≥ 3 Q#bi+H2b bi`B+i2K2Mi +QMp2t2b pû`B}Mi H

+QM/BiBQM /ǶAFr X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X jk

kX bvKTiQiB[m2 2M i2KTb HQM; /Ƕû[miBQMb /ǶQM/2b MQM HBMûB`2b j8

kXRX GǶû[miBQM /2b QM/2b MQM HBMûB`2 /Mb /2b /QKBM2b 2tiû`B2m`b X X X X X X X j8 kXkX GǶû[miBQM /2 a+?`ƺ/BM;2` MQM HBMûB`2 p2+ mM TQi2MiB2H X X X X X X X X X X X 9k

jX S2`bT2+iBp2b 9N

AAX AMi`Q/m+iBQM U1M;HBb?V 8R

9X ai`B+?`ix 2biBKi2b QmibB/2 KMv +QMp2t Q#bi+H2b 88

9XRX ai`B+?`ix 2biBKi2b BM i?2 7`22 bT+2 X X X X X X X X X X X X X X X X X X X X X X 88 9XkX ai`B+?`ix 2biBKi2b M/ ;2QK2i`v X X X X X X X X X X X X X X X X X X X X X X X 8d 9XjX aKQQi?BM; 2z2+i- ai`B+?`ix 2biBKi2b M/ MQM@+TiBpBiv X X X X X X X X X X X ey 9X9X aKQQi?BM; 2z2+i M/ i`TT2/ i`D2+iQ`B2b X X X X X X X X X X X X X X X X X X X eR 9X8X *QKT2Mbi2 i?2 HQbb BM/m+2/ #v M ?vT2`#QHB+ i`TT2/ i`D2+iQ`v X X X X X X ej 9XeX _2bmHi, ai`B+?`ix 2biBKi2b BM i?2 2ti2`BQ` Q7 irQ bi`B+iHv +QMp2t Q#bi+H2b X e9 9XdX _2bmHi, h?2 2ti2`BQ` Q7 N ≥ 3 bi`B+iHv +QMp2t Q#bi+H2b p2`B7vBM; AFrǶb

+QM/BiBQM X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X eN N

h#H2 /2b KiBĕ`2b

8X GQM; iBK2 bvKTiQiB+b Q7 MQM@HBM2` rp2 2[miBQMb dj

8XRX h?2 MQM@HBM2` rp2 2[miBQM BM 2ti2`BQ` /QKBMb X X X X X X X X X X X X X X X dj 8XkX h?2 MQM@HBM2` a+?`ƺ/BM;2` 2[miBQM rBi?  TQi2MiBH X X X X X X X X X X X X X 3y

eX PmiHQQFb 3d

AAAX .BbT2`bBp2 2biBKi2b 3N

dX ai`B+?`ix 2biBKi2b rBi?Qmi HQbb QmibB/2 irQ bi`B+iHv +QMp2t Q#bi+H2b NR

dXRX AMi`Q/m+iBQM X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X NR dXkX _2/m+iBQM iQ HQ;`Bi?KB+ iBK2b M2` i?2 T2`BQ/B+ `v X X X X X X X X X X X X X Nj dXjX _2/m+iBQM iQ i?2 i`TT2/ `vb X X X X X X X X X X X X X X X X X X X X X X X X X X Ryy dX9X TT`QtBKiBQM Q7 i?2 bQHmiBQM M2` i?2 i`TT2/ b2i X X X X X X X X X X X X X X RRk dX8X S`QQ7 Q7 i?2Q`2K R X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Rke

3X h?2 +b2 Q7 i?2 rp2 2[miBQM RjR

3XRX AMi`Q/m+iBQM X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X RjR 3XkX _2/m+iBQM Q7 i?2 T`Q#H2K X X X X X X X X X X X X X X X X X X X X X X X X X X X X Rjj 3XjX *QMbi`m+iBQM Q7 M TT`QtBKi2 bQHmiBQM X X X X X X X X X X X X X X X X X X X X R9y 3X9X S`QQ7 Q7 i?2 KBM `2bmHi X X X X X X X X X X X X X X X X X X X X X X X X X X X X X R9N

NX h?2 2ti2`BQ` Q7 KMv +QMp2t Q#bi+H2b R88

NXRX AMi`Q/m+iBQM X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X R88 NXkX aKQQi?BM; 2z2+i rBi?Qmi HQbb QmibB/2 i?2 i`TT2/ b2i X X X X X X X X X X X X R83 NXjX _2/m+iBQM iQ i?2 HQ;`Bi?KB+ i`TT2/ b2i X X X X X X X X X X X X X X X X X X X Re9 NX9X *QMbi`m+iBQM Q7 M TT`QtBKi2 bQHmiBQM X X X X X X X X X X X X X X X X X X X X ReN NX8X S`QQ7 Q7 i?2 KBM `2bmHi X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Rd9 NXeX h?2 rp2 2[miBQM X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X RdN

AoX a+ii2`BM; 7Q` MQMHBM2` 2[miBQMb R3R

RyXa+ii2`BM; 7Q` LGa rBi?  TQi2MiBH QM i?2 HBM2 R3j

RyXRX AMi`Q/m+iBQM X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X R3j RyXkX S`2HBKBM`B2b X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X R38 RyXjX S`Q}H2 /2+QKTQbBiBQM X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X R3d RyX9X LQM HBM2` T`Q}H2b X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X RNk RyX8X *QMbi`m+iBQM Q7  +`BiB+H 2H2K2Mi X X X X X X X X X X X X X X X X X X X X X X X X RNe RyXeX _B;B/Biv X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X RN3

RRXa+ii2`BM; 7Q` i?2 MQMHBM2` rp2 2[miBQM QmibB/2 Q#bi+H2b kyR

RRXRX AMi`Q/m+iBQM X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X kyR RRXkX  b+ii2`BM; +`Bi2`BQM X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X kyj RRXjX  T`iBH `2bmHi BM i?2 2ti2`BQ` Q7 irQ bi`B+iHv +QMp2t Q#bi+H2b X X X X X X X X ky8 Ry

h#H2 /2b KiBĕ`2b RRX9X P#bi+H2b BHHmKBMi2/ #v M 2HHBTbQB/ X X X X X X X X X X X X X X X X X X X X X X kRj

RkXa+ii2`BM; 7Q` LGa rBi?  bmK Q7 irQ `2TmHbBp2 TQi2MiBHb kRN

RkXRX AMi`Q/m+iBQM X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X kRN RkXkX S`2HBKBM`B2b X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X kkR RkXjX *QMbi`m+iBQM Q7  +`BiB+H bQHmiBQM X X X X X X X X X X X X X X X X X X X X X X X X kkj RkX9X 1tiBM+iBQM Q7 i?2 +`BiB+H bQHmiBQM X X X X X X X X X X X X X X X X X X X X X X X X kjR

S`i AX

AMi`Q/m+iBQM U6`MÏBbV

h#H2 /2b KiBĕ`2b

RX 1biBKiBQMb /2 ai`B+?`ix ¨ HǶ2tiû`B2m` /2 THmbB2m`b Q#bi+H2b +QMp2t2b Rd

RXRX 1biBKiBQMb /2 ai`B+?`ix /Mb HǶ2bT+2 HB#`2 X X X X X X X X X X X X X X X X X X Rd RXkX 1biBKiBQMb /2 ai`B+?`ix 2i ;ûQKûi`B2 X X X X X X X X X X X X X X X X X X X X X RN RXkXRX GǶû[miBQM /2b QM/2b X X X X X X X X X X X X X X X X X X X X X X X X X X X RN RXkXkX GǶû[miBQM /2 a+?`ƺ/BM;2` X X X X X X X X X X X X X X X X X X X X X X X X ky RXjX 1z2i `û;mH`BbMi- 2biBKiBQMb /2 ai`B+?`ix 2i MQM@+TiBpBiû X X X X X X X X X kk RX9X 1z2i `û;mH`BbMi 2i i`D2+iQB`2b +Tiû2b X X X X X X X X X X X X X X X X X X X X kj RX8X *QKT2Mb2` H T2`i2 /m2 ¨ mM2 i`D2+iQB`2 +TiBp2 ?vT2`#QHB[m2 X X X X X X X k8 RXeX _ûbmHii , 1biBKiBQMb /2 ai`B+?`ix 2M /2?Q`b /2 /2mt Q#bi+H2b bi`B+i2K2Mi

+QMp2t2b X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X ke RXeXRX _û/m+iBQM /m T`Q#HĕK2 X X X X X X X X X X X X X X X X X X X X X X X X X kd RXeXkX *QMbi`m+iBQM /ǶmM2 bQHmiBQM TT`Q+?û2 X X X X X X X X X X X X X X X X X k3 RXeXjX G2b /û`Bpû2b /B`2+iBQMM2HH2b /2 H T?b2 X X X X X X X X X X X X X X X X X jy RXeX9X S?b2 biiBQMMB`2 2i +QM+HmbBQM X X X X X X X X X X X X X X X X X X X X jR RXeX8X GǶû[miBQM /2b QM/2b X X X X X X X X X X X X X X X X X X X X X X X X X X X jk RXdX _ûbmHii , GǶ2tiû`B2m` /2 N ≥ 3 Q#bi+H2b bi`B+i2K2Mi +QMp2t2b pû`B}Mi H

+QM/BiBQM /ǶAFr X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X jk

kX bvKTiQiB[m2 2M i2KTb HQM; /Ƕû[miBQMb /ǶQM/2b MQM HBMûB`2b j8

kXRX GǶû[miBQM /2b QM/2b MQM HBMûB`2 /Mb /2b /QKBM2b 2tiû`B2m`b X X X X X X X j8 kXRXRX 1tBbi2M+2 ;HQ#H2 2i +`Biĕ`2 /2 /BzmbBQM X X X X X X X X X X X X X X X X je kXRXkX _ûbmHii , .BzmbBQM ¨ HǶ2tiû`B2m` /ǶmM Q#bi+H2 BHHmKBMû T` mM 2HHBT@

bQś/2 X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X jd kXRXjX _ûbmHii , h?ûQ`ĕK2 /2 `B;B/Biû ¨ HǶ2tiû`B2m` /2 /2mt +QMp2t2b bi`B+ib jN

h"G1 .1a JhAĔ_1a

kXkX GǶû[miBQM /2 a+?`ƺ/BM;2` MQM HBMûB`2 p2+ mM TQi2MiB2H X X X X X X X X X X X 9k kXkXRX *QM+2Mi`iBQM@+QKT+Biûf`B;B/Biû X X X X X X X X X X X X X X X X X X X 9j kXkXkX _ûbmHii , .BzmbBQM TQm` LGa p2+ mM TQi2MiB2H X X X X X X X X X X X 98 kXkXjX _ûbmHii , .BzmbBQM TQm` LGa p2+ mM2 bQKK2 /2 /2mt TQi2MiB2Hb

`ûTmHbB7b X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X 9d

jX S2`bT2+iBp2b 9N

1biBKiBQMb /2 ai`B+?`ix ¨ HǶ2tiû`B2m` /2

THmbB2m`b Q#bi+H2b +QMp2t2b

1

aQBi Ω ⊂ Rd mM /QKBM2 ¨ #Q`/X *QMbB/û`QMb HǶû[miBQM /2 a+?`ƺ/BM;2` p2+ +QM/BiBQMb

m #Q`/ /2 .B`B+?H2i ! i∂tu− ∆u = 0, u = 0 bm` ∂Ω, u(0) = u0, UGaV BMbB [m2 HǶû[miBQM /2b QM/2b ! ∂2<sub>t</sub>u<sub>− ∆u = 0, u = 0 bm` ∂Ω,</sub> (u, ∂tu)(0) = (u0, u1). UGqV }M /2 +QKT`2M/`2 H i?ûQ`B2 T2`im`#iBp2- BMbB [m2 H2b û[miBQMb MQM HBMûB`2b bbQ+Bû2b ¨ /2 i2HH2b û[miBQMb- BH 2bi +`m+BH /2 +QKT`2M/`2 +QKK2Mi H2b bQHmiBQMb /2b û[miBQMb

HBMûB`2b /û+`QBbb2MiX 1M T`iB+mHB2`- H K2bm`2 /2b MQ`K2b /2 G2#2b;m2 2bT+2@i2KTb Lp tL

q x

-Qm 2biBKiBQMb /2 ai`B+?`ix- bǶpĕ`2 T`iB+mHBĕ`2K2Mi T2`iBM2Mi2X GǶQ#D2i /2 H T`2KBĕ`2 T`iB2 /2 +2ii2 i?ĕb2 2bi HǶQ#i2MiBQM /2 i2HH2b 2biBKiBQMb- mbbB #QMM2b [m2 /Mb HǶ2bT+2 HB#`2- /Mb mM2 ;ûQKûi`B2 +TiMi2 BMbi#H2 , HǶ2tiû`B2m` /2 THmbB2m`b Q#bi+H2b bi`B+i2K2Mi +QMp2t2bX

RXRX 1biBKiBQMb /2 ai`B+?`ix /Mb HǶ2bT+2 HB#`2

.Mb H2 +b /2 HǶ2bT+2 HB#`2 Ω = RdTQm` HǶû[miBQM /2 a+?`ƺ/BM;2`- T`2M/`2 H i`Mb7Q`@

Kû2 /2 6Qm`B2` 2M 2bT+2 /Mb UGaV 2i BMiû;`2` 2M i2KTb 7Qm`MBi H `2T`ûb2MiiBQM 2tTHB+Bi2 u(x, t) = 1

4iπ|t|d/2

"

Rd

ei|x−y|24t u₀(y)dy,

/ǶQɍ QM T2mi iB`2` HǶ2biBKiBQM /2 /BbT2`bBQM TQM+im2HH2 ∥eit∆∥L1_−→L∞ !

1 td/2,

RX 1biBKiBQMb /2 ai`B+?`ix ¨ HǶ2tiû`B2m` /2 THmbB2m`b Q#bi+H2b +QMp2t2b

[mB 2bi mM2 T`2KBĕ`2 K2bm`2 /2 H /û+`QBbbM+2 /m ~Qi HBMûB`2X 1HH2 T2`K2i 2M T`iB+mHB2`-+QK#BMû2 ¨ /2b `;mK2Mib /2 /mHBiû- /ǶQ#i2MB` /2b 2biBKiBQMb /Bi2b /2 ai`B+?`ix ,

∥eit∆u0∥Lq_(R,Lr_(Rd₎₎!∥u₀∥_L2,

Qɍ H2 +QmTH2 (q, r)- /QBi `2bT2+i2` H +QM/BiBQM /Ƕ/KBbbB#BHBiû /QMMû2 T` HǶBMp`BM+2 /Ƕû+?2HH2 /2 HǶû[miBQM- bQBi 2 q + d r = d 2, (q, r, d)̸= (2, ∞, 2).

GQ+H2K2Mi 2M i2KTb- mM2 i2HH2 2biBKiBQM `2~ĕi2 mM ;BM /ǶBMiû;`#BHBiû , bB H /QMMû2 2bi L2- H bQHmiBQM 2bi /Mb Lr- r > 2 TQm` T`2b[m2 iQmi i2KTbX :HQ#H2K2Mi- 2HH2 /û+`Bi mM 2z2i /2 /û+`QBbbM+2 , H MQ`K2 Lr <sub>/2 H bQHmiBQM /QBi i2M/`2 p2`b xû`Q- m KQBMb /Mb mM</sub>

b2Mb 2M KQv2MM2 Lq 2M i2KTbX

SQm` HǶû[miBQM /2b QM/2b /Mb Rd- H2b 2biBKiBQMb /2 ai`B+?`ix T2mp2Mi bǶû+`B`2Mi

∥u∥Lq_(R,Lr_(Rd₎₎ !∥u₀∥_H˙s+∥u1∥H˙s−1

Qɍ 2 ≤ q ≤ ∞ 2i 2 ≤ r < ∞ `2bT2+i2Mi HǶBMp`BM+2 /Ƕû+?2HH2 1 q + d r = d 2 − s, 2i H +QM/BiBQM /Ƕ/KBbbB#BHBiû TQm` H2b QM/2b 1 q + d_{− 1} 2r ≤ d_{− 1} 4 .

1HH2b T2mp2Mi āi`2 Q#i2Mm2b T` bQKKiBQM pB H i?ûQ`B2 /2 GBiiH2rQQ/@SH2v /Ƕ2biBK@ iBQMb HQ+HBbû2b 2M 7`û[m2M+2- 2HH2b KāK2b T` 2t2KTH2 +QMbû[m2M+2b /2 HǶ2biBKiBQM /2 /BbT2`bBQM TQm` H2b QM/2b

∥e−it√−∆Ψ_(h√_{−∆)u0∥L}∞ ! h− d+1

2 1

td−12

∥Ψ(h√−∆)u0∥L1,

Ψ_(h√_{−∆) HQ+HBbMi 2M 7`û[m2M+2b ∼ h}−1X

SQm` a+?`ƺ/BM;2` +QKK2 TQm` H2b QM/2b- /2 i2HH2b 2biBKiBQMb `2~ĕi2Mi H Mim`2 /BbT2`@ bBp2 /2 HǶû[miBQM , +?[m2 7`û[m2M+2 b2 T`QT;2 b2HQM /2b /B`2+iBQMb /Bzû`2Mi2b- 2i bQMi KQBMb #QMM2b TQm` H2b QM/2b- TQm` H[m2HH2 H /BbT2`bBQM 2bi THmb 7B#H2- +?[m2 7`û[m2M+2 b2 /ûTH+Mi ¨ H KāK2 pBi2bb2X 1M THmb /2 H2m` BMi2`āi BMi`BMbĕ[m2- 2HH2b 7Qm`MBbb2Mi mM QmiBHb i`ĕb `Q#mbi2 TQm` HǶûim/2 /2b û[miBQMb MQM@HBMûB`2b +Q``2bTQM/Mi2b- 2M /QMMMi-T` 2t2KTH2- /2 #QMb 2bT+2b /Mb H2b[m2Hb i`pBHH2` TQm` Q#i2MB` /2b `ûbmHiib /Ƕ2tBbi2M+2 HQ+H2 pB /2b Kûi?Q/2b /2 TQBMi }t2- BMbB [mǶmM +/`2 miBH2 ¨ HǶûim/2 /2 HǶbvKTiQiB[m2 2M i2KTb HQM; /2 i2HH2b û[miBQMbX

G2m` HQM;m2 ?BbiQB`2 `2KQMi2 mt i`pmt 7QM/i2m`b /2 ai`B+?`ix (ai`dd) TQm` H2 +b p = q TQm` H2b QM/2b /Mb Rd- ûi2M/m ¨ iQmb H2b 2tTQbMib q ̸= 2 TQm` a+?`ƺ/BM;2` T`

RXkX 1biBKiBQMb /2 ai`B+?`ix 2i ;ûQKûi`B2 :BMB#`2 2i o2HQ (:o38#)- 2i ¨ iQmb H2b 2tTQbMib r ̸= ∞ TQm` H2b QM/2b T` H2b KāK2b mi2m`b (:o38- :o3d- :oN8) BMbB [m2 GBM/#H/ 2i aQ;;2 (GaN8)X G2 +b HBKBi2 UQm

2M/TQBMiV q = 2 TQm` a+?`ƺ/BM;2`- `2bT2+iBp2K2Mi r = ∞ TQm` H2b QM/2b- 2bi THmb /ûHB+iX

AH  ûiû Q#i2Mm /Mb mM +/`2 ;ûMû`H T` (EhN3)- 2i 2bi 2M T`iB+mHB2` 7mt TQm` a+?`ƺ/BM;2` 2M /BK2MbBQM /2mt (p, q, r) = (2, ∞, 2) Ĝ pQB` T` 2t2KTH2 (JaN3)- 2i TQm` H2b QM/2b 2M /BK2MbBQM i`QBb (EJNj)X

RXkX 1biBKiBQMb /2 ai`B+?`ix 2i ;ûQKûi`B2

LQmb /QMMQMb- /Mb +2 T`;`T?2 [mB MǶ Tb pQ+iBQM ¨ āi`2 2t?mbiB7- mM #`27 ?Bb@ iQ`B[m2 /2 [m2H[m2b@mMb /2b T`BM+BTmt `ûbmHiib Q#i2Mmb /Mb /2b ;ûQKûi`B2b MQM 2m+HB@ /B2MM2b [mB MQmb b2K#H2Mi T2`iBM2Mib TQm` H bmBi2 /2 HǶ2tTQbBiBQMX

RXkXRX GǶû[miBQM /2b QM/2b

SQm` HǶû[miBQM /2b QM/2b /Mb mM2 p`Bûiû bMb #Q`/- H2b bBM;mH`Biûb b2 T`QT;2Mi b2HQM H2b ;ûQ/ûbB[m2b ¨ pBi2bb2 +QMbiMi2 2i H pBi2bb2 }MB2 /2 T`QT;iBQM KQMi`2 [mǶBH bm{i /2 b2 TH+2` 2M +QQ`/QMMû2b HQ+H2b TQm` Q#i2MB` /2b 2biBKiBQMb HQ+H2bX *Ƕ2bi H2 +?2KBM bmBpBi T` 2t2KTH2 T` (ET3N)- (JaaNj)- (aKBN3)- 2i (hiyk)X G2 +b /ǶmM /QKBM2 ¨ #Q`/-Qɍ HǶQM /QBi i`Bi2` H T`QT;iBQM /2b bBM;mH`Biûb m #Q`/- 2bi THmb /ûHB+iX GǶûim/2 /2 H T`QT;iBQM /2b bBM;mH`Biûb /Mb mM i2H +/`2  ûiû K2Mû2 T` J2H`Qb2 2i aDƺbi`M/ (Jad3- Ja3k) [mB /û}MBbb2Mi H MQiBQM /2 `vQM #B+`+iû`BbiB[m2 ;ûMû`HBbû- ;ûMû`HBbMi H MQiBQM /2 `vQM QTiB[m2- b2HQM H2b[m2Hb QMi HB2m H T`QT;iBQMX

 HǶ2tiû`B2m` /ǶmM +QMp2t2 bi`B+i- H T`Kûi`Bt /2 J2H`Qb2 2i hvHQ` (Jh3d) 7Qm`MBi mM2 /2b+`BTiBQM 2tTHB+Bi2 /2 H bQHmiBQM T`ĕb /2b TQBMib /Bz`+iB7bX 1HH2 2bi MQiKK2Mi miBHBbû2 T` aKBi? 2i aQ;;2 (aaN8) TQm` KQMi`2` [m2 H2b 2biBKiBQMb /2 HǶ2bT+2 HB#`2 bQMi p`B2b 2M /2?Q`b /ǶmM Q#bi+H2 bi`B+i2K2Mi +QMp2t2X

.Mb H2 +/`2 /ǶmM2 ;ûQKûi`B2 [m2H+QM[m2- mM2 i2HH2 T`Kûi`Bt MǶ2bi THmb miBHBb#H2 ¨ +mb2 /2 H T`ûb2M+2 /2 `vQMb `û~û+?Bb THmbB2m`b 7QBb 2i /2 H2m` HBKBi2b- H2b `vQMb /Bib

;HBbbMibX aB QM  HQM;i2KTb T2Mbû [m2 H2b 2biBKiBQMb /2 HǶ2bT+2 HB#`2 /2pB2Mi āi`2 p`B2b

¨ HǶBMiû`B2m` /ǶmM +QMp2t2 bi`B+i TmBb[m2 H2b KQ/2b /2 ;H2`B2- ++mKmHMi HǶûM2`;B2 T`ĕb /m #Q`/- M2 bQMi `2bTQMb#H2b /Ƕm+mM2 T2`i2- ApMQpB+B (ApRk) /QMM2 mM +QMi`2 2t2KTH2 7Qm`MB T` H2b +mbiB[m2b [mB T2mp2Mi TT`ŗi`2 T`ĕb /m #Q`/ 2M i2KTb T2iBiX

G2b T`2KBĕ`2b 2biBKiBQMb HQ+H2b TQm` HǶû[miBQM /2b QM/2b /Mb mM /QKBM2 ;ûMû`H QM ûiû Q#i2Mm2b T` "m`[- G2#2m 2i SHM+?QM ("GSy3)X 1HH2b bQMi H +QMbû[m2M+2 /Ƕ2biBK@ iBQMb /2 T`QD2+i2m`b bT2+i`mt /mb ¨ aKBi? 2i aQ;;2 (aayd)- 2i bQMi ;ûMû`HBbû2b 2MbmBi2 T` "HB`- aKBi? 2i aQ;;2 ("aayN)X 1HH2b b2 7QMi bMb T2`i2 /2 /û`Bpû2b UH T2`i2 p2MMi /2b /B`2+iBQMb iM;2Mi2b ûiMi +QKT2Mbû2 T` H2 7Bi [m2 +2b /B`2+iBQMb pBp2Mi /Mb mM T2iBi +ƬM2 /2 HǶ2bT+2 /2b 7`û[m2M+2bV- KBb bm` /2b TH;2b /ǶBM/B+2b `2bi`2BMi2b T` `TTQ`i ¨ HǶ2bT+2 HB#`2X

_û+2KK2Mi- ApMQpB+B- G2#2m 2i SHM+?QM (AGSR9) /Mb mM +b KQ/ĕH2- 2i ApMQpB+B-Gb+`- G2#2m 2i SHM+?QM (AGGS) KûHBQ`2Mi iQmb H2b `ûbmHiib +QMMmb TQm` d ≥ 3 ¨ HǶBMiû`B2m` /ǶmM +QMp2t2 bi`B+i [m2H+QM[m2X AHb Q#iB2MM2Mi mM2 2biBKiBQM /2 /BbT2`bBQM RN

RX 1biBKiBQMb /2 ai`B+?`ix ¨ HǶ2tiû`B2m` /2 THmbB2m`b Q#bi+H2b +QMp2t2b

TQM+im2HH2 p2+ mM2 T2`i2 /Ƕ1/4 /2 /û`Bpû2b- 2i KQMi`2Mi [m2 +2ii2 T2`i2- /m2 ¨ HǶTT`BiBQM /2 +mbiB[m2b- 2bi BMûpBi#H2 /Mb HǶ2biBKiBQM /BbT2`bBp2X .2 THmb- 2M KQMi`Mi [mǶ2HH2b TT`Bbb2Mi /2 KMBĕ`2 bm{bKK2Mi 2t+2TiBQMM2HH2 TQm` [m2 +2i 2z2i TmBbb2 āi`2 iiûMmû T` mM2 KQv2MM2 2M i2KTb- BHb Q#iB2MM2Mi /Mb (AGSR9#) /2b 2biBKiBQMb /2 ai`B+?`ix bMb T2`i2 /2 /û`Bpû2- p2+ /2b 2tTQbMib bm` mM2 TH;2 `û/mBi2 [mB +Q``2bTQM/`Bi ¨ mM2 T2`i2 /Ƕ1/6 /Mb HǶ2biBKiBQM /BbT2`bBp2X

RXkXkX GǶû[miBQM /2 a+?`ƺ/BM;2`

oBi2bb2 }MB2 /2 T`QT;iBQM TQm` HǶû[miBQM b2KB@+HbbB[m2

.Mb H2 +b /2 HǶû[miBQM /2 a+?`ƺ/BM;2`- H pBi2bb2 BM}MB2 /2 T`QT;iBQM `2M/ H2b +?Qb2b #2m+QmT THmb /ûHB+i2bX *2T2M/Mi- H2 +?M;2K2Mi /Ƕû+?2HH2 b2KB@+HbbB[m2 T2`K2i ;`+2 ¨ H Kûi?Q/2 /2 G2#2m (G2#Nk) /2 b2 `K2M2` ¨ mM2 pBi2bb2 }MB2 UKBb MQM +QMbiMi2V /2 T`QT;iBQMX PM +QMbB/ĕ`2 mM2 /QMMû2 HQ+HBbû2 2M 7`û[m2M+2b /ǶQ`/`2 h−1_{- 2i QM miBHBb2 H2}

+?M;2K2Mi /2 p`B#H2 2M i2KTb t → htX PM +QMbB/û`2` HǶû[miBQM /Bi2 b2KB@+HbbB[m2 , !

i∂tu− h∆u = 0, u = 0 bm` ∂Ω,

u(0) = Ψ(−h2_∆)u 0.

UGascV

JBMi2MMi- bQBi (en)n≥0 mM bvbiĕK2 Q`i?QMQ`KH /Mb L2(Ω) /2 7QM+iBQMb T`QT`2b /2

−∆D bm` Ω- /2 pH2m`b T`QT`2b (λn)n≥0X PM B/2MiB}2 H2b /QMMû2b u_{k,0 ∈ o2+i}#en, 2kα≤ $ λn≤ 2kβ % p2+ H2b bQHmiBQMb /m bvbiĕK2- TQm` hk= 2−k ! i∂tuk− hk∆uk= 0, uk= 0 bm` ∂Ω, uk(0) = uk,0. UGascV PM /B` [m2 ρ0 = (y0, η0) ∈ T⋆(Ω× R) ∪ T⋆(∂Ω× R) MǶ2bi Tb /Mb H2 7`QMi /ǶQM/2 m

#Q`/ W Fb(U ) /2 U = (uk,0)k<sub>≥0</sub>, bǶBH 2tBbi2 mM QTû`i2m` hk@Tb2m/Q/Bzû`2MiB2H iM;2MiB2H

m #Q`/- /2 bvK#QH2 p +QKT+i2K2Mi bmTTQ`iû- 2HHBTiB[m2 2M ρ0- 2i ψ ∈ Cc∞ û;H2 ¨ 1 T`ĕb

/2 y0 i2Hb [m2 ∀σ, ∀N, ∥PThk(p)ψuk∥Hσ(R×Ω)≤ CN,σh N k. HQ`b- /ǶT`ĕb (G2#Nk )-W Fb(uk)⊂ Σb∩ {τ ∈ [α, β], |ξ| ≤ D} , Qɍ Σb2bi HǶ2Mb2K#H2 +`+iû`BbiB[m2 Σb = & τ_{− |ξ|}2= 0'X .2 THmb- 2M TQbMi Θ(U)(x, t, s) = ( ke−ih −1 k su_k(x, t, s)- Θ(U) pû`B}2 ∂<sub>t,s</sub>2 Θ<sub>(U )− ∆(U) = 0, u = 0 bm` ∂Ω × R × R,</sub> TQm` H2[m2H QM /BbTQb2 /QM+ /m i?ûQ`ĕK2 /2 J2H`Qb2 2i aDƺbi`M/X .ǶT`ĕb (G2#Nk)- QM  ky

RXkX 1biBKiBQMb /2 ai`B+?`ix 2i ;ûQKûi`B2

G2KK2 RXRX SQm` iQmi s0 ∈ R- TQm` iQmi ρ0 ∈ T⋆(Ω× R) ∪ T⋆(Ω×

R)-ρ0∈ W Fb(U ) ⇐⇒ θ(ρ0, s0)∈ W Fb(Θ(U ))

Qɍ θ(ρ, s) := (ρ; s, 1)X

G2 7`QMi /ǶQM/2 m #Q`/ /2 H bmBi2 U 2bi /QM+ BMp`BMi T` H2 ~Qi #B+`+iû`Bb[m2 ;ûMû`HBbû bm` T⋆_{(Ω× R) ∪ T}⋆_{(∂Ω× R)X}

1biBKiBQMb /2 ai`B+?`ix

1M i`pBHHMi bm` HǶû[miBQM b2KB@+HbbB[m2 p2+ /2b /QMMû2b bT2+i`H2K2Mi HQ+HBbû2b-QM T2mi /HQ+HBbû2b-QM+ iB`2` T`iB /2 H pBi2bb2 }MB2 /2 T`QT;iBHQ+HBbû2b-QM 2i- /Mb H2 +b /ǶmM2 p`Bûiû bMb #Q`/- i`pBHH2` 2M +QQ`/QMMû2b HQ+H2bX *Ƕ2bi T` 2t2KTH2 H bi`iû;B2 bmBpB2 T` "m`[- :û``/ 2i hx2iFQp (":hy9#)- [mB- bǶBMiû`2bb2Mi ¨ mM2 p`Bûiû +QKT+i2 bMb #Q`/X 1M i`pBHHMi bm` HǶû[miBQM b2KB@+HbbB[m2 p2+ /QMMû2 HQ+HBbû2 2M 7`û[m2M+2 ∼ h−1_{- BHb}

+QMbi`mBb2Mi mM2 T`Kûi`Bt 2M i2KTb h /QMi BHb /û/mBb2Mi /2b 2biBKiBQMb /2 ai`B+?`ix p2+ T2`i2 /Ƕ1/p /û`Bpû2b ,

∥eit∆_u

0∥Lp_{([0,T ],L}q_{(M ))}≤ C(T )∥u₀∥_H1/p_{(M )}.

LQiQMb [mǶmM2 i2HH2 T2`i2 2bi Mim`2HH2 m b2Mb Qɍ HǶûM2`;B2 b2 T`QT;2Mi ¨ H pBi2bb2 h−1

-QM bǶii2M/ ¨ TQmpQB` K-QMi`2` /2b 2biBKiB-QMb bMb T2`i2 2M i2KTb h- [mB T`ĕb bQKKiB-QM- bQKKiBQM-/QMM2Mi mM2 T2`i2 1/pX J2MiBQMMQMb û;H2K2Mi- /Mb H2 +b /ǶmM QTû`i2m` /2 a+?`ƺ/BM;2` ¨ +Q2{+B2Mib p`B#H2b- H2b i`pmt /2 ai{HMB 2i hi`m (ahyk)X

G2 +b /ǶmM2 p`Bûiû p2+ #Q`/- Qɍ BH 7mi i2MB` +QKTi2 /2b 2z2ib /2 `2~2tBQM- 2bi- 2M+Q`2 mM2 7QBb- THmb /ûHB+iX lM2 bi`iû;B2- bmBpB2 T` MiQM (Miy3) TmBb "HB`- aKBi? 2i aQ;;2 ("aay3)- +QMbBbi2- 2M `2/`2bbMi H2 #Q`/ 2i 2M 7BbMi mM2 bvKûi`B2 T` `TTQ`i ¨ +2HmB@ +B- ¨ b2 `K2M2` m +b /ǶmM2 p`Bûiû bMb #Q`/b KBb p2+ +Q2{+B2Mib GBTb?BixB2MbX AHb Q#iB2MM2Mi BMbB /2b 2biBKiBQMb p2+ T2`i2 /2 4

3p /û`Bpû2b- bQBi

∥eit∆u_0∥Lp_{([0,T ],L}q_{(M ))}≤ C(T )∥u₀∥_H4/3p_{(M )}.

*2T2M/Mi- mM2 i2HH2 Kûi?Q/2 M2 iB2Mi Tb +QKTi2 /2 H ;ûQKûi`B2 T`iB+mHBĕ`2 /m #Q`/ 2i MǶ Tb /Ƕ2bTQB` /Ƕāi`2 miBHBbû2b /Mb /2b ;ûQKûi`B2b THmb T`iB+mHBĕ`2b Qɍ /2b 2biBKiBQMb K2BHH2m`2b bQMi 2bTû`û2bX .Mb H2 +b /ǶmM2 p`Bûiû +QKT+i2 p2+ #Q`/ bi`B+i2K2Mi +QM+p2-/QMi mM 2t2KTH2 2bi /QMMû T` H2 #BHH`/ /2 aBMś- ApMQpB+B iB`2 T`iB /Mb (ApRy) /2 H pBi2bb2 }MB2 /2 T`QT;iBQM TQm` HǶû[miBQM b2KB@+HbbB[m2 2i /2 H T`Kûi`Bt /2 J2H`Qb2 2i hvHQ`- TQm` KQMi`2` /2b 2biBKiBQMb 2M i2KTb h bm` H2b /QMMû2b HQ+HBbû2b 2M 7`û[m2M+2-[mB mM2 MQmp2HH2 7QBb 7Qm`MBbb2Mi T`ĕb bQKKiBQM /2 GBiiH2rQQ/@SH2v- /2b 2biBKiBQMb HQ+H2b 2M i2KTb p2+ T2`i2 /Ƕ1/p /û`Bpû2b bm` HǶû[miBQM +HbbB[m2X  HǶB/2 /ǶmM `;mK2Mi /QMi MQmb /Bb+mi2`QMb m T`;`T?2 bmBpMi- 2HH2 2M /û/mBi /2b 2biBKiBQMb ;HQ#H2b mbbB #QMM2b [m2 /Mb HǶ2bT+2 HB#`2 ¨ HǶ2tiû`B2m` /Mb +QMp2t2 bi`B+iX S` BHH2m`b- 2HH2 KQMi`2 /Mb (ApRk) [mǶ¨ HǶBMiû`B2m` /ǶmM +QMp2t2 bi`B+i- mM2 T2`i2 /Ƕ1/6 /2 /û`Bpû2b 2bi BMûpBi#H2X aQM +QMi`2 2t2KTH2 2bi /QMMû T` H2b KQ/2b /2 ;H2`B2- [mB- `TT2HQMb@H2- MǶBM/mBb2Mi m+mM2 T2`i2 /Mb H2 +b /2 HǶû[miBQM /2b QM/2bX

RX 1biBKiBQMb /2 ai`B+?`ix ¨ HǶ2tiû`B2m` /2 THmbB2m`b Q#bi+H2b +QMp2t2b

1M}M- MQiQMb [m2 ;ûMû`HBb2` H2b i`pmt (AGSR9) 2i (AGGS) ¨ HǶû[miBQM /2 a+?`ƺ/BM;2` }M /ǶQ#i2MB` /2b 2biBKiBQMb }M2b ¨ HǶBMiû`B2m` /ǶmM +QMp2t2 bi`B+i b2K#H2 TQm` H2 KQK2Mi ?Q`b /2 TQ`iû2 , H pBi2bb2 MQM@+QMbiMi2 /2 T`QT;iBQM `2M/ H2b `û~2tBQMb KmHiBTH2b m #Q`/ 2ti`āK2K2Mi /B{+BH2b ¨ ;û`2`X

_2K`[m2X .2b 2biBKiBQMb /2 ai`B+?`ix bMb T2`i2 MǶBKTHB[m2Mi Tb Mû+2bbB`2K2Mi mM2

2biBKiBQM /2 /BbT2`bBQM TQM+im2HH2 mbbB #QMM2 [m2 /Mb HǶ2bT+2 HB#`2X lM 2t2KTH2 `2@ K`[m#H2 /2 +2 7Bi 2bi 7Qm`MBi T` ApMQpB+B 2i G2#2m- [mB KQMi`2Mi 2M T`iB+mHB2` /Mb (AGRd) [m2 HǶ2biBKiBQM /2 /BbT2`bBQM TQM+im2HH2 2bi 7mbb2 ¨ HǶ2tiû`B2m` /ǶmM2 #QmH2 2M /B@ K2MbBQMb d ≥ 4X *QKK2 BH MǶv  Tb /Ƕ2z2i +QM+2Mi`iBQM 2tTHB+Bi2 /2 HǶûM2`;B2 /Mb mM i2H T`Q#HĕK2- +2 `ûbmHii 2bi bm`T`2MMiX *2T2M/Mi- /ǶT`ĕb H2 T`BM+BT2 /2 >mv;2Mb- +?[m2 TQBMi /m #Q`/ ii2BMi T` HǶQM/2 b2 +QKTQ`i2 +QKK2 mM2 MQmp2HH2 bQm`+2 /2 HmKBĕ`2- [mB T2mi /QM+ `27Q+HBb2` /Mb HǶQK#`2 /2 HǶQ#bi+H2 2M mM TQBMi 2tTû`BK2MiH2K2Mi Q#b2`pû /Bi /2 SQBbbQM@`;Q- [mB 2bi ¨ HǶQ`B;BM2 /m +QMi`2 2t2KTH2 /QMMû T` H2b mi2m`bX

RXjX 1z2i `û;mH`BbMi- 2biBKiBQMb /2 ai`B+?`ix 2i MQM@+TiBpBiû

lM2 Q#bi`m+iBQM BMimBiBp2 ¨ HǶQ#i2MiBQM /Ƕ2biBKiBQMb /2 ai`B+?`ix ;HQ#H2b 2i bMb T2`i2b 2bi H T`ûb2M+2 /2 `vQMb +Tiûb- [mB /2p`B2Mi- m KQBMb 2M mM +2`iBM b2Mb- BM/mB`2 /2b 2z2ib /2 +QM+2Mi`iBQM /2 HǶûM2`;B2X SQm` +2ii2 `BbQM- BH 2bi Mim`2H /2 b2 TH+2` bQmb HǶ?vTQi?ĕb2 /2 MQM@+TiBpBiû ,

.û}MBiBQMX PM /Bi [mǶmM Q#bi+H2 Θ ⊂ Rd2bi MQM +TiMi bB iQmb H2b `vQMb /2 HǶQTiB[m2

;ûQKûi`B[m2 /2 Rd<sub>\Θ</sub> bQ`i2Mi /2 iQmi +QKT+i 2M i2KTb }MBX .2 H KāK2 KMBĕ`2- QM /Bi

[mǶmM2 p`Bûiû bMb #Q`/ (M, g) 2bi MQM@+TiMi2 bB iQmi2b H2b ;ûQ/ûbB[m2b bQ`i2Mi /2 iQmi +QKT+i 2M i2KTb }MBX

SHmbB2m`b mi2m`b Q#iB2MM2Mi /2b `ûbmHiib 2M b2 TH+Mi bQmb +2ii2 ?vTQi?ĕb2 , +BiQMb MQiKK2Mi H2b i`pmt /2 aKBi? 2i aQ;;2 (aayy)- ai{HMB 2i hi`m (ahyk)- "m`[ ("m`yj )-J2i+H72 (J2iy9)- >bb2HH- hQ 2i qmMb+? (>hqye)- "Qm+H2i 2i hx2iFQp ("hyd)- BMbB [m2 +2mt /2 "Qm+H2i ("QmRR)X

1M T`iB+mHB2`- aKBi? 2i aQ;;2 KQMi`2Mi /Mb (aayy) 2M /BK2MbBQM BKTB`2 TQm` H2b QM/2b- [m2 bQmb HǶ?vTQi?ĕb2 /2 MQM@+TiBpBiû- /2b 2biBKiBQMb /2 ai`B+?`ix HQ+H2b 2M i2KTb BKTHB[m2Mi /2b 2biBKiBQMb ;HQ#H2bX *2 `ûbmHii 2bi ûi2M/m ¨ iQmi H2b /BK2MbBQMb /Ƕ2bT+2 BM/ûT2M/KK2Mi T` "m`[ ("m`yj) 2i J2i+H72 (J2iy9)X lM 7Bi `2K`[m#H2 TT`BbbMi MQiKK2Mi /Mb ("m`yj) 2bi [m2 /2b 2biBKiBQMb /2 /û+`QBbbM+2 HQ+H2 /2 HǶûM2`;B2 TQm` H2b QM/2b

∥(χu, χ∂tu)∥_L2_{(R, ˙H}1_×L2₎!∥u0∥_H˙1+∥u1∥L2, ∀χ ∈ C_c∞,

[mB bQMi 2M T`iB+mHB2` p`B2b /Mb HǶ2bT+2 HB#`2- T2`K2ii2Mi /2 K2ii`2 #Qmi ¨ #Qmi /2b 2biBKiBQMb /2 ai`B+?`ix HQ+H2b 2M i2KTb TQm` Q#i2MB` mM2 2biBKiBQM ;HQ#H2X .2 H KāK2 KMBĕ`2- H +QMi`2T`iB2 TQm` a+?`ƺ/BM;2` /2 H /û+`QBbbM+2 HQ+H2 /2 HǶûM2`;B2- bQBi HǶ2z2i `û;mH`BbMi

∥χu∥L2_(R,H1/2₎!∥u0∥L2, ∀χ ∈ C_c∞,

RX9X 1z2i `û;mH`BbMi 2i i`D2+iQB`2b +Tiû2b [mB `2~ĕi2 mM ;BM /ǶmM2 /2KB@/û`Bpû2- T2`K2i /2 K2ii`2 #Qmi ¨ #Qmi /2b 2biBKiBQMb /2 ai`B+?`ix b2KB@+HbbB[m2b 2M i2KTb hX >2m`BbiB[m2K2Mi- HǶ2biBKiBQM `û;mH`BbMi2 bB;MB}2 [mǶmM2 bQHmiBQM /2 HǶû[miBQM /2 a+?`ƺ/BM;2` ¨ 7`û[m2M+2 ∼ h−1 _{Tbb2 mM i2KTb ∼ h T`ĕb}

/2 HǶQ#bi+H2- BH bm{i /QM+ /ǶQ#i2MB` /2b 2biBKiBQMb TQm` /2b i2KTb /2 HǶQ`/`2 /2 hX *Ƕ2bi T` 2t2KTH2 H bi`iû;B2 bmBpB2 T` (ApRy) TQm` /û/mB`2 /2b 2biBKiBQMb b2KB@+HbbB[m2b 2M i2KTb h bm` H2 #BHH`/ /2 aBMś- /2b 2biBKiBQMb ;HQ#H2b 2i bMb T2`i2 ¨ HǶ2tiû`B2m` /ǶmM Q#bi+H2 +QMp2t2X

LQiQMb [mǶmM T`û+m`b2m` `2K`[m#H2 ¨ /2b `ûbmHiib /Ƕ2biBKiBQMb HQ+H2 /2 HǶûM2`;B2 ¨ HǶ2tiû`B2m` /ǶQ#bi+H2b MQM +TiB7b TQm` H2b QM/2b bQMi H2b i`pmt /2 JQ`r2ix (JQ`eR )-JQ`r2ix- _HibQM 2i ai`mbb (J_add)- [mB bǶTTmB2Mi bm` H +QMbi`m+iBQM /ǶmM2 7QM+iBQM /Ƕû+?TT2K2Mi- +Ƕ2bi ¨ /B`2 mM2 7QM+iBQM /2 HǶ2bT+2 /2b T?b2b [mB +`Qŗi b2HQM H2b `vQMb 2i `2bT2+i2 H `û~2tBQM m #Q`/X lM2 i2HH2 7QM+iBQM  ûiû +QMbi`mBi2 T` (J_add) /Mb H2 +b /ǶmM Q#bi+H2 MQM +TiMiX a +QMbi`m+iBQM /Mb H2 +/`2 /2 p`Bûiûb bMb #Q`/b 2bi m +Q2m` /2 i`pmt THmb `û+2Mib , +BiQMb- T` 2t2KTH2 H2 `ûbmHii /2 .i+?2p 2i obv (.oRk)X "m`[- :û``/ 2i hx2iFQp (":hy9) KQMi`2Mi H2 KāK2 2z2i `û;mH`BbMi [m2 /Mb HǶ2bT+2 HB#`2 ¨ HǶ2tiû`B2m` /ǶmM Q#bi+H2 MQM@+TiMi ;ûMû`HX AHb HǶQ#iB2MM2Mi +QKK2 +QMbû[m2M+2 /2 HǶ2biBKiBQM /2 `ûbQHpMi2

∥χ(−∆D− (λ ± i.)2)−1χ∥L2_(Ω)−→L2_(Ω)!|λ|−1,

|λ|2 1, 0 < . 3 1,

TQm` H[m2HH2 HǶ?vTQi?ĕb2 /2 MQM@+TiBpBiû DQm2 mM `ƬH2 +`m+BH- KQMi`û2 2M ;`M/2 ;ûMû`@ HBiû 2Mi`2 mi`2b T` Gt 2i S?BHBTb (GS3N)- J2H`Qb2 2i aDƺbi`M/ (Jad3-Ja3k)- oBM#2`; (oM33)- obv 2i wrQ`bFB (owyy)- 2i [mB T2`K2i û;H2K2Mi- p2+ mM2 Kûi?Q/2 bBKBHB`2- /2 /û/mB`2 H /û+`QBbbM+2 HQ+H2 /2 HǶûM2`;B2 TQm` H2b QM/2bX .Ƕmi`2 T`i- "m`[ KQMi`2 2M T`iB+mHB2` /Mb ("m`y9)- /Mb mM +/`2 i`ĕb ;ûMû`H- [m2 HǶ2z2i `û;mH`BbMi bMb T2`i2- v +QKT`Bb HQ+H 2M i2KTb- 2bi 7mt bǶBH 2tBbi2 mM2 i`D2+iQB`2 +Tiû2X SHmb T`û+BbûK2Mi- /Mb H2 +/`2 [mB MQmb BMiû`2bb2- TQm` iQmi t0 >0- HǶTTHB+iBQM

u0 ∈ Cc∞⊂ L2 −→ χeit∆u0 ∈ L2([0, t0], H

1 2)

MǶ2bi DKBb #Q`Mû2 HQ`b[m2 χ MǶ2bi Tb B/2MiB[m2K2Mi MmHH2 bm` HǶ2Mb2K#H2 +TiûX

RX9X 1z2i `û;mH`BbMi 2i i`D2+iQB`2b +Tiû2b

lM2 [m2biBQM Mim`2HH2 bmBi2 m `ûbmHii /2 ("m`y9) ûpQ[mû T`û+û/2KK2Mi 2bi /2 b2 /2@ KM/2` bB mM 2z2i `û;mH`BbMi p2+ T2`i2 b2`Bi pH#H2 /Mb +2`iBM2b ;ûQKûi`B2b +TiBp2bX PM T2mi- 2bb2MiB2HH2K2Mi- /BbiBM;m2` i`QBb ivT2b /2 i`D2+iQB`2b +TiBp2b , 2HHBTiB[m2b- T`@

#QHB[m2b- 2i ?vT2`#QHB[m2bX *QMbB/û`Mi HǶTTHB+iBQM /2 SQBM+`û /2 T`2KB2` `2iQm` P /Mb

mM pQBbBM;2 /2 HǶ2bT+2 /2b T?b2b /ǶmM TQBMi ρ ∈ T⋆_Ω /ǶmM2 i`D2+iQB`2 +Tiû2- QM /Bi

[m2 H i`D2+iQB`2 2bi 2HHBTiB[m2 Ubi#H2V bB iQmi2b H2b pH2m`b T`QT`2b /2 H /Bzû`2MiB2HH2 /2 P 2M ρ bQMi /2 KQ/mH2b THmb T2iBib [m2 mM- T`#QHB[m2 UM2mi`2V- bB +2b pH2m`b T`QT`2b bQMi iQmi2b /2 KQ/mH2 û;mt ¨ mM- Qm ?vT2`#QHB[m2 UBMbi#H2V bB BH 2M 2tBbi2 m KQBMb mM2 /2 KQ/mH2 THmb ;`M/ [m2 mMX

RX 1biBKiBQMb /2 ai`B+?`ix ¨ HǶ2tiû`B2m` /2 THmbB2m`b Q#bi+H2b +QMp2t2b

aǶBH 2tBbi2 mM2 i`D2+iQB`2 2HHBTiB[m2- BH 2bi TQbbB#H2 /2 +QMbi`mB`2 /2b [mbBKQ/2b ¨ bmTTQ`i +QKT+i (en)n<sub>≥1</sub> bbQ+Bûb ¨ λn−→ ∞ −∆en= λnen+ rn, ∥rn∥Hs = O(λ−∞<sub>n</sub> ), HQ`b- TQm` iQmi t0 >0 2i iQmi s > 0 ∥eit∆en∥L1_([0,t0[,Hs₎≥ ∥ " t0 0 eit∆en∥Hsdt= t₀∥λ_ne_n+ r_n∥_Hs ≥ t0λn− O(λ−∞n ),

2i m+mM ivT2 /Ƕ2z2i `û;mH`BbMi M2 T2mi āi`2 p`BX

GǶ2z2i `û;mH`BbMi ¨ HǶ2tiû`B2m` /2 THmbB2m`b Q#bi+H2b bi`B+i2K2Mi +QMp2t2b LQmb MQmb BMiû`2bbQMb B+B ¨ HǶ2t2KTH2 +MQMB[m2 /m +b ?vT2`#QHB[m2- bQBi HǶ2tiû`B2m` /2 /2mt Q#bi+H2b bi`B+i2K2Mi +QMp2t2b- 2i b ;ûMû`HBbiBQM ¨ HǶ2tiû`B2m` /2 THmbB2m`b (N ≥ 3V Q#bi+H2b bi`B+i2K2Mi +QMp2t2bX lM i2H +/`2  ûiû ûim/Bû /Mb mM T`2KB2` i2KTb T` JXAFr (AF3k- AF33) 2i *X :û``/ (:3œ3) /Mb H2b MMû2b RN3yX AFr KQMi`2 2M T`iB@ +mHB2` /Mb (AF3k) ¨ HǶ2tiû`B2m` /2 /2mt Q#bi+H2b- `ûbmHii [mǶBH ;ûMû`HBb2 THmb i`/ /Mb (AF33) ¨ HǶ2ti2`B2m` /ǶmM MQK#`2 `#Bi`B`2 /ǶQ#bi+H2b- [m2 H `ûbQHpMi2 χ(−∆D−τ2)−1

χ-?QHQKQ`T?2 bm` {AKτ=y} - /K2i mM T`QHQM;2K2Mi χ-?QHQKQ`T?2 bm` bm` mM2 #M/2 /m /2KB@2bT+2 bmTû`B2m` {|τ| > 1, AKτ ≤ α} 2i v biBb7Bi

∥χ(−∆ − τ2)−1χ_∥L2_(Ω)→L2_(Ω)≤ C|τ|N,

TQm` mM +2`iBM N 2 1- +2 [mB BKTHB[m2 2M T`iB+mHB2` /2b 2biBKiBQMb /2 /û+`QBbbM+2 HQ@ +H2b /2 HǶûM2`;B2 2i 2z2ib `û;mH`BbMi p2+ T2`i2 /ǶmM MQK#`2 }MB /2 /û`Bpû2bX  HǶ2tiû`B2m` /2 THmbB2m`b (N ≥ 3V Q#bi+H2b- BH MǶv  MQM THmb mM2 b2mH2- KBb mM2 BM}MBiû /2 i`D2+iQB`2b +TiBp2bX GǶ?vT2`#QHB+Biû /2b i`D2+iQB`2b +QMbB/û`û2b bûT`2KK2Mi M2 bm{i Tb 2i BH 7mi bǶBMiû`2bb2` ¨ H2m` BMi2`+iBQMX AFr 7Bi /Mb (AF33) mM2 ?vTQi?ĕb2 /Ƕ?vT2`#QHB+Biû 7Q`i2 ,

.û}MBiBQM RXk U*QM/BiBQM /ǶAFr- R , ?vT2`#QHB+Biû 7Q`i2VX AH 2tBbi2 α > 0 i2H [m2

) γ_∈P λγdγeαdγ <∞, URX9XRV Qɍ P /ûbB;M2 HǶ2Mb2K#H2 /2b i`D2+iQB`2b +Tiû2b- dγ H HQM;m2m` /2 γ ∈ P 2i λγ = * µ1 γµ2γ Qɍ µ1,2

γ bQMi H2b /2mt pH2m`b T`QT`2b /2 KQ/mH2 THmb T2iBi [m2 mM /2 HǶTTHB+iBQM /2

SQBM+`û bbQ+Bû2 ¨ γX

*2ii2 ?vTQi?ĕb2 b2` THmb i`/ +QMMMm2 bQmb H2 MQK /2 +QM/BiBQM /ǶAFrX aQM i`pBH bǶTTmB2 bm` mM2 +QMbi`m+iBQM /2 T`Kûi`Bt TQm` HǶû[miBQM /2b QM/2b [mB b2` `2T`Bb2 T` "m`[ ("m`Nj) 2i MQmb KāK2- 2i b2` /û+`Bi2 /Mb H bmBi2 /2 +2ii2 2tTQbBiBQMX *2ii2 +QMbi`m+iBQM b2 7Bi m T`Bt /2 HǶ?vTQi?ĕb2 i2+?MB[m2 bmBpMi2

RX8X *QKT2Mb2` H T2`i2 /m2 ¨ mM2 i`D2+iQB`2 +TiBp2 ?vT2`#QHB[m2

.û}MBiBQM RXj U*QM/BiBQM /ǶAFr- k , Tb /ǶQ#bi+H2 /Mb HǶQK#`2VX SQm` iQmb i, j, k

/2mt ¨ /2mt

/BbiBM+ib-*QMp(Θi∪ Θj)∩ Θk=∅, URX9XkV

"m`[ `2K`[m2 /Mb ("m`y9) [m2 H2 `ûbmHii /ǶAFr- +QK#BMû ¨ mM2 2biBKiBQM biM/`/ TQm` H2b QTû`i2m`b miQ@/DQBMib 2i ¨ mM T`BM+BT2 /m KtBKmK b2KB+HbbB[m2 BMbTB`û /2 h?M; 2i wrQ`bFB (hwyy)- BKTHB[m2 2M 7Bi HǶ2biBKiBQM THmb }M2

∥χ(−∆D− (λ ± i.))−1χ∥L2_(Ω)−→L2_(Ω)! HQ;(2 +

√ λ) 1 +√λ , λ_{∈ R, 0 < . 3 1,}

2i /QM+ mM2 2biBKiBQM `û;mH`BbMi2 p2+ mM2 T2`i2 `#Bi`B`2K2Mi T2iBi2 /2 . > 0 /û`B@ pû2bX SHmb T`û+BbûK2Mi- HǶ2biBKiBQM /2 `ûbQHpMi2 T`û+û/2Mi2 BKTHB[m2 TQm` /2b /QMMû2b HQ+HBbû2b 2M 7`û[m2M+2 mM2 T2`i2 HQ;`Bi?KB[m2 T` `TTQ`i ¨ H 7`û[m2M+2- bQBi ,

∥χeit∆ψ(_−h2∆_)u0∥L2

(R,H1/2₎!|HQ; h| 1

2∥ψ(−h2∆)u₀∥_L2, ∀χ ∈ C_c∞,

Qɍ ψ(−h2_{∆)HQ+HBb2 ¨ H 7`û[m2M+2 hX}

_2K`[m2X G2 +b BMi2`Kû/BB`2 2Mi`2 H T2`i2 iQiH2 2M;2M/`û2 T` mM2 i`D2+iQB`2 2HHBT@

iB[m2 2i H T2`i2 HQ;`Bi?KB[m2 BM/mBi2 T` mM 2Mb2K#H2 +Tiû /2 ivT2 ?vT2`#QHB[m2 2bi ûim/Bû T` *?`BbiBMbQM 2i qmMb+? (*qRj)- [mB KQMi`2Mi mM2 2biBKiBQM `û;mH`BbMi2 QTiBKH2 p2+ T2`i2 TQHvMQKBH2 /Mb mM2 p`Bûiû vMi mM2 b2mH2 i`D2+iQB`2 Tû`BQ/B[m2 7B#H2K2Mi ?vT2`#QHB[m2X

RX8X *QKT2Mb2` H T2`i2 /m2 ¨ mM2 i`D2+iQB`2 +TiBp2

?vT2`#QHB[m2

aB H2 HB2M T`û+Bb 2Mi`2 2z2i `û;mH`BbMi bMb T2`i2 2i T`ûb2M+2 /ǶmM2 i`D2+iQB`2 +Tiû2 2bi KBMi2MMi #B2M +QKT`Bb- bmBi2 MQiKK2Mi m `ûbmHii /2 ("m`y9)- +2 MǶ2bi Tb 2M+Q`2 H2 +b TQm` H2b 2biBKiBQMb /2 ai`B+?`ixX 1M T`iB+mHB2`- "m`[- :mBHH`KQm 2i >bb2H ûi#HBbb2Mi /Mb (":>Ry) /2b 2biBKiBQMb /2 ai`B+?`ix bMb T2`i2 /Mb mM2 p`Bûiû (M, g) bMb #Q`/-+QMB[m2 ¨ HǶBM}MB- /2 +Qm`#m`2 Mû;iBp2 2i vMi mM 2Mb2K#H2 +Tiû /2 ivT2 ?vT2`#QHB[m2 bm{bKK2Mi T2iBi- ûi#HBbbMi BMbB [m2 H T`ûb2M+2 /2 i`D2+iQB`2b +Tiû2b- TQm`pm [mǶ2HH2b bQB2Mi bm{bKK2Mi BMbi#H2b- MǶBM/mBb2Mi Tb Mû+2bbB`2K2Mi mM2 T2`i2 m MBp2m /2b 2biBKiBQMb /2 ai`B+?`ixX SHmb T`û+BbûK2Mi- H2b mi2m`b +QMbB/ĕ`2Mi mM2 p`Bûiû +QMB[m2 ¨ HǶBM}MB /2 +Qm`#m`2 bi`B+i2K2Mi Mû;iBp2 /Mb mM T`iB2 +QKT+i2 +QMp2t2 [mB BM+Hmi HǶ2Mb2K#H2 +Tiû- +2 /2`MB2` ûiMi bmTTQbû bb2x T2iBi m b2Mb Qɍ b T`2bbBQM iQTQHQ;B[m2 2M s = 1

2 2bi bi`B+i2K2Mi Mû;iBp2 ,

P(1 2) < 0.

G T`2bbBQM iQTQHQ;B[m2 P (s)- /QMi QM TQm`` i`Qmp2` mM2 /û}MBiBQM T`û+Bb2 /Mb (Lwye) 2i (":>Ry)- K2bm`2 H +QKTûiBiBQM 2Mi`2 H /2MbBiû /2 HǶ2Mb2K#H2 +Tiû 2i HǶBMbi#BHBiû /m k8

RX 1biBKiBQMb /2 ai`B+?`ix ¨ HǶ2tiû`B2m` /2 THmbB2m`b Q#bi+H2b +QMp2t2b

~QiX GǶ?vTQi?ĕb2 P (1/2) < 0 bB;MB}2 [m2 +Ƕ2bi HǶBMbi#BHBiû [mB HǶ2KTQ`i2X AH bǶ;Bi 2M 7Bi /2 HǶ?vTQi?ĕb2 /ǶAFr /Ƕ?vT2`#QHB+Biû 7Q`i2 URX9XRV- û+`Bi2 /Mb H2 T`ûb2Mi +QMi2ti2 , +2 T`Q#HĕK2 2bi /QM+ mM ?QKQHQ;m2 bMb #Q`/ m T`Q#HĕK2 ¨ HǶ2tiû`B2m` /2 THmbB2m`b Q#bi+H2b bi`B+i2K2Mi +QMp2t2b pû`B}Mi H +QM/BiBQM /ǶAFrX

.Mb mM2 i2HH2 ;ûQKûi`B2- HǶ2z2i `û;mH`BbMi bMb T2`i2 MǶ2bi THmb pH#H2 /ǶT`ĕb ("m`y9 )-2i M2 T2mi /QM+ THmb āi`2 miBHBbû TQm` K)-2ii`2 #Qmi ¨ #Qmi /2b 2biBKiBQMb /2 ai`B+?`ix b2KB@+HbbB[m2b 2M i2KTb hX LQMM2MK+?2` 2i wrQ`bFB (Lwye)- .i+?2p (.iyN) KQMi`2Mi [m2 /2 H KāK2 KMBĕ`2 [m2 TQm` H2 T`Q#HĕK2 /ǶAFr- mM 2z2i `û;mH`BbMi p2+ T2`i2 HQ;`Bi?KB[m2 2bi pH#H2 /Mb mM2 i2HH2 p`Bûiû ,

∥χeit∆ψ(_−h2∆_)u0∥

L2_(R,H1/2₎!|HQ; h| 1

2∥ψ(−h2∆)u₀∥_L2, ∀χ ∈ C∞

c .

G `2K`[m2 7QM/K2MiH2 /2 (":>Ry) 2bi [m2 +2ii2 T2`i2 HQ;`Bi?KB[m2 T2mi āi`2 +QK@ T2Mbû2 bB HǶQM KQMi`2 /2b 2biBKiBQMb /2 ai`B+?`ix 2M i2KTb HQ;`Bi?KB[m2 h| HQ; h|- bQBi

∥eit∆ψ(_−h2∆_)u0∥Lp_([0,h|HQ; h|])Lq_{(M )}!∥ψ(−h2∆)u₀∥_L2.

AMimBiBp2K2Mi- H T2`i2 HQ;`Bi?KB[m2 /Mb HǶ2biBKiBQM `û;mH`BbMi2 bB;MB}2 [m2 HǶQM/2 Tbb2 /m 7Bi /2b T?ûMQKĕM2b /2 +TiBpBiû mM i2KTb THmb ;`M/ T`ĕb /2 HǶQ#bi+H2- /2 HǶQ`/`2 KBMi2MMi /2 h| HQ; h| , BH bǶ;Bi /QM+ /2 KQMi`2` /2b 2biBKiBQMb Dmb[mǶmt i2KTb HQ;`Bi?KB[m2bX 1M /2 i2Hb i2KTb- H2b mi2m`b /Ti2Mi H +QMbi`m+iBQM /2 T`Kûi`Bt /2 MMi?`KM (My3) TQm` Q#i2MB` mM2 2biBKiBQM /2 /BbT2`bBQM TQM+im2HH2 T`ĕb /2 HǶ2Mb2K#H2 +TiûX LQiQMb [m2- /Mb H2 +b /2 HǶû[miBQM /2b QM/2b /Mb mM2 ;ûQKûi`B2 bBKBHB`2- mM2 i2HH2 T`Kûi`Bt  ûiû +QMbi`mBi2 T` "û``/ ("dœd)X

RXeX _ûbmHii , 1biBKiBQMb /2 ai`B+?`ix 2M /2?Q`b /2 /2mt

Q#bi+H2b bi`B+i2K2Mi +QMp2t2b

JQiBpûb T` H2b i`pmt /2 "m`[- :mBHH`KQm 2i >bb2H (":>Ry) 2i /ǶAFr (AF3k

-AF33)- MQmb MQmb BMiû`2bbQMb mt 2biBKiBQMb /2 ai`B+?`ix TQm` H2b û[miBQMb /2 a+?`ƺ@ /BM;2` 2i /2b QM/2b ¨ HǶ2tiû`B2m` /2 THmbB2m`b Q#bi+H2b bi`B+i2K2Mi +QMp2t2b pû`B}Mi H +QM/BiBQM /ǶAFrX LQmb i`BiQMb iQmi /Ƕ#Q`/ H2 +b /2 /2 /2mt Q#bi+H2b bi`B+i2K2Mi +QMp2t2b- ¨ HǶ2tiû`B2m` /2b[m2Hb MQmb Q#i2MQMb /2b 2biBKiBQMb /2 ai`B+?`ix ;HQ#H2b 2i bMb T2`i2 T` `TTQ`i ¨ HǶ2bT+2 HB#`2 ,

h?ûQ`ĕK2 RX9 U(G7Rd#-G7Rd)VX aQB2Mi Θ1, Θ2⊂ R3/2mt Q#bi+H2b bi`B+i2K2Mi +QMp2t2bX

HQ`b- /2b 2biBKiBQMb /2 ai`B+?`ix ;HQ#H2b 2i bMb T2`i2b bQMi pH#H2b /Mb Ω = R3_{\ (Θ}

1∪ Θ2)

TQm` H2b û[miBQMb /2 a+?`ƺ/BM;2` 2i /2b QM/2b ,

∥u∥Lp_(R,Lq_(Ω))!∥u₀∥_H˙s+∥u1∥H˙s−1, (Ondes)

∥u∥Lp_(R,Lq_(Ω))!∥u₀∥_L2, (Schr¨odinger)

H2 +QmTH2 (p, q) pû`B}Mi H +QM/BiBQM /Ƕ/KBbbB#BHBiû ?Q`b TQBMi HBKBi2 TQm` HǶû[miBQM bbQ@ +Bû2X

RXeX _ûbmHii , 1biBKiBQMb /2 ai`B+?`ix 2M /2?Q`b /2 /2mt Q#bi+H2b bi`B+i2K2Mi +QMp2t2b LQmb KQMi`QMb H2 `ûbmHii TQm` HǶû[miBQM /2 a+?`ƺ/BM;2` /Mb (G7Rd#)X *2 `ûbmHii 2bi B+B T`ûb2Miû /Mb H T`iB2 AAXRX AH 2bi 2MbmBi2 ;ûMû`HBbû ¨ HǶû[miBQM /2b QM/2b /Mb (G7Rd )-;ûMû`HBbiBQM B+B T`ûb2Miû2 2M T`iB2 AAXkX LQmb bmBpQMb H Kûi?Q/2 bmBpMi2- BMbTB`û2 T` H2b ûHûK2Mib T`û+û/2Mib /2 MQi`2 2tTQbBiBQM ,

RX :`+2 ¨ HǶ2biBKiBQM `û;mH`BbMi2 p2+ T2`i2 HQ;`Bi?KB[m2 /m2 ¨ ("m`y9)- mt i`@ pmt /ǶApMQpB+B 2M /2?Q`b /ǶmM +QMp2t2 bi`B+i (ApRy)- 2i ¨ H `2K`[m2 /2 "m`[-:mBHH`KQm 2i >bb2H (":>Ry)- QM b2 `û/mBi ¨ /2b 2biBKiBQMb /2 ai`B+?`ix 2M i2KTb HQ;`Bi?KB[m2 T`ĕb /2 H i`D2+iQB`2

+Tiû2-kX QM bmBi H +QMbi`m+iBQM /ǶAFr (AF3k-AF33)- `2T`Bb2 T` "m`[ ("m`Nj)- TQm` +QMbi`mB`2 mM2 bQHmiBQM TT`Q+?û2 T`ĕb /2 H i`D2+iQB`2

+Tiû2-jX mM `;mK2Mi /2 T?b2 biiBQMMB`2- +QK#BMû ¨ mM `ûbmHii /ǶAFr (AF3k- AF33) +QM+2`MMi H /û+`QBbbM+2 /ǶmM2 i2HH2 bQHmiBQM- /QMM2 H2 `ûbmHiiX

_ûbmKQMb #`Bĕp2K2Mi MQi`2 `;mK2Mi- 2M T`2KB2` HB2m TQm` HǶû[miBQM /2 a+?`ƺ/BM;2` , RXeXRX _û/m+iBQM /m T`Q#HĕK2

amTTQbQMb [m2 /2b 2biBKiBQMb /2 ai`B+?`ix TQm` a+?`ƺ/BM;2` bQB2Mi pû`B}û2b 2M i2KTb HQ;`Bi?KB[m2 T`ĕb /2 H i`D2+iQB`2 +Tiû2 R- bQBi

∥χe−it∆D_ψ(−h2_∆)u

0∥Lp(0,h|HQ; h|)Lq_(Ω) ≤ C∥u₀∥_L2,

Qɍ χ 2bi bmTTQ`iû2 T`ĕb /2 H i`D2+iQB`2 +Tiû2- 2i KQMi`QMb +QKK2Mi Q#i2MB` H2 `ûbmHiiX aQB2Mi χQ#bi, χ<sub>`v∈ C0</sub>∞i2H [m2 χ_Q#bi = 1/Mb mM pQBbBMM;2 /2 Θ1∪Θ2∪R- 2i χ`v∈ C0∞

i2H [m2χ`v= 1 /Mb mM pQBbBM;2 /2 R- 2i bQBi ˜ψ i2H [m2 ˜ψ= 1 bm` H2 bmTTQ`i /2 ψX PM /û+QKTQb2 , ψ(<sub>−h</sub>2∆<sub>)e</sub>it∆D<sub>u</sub> 0 = ˜ψ(−h2∆)(1− χQ#bi)ψ(−h2∆)eit∆Du0 + ˜ψ(<sub>−h</sub>2∆<sub>)χQ#bi(1− χ`v)ψ(−h</sub>2∆_)eit∆D_u 0 + ˜ψ(−h2∆_{)χQ#biχ`vψ(−h}2∆_)eit∆D_u 0,

[mB +Q``2bTQM/2Mi `2bT2+iBp2K2Mi mt T`iB2b HQBM /2b Q#bi+H2b- T`ĕb /2b Q#bi+H2b KBb HQBM /2 H i`D2+iQB`2 +Tiû2- 2i T`ĕb ¨ H 7QBb /2 HǶQ#bi+H2 2i /2 H i`D2+iQB`2 +Tiû2X *2 /2`MB2` i2`K2 2bi ;û`û 2M bmBpMi H `2K`[m2 /2 "m`[- :mBHH`KQm 2i >bb2H T`û+û/2KK2Mi T`ûb2Miû2 TQm` T2m [m2 HǶQM /BbTQb2 /ǶmM2 2biBKiBQM bMb T2`i2 2M /2?Q`b /2 H i`D2+iQB`2 +Tiû2- [mB- /Mb +2ii2 ;ûQKûi`B2 T`iB+mHBĕ`2- 2bi T` 2t2KTH2 +QMbû[m2M+2 /Ƕ2biBKiBQMb /2 `ûbQHpMi2 /2 (owyy) 2M /2?Q`b /ǶmM Q#bi+H2 MQM +TiMi- TmBb[m2 Θ1∪ Θ2

+QśM+B/2-2M /2?Q`b /2 H i`D2+iQB`2 +Tiû2- p2+ mM i2H Q#bi+H2X

G2b /2mt T`2KB2`b i2`K2b bQMi +QMi`QHûb 2M /TiMi /2b `;mK2Mib /2 (ApRy)X G2 T`2@ KB2` i2`K2- w = (1 − χQ#bi)ψ(−h2_∆)eit∆D_u

0- pû`B}2 H2 T`Q#HĕK2

!

i∂tw+ ∆Dwh= −[χQ#bi, ∆D]ψ(−h2∆)eit∆Du0

w(t = 0) = (1− χQ#bi)ψ(−h2_∆)u 0

RX 1biBKiBQMb /2 ai`B+?`ix ¨ HǶ2tiû`B2m` /2 THmbB2m`b Q#bi+H2b +QMp2t2b

KBb +QKK2 1 − χQ#bi = 1 T`ĕb /2 ∂Ω- BH bǶ;Bi /ǶmM T`Q#HĕK2 /Mb HǶ2bT+2 HB#`2 5 G 7Q`KmH2 /2 .m?K2H +QK#BMû2 p2+ H2b 2biBKiBQMb /2 ai`B+?`ix /Mb Rn 2i mM H2KK2 /2

*?`Bbi@EBb2H2p 7Qm`MBbb2Mi HQ`b

∥w∥Lp_Lq !∥(1 − χQ#bi)ψ(−h2∆)eit∆Du₀∥_L2_(Rn₎+∥[χQ#bi, ∆_D]ψ(−h2∆)eit∆Du₀∥_L2_H−1/2.

KBb [χ, ∆D] 2bi bmTTQ`iû HQBM /2 R 2i HǶ2biBKiBQM `û;mH`BbMi2 bMb T2`i2 T2`K2i /2

+QMi`ƬH2` H2 K2K#`2 /2 /`QBi2X G2 b2+QM/ i2`K2- χQ#bi(1−χ`v)ψ(−h2_∆)eit∆D_u

0+Q``2bTQM/

¨ mM T`Q#HĕK2 T`ĕb /2 HǶQ#bi+H2 KBb HQBM /2 H i`D2+iQB`2 +Tiû2- BH T2mi /QM+ āi`2 +QMi`QHû 2M K2iiMi #Qmi ¨ #Qmi H2b 2biBKiBQMb 2M i2KTb h /Mb mM2 p`Bûiû +QKT+i2 ¨ #Q`/ bi`B+i2K2Mi +QM+p2 Q#i2Mm2b T` (ApRy)- ;`+2 ¨ HǶ2biBKiBQM bMb T2`i2X

RXeXkX *QMbi`m+iBQM /ǶmM2 bQHmiBQM TT`Q+?û2

lM `;mK2Mi /2 i`MbHiBQM BMbTB`û /2 (ApRy) KQMi`2 HQ`b [m2 HǶQM T2mi /2 THmb b2 `û/mB`2 ¨ HǶûim/2 /2 /QMMû2b KB+`QHQ+H2K2Mi bmTTQ`iû2b T`ĕb /2 H i`D2+iQB`2 +Tiû2X SHmb T`û+BbûK2Mi- MQmb /û}MBbbQMb HǶ2Mb2K#H2 +Tiû 2M i2KTb T /ǶmM pQBbBM;2 D /2 R-MQiû ˆTT(D)⊂ T⋆Ω+QKK2 HǶ2Mb2K#H2 /2b TQBMib /2 D × {|ξ| ∈ [α0, β0]}- [mB `2bi2Mi /Mb

D 2M i2KTb T - 2i QM b2 `û/mBi ¨ HǶûim/2 /2 /QMMû2b u0 =PT(q)u0- Qɍ q 2bi bmTTQ`iû /Mb

mM pQBbBM;2 /2 ˆT2%|HQ; h|( ˜D) 2i q = 1 bm` ˆT2%|HQ; h|(D)- Qɍ D ⊂ ˜DX

LQiQMb [m2 H Mû+2bbBiû MQiKK2Mi /2 /2pQB` T` H bmBi2 +QMi`ƬH2` H2b /û`Bpû2b /2 q MQmb KQiBp2 ¨ ûim/B2` H `û;mH`Biû /m ~Qi /2 #BHH`/- 2i HǶQM KQMi`2 [m2

d(Φt(ρ), Φt(˜ρ))≤ Ctd(ρ, ˜ρ)µ

bm7 bm` /2 T2iBib BMi2`pHH2b /2 i2KTb- H T2`i2 /2 `û;mH`Biû >ƺH/û`B2MM2 ûiMi /m2 mt `vQMb [mB 7`ƬH2Mi /2b i`D2+iQB`2b iM;2Mi2b- 2i mM2 i2HH2 `û;mH`Biû bm{` TQm` +?QBbB` q /2 i2HH2 bQ`i2 [m2 |∂α

x,ξq| ! h−c|α|%- +2 [mB MǶBM/mB` m+mM2 T2`i2 TmBb[m2 +2b /û`Bpû2b b2`QMi

T` H bmBi2 +QM7`QMiû2b ¨ /2b i2`K2b /ǶQ`/`2 m KQBMb hX

lM `;mK2Mi /2 /mHBiû KQMi`2 [mǶBH bm{i /ǶQ#i2MB` HǶ2biBKiBQM /2 /BbT2`bBQM- TQm` t_{≤ .| HQ; h|} |eiht∆δy| ! (ht)−3/2, Qɍ δy = 1 (2πh)3 " ei(x−y)·ξ/hq(x, ξ)dξ,

2i H2 ~Qi ûiMi HBMûB`2- BH bm{i TQm` Q#i2MB` mM2 bQHmiBQM TT`Q+?û2 p2+ /QMMû2 δy /Ƕ2M

+QMbi`mB`2 mM2 p2+ /QMMû2 ei(x−y)·ξ_{q(x, ξ)- y 2i ξ ûiMi }tûbX LQmb bmBpQMb TQm` +2H H}

Kûi?Q/2 /ǶAFr (AF3k- AF33)- `2T`Bb2 T` "m`[ ("m`Nj) TQm` H2 +QMi`ƬH2 /2 HǶû[miBQM /2b TH[m2bX

PM /û}MBi H2b ?BbiQB`2b /2 `û~2tBQMb I +QKK2 HǶ2Mb2K#H2 /2b n@mTH2i J = (j1,· · · , jn)

-n_{≥ 0- Qɍ j}i+1̸= ji- QM MQi2 J′= (j1,· · · , jn−1) 2i mM2 bQHmiBQM 2bi +?2`+?û2 bQmb H 7Q`K2

/ǶmM2 bû`B2 /2 L2mKMM eith∆D,Ω<sub>q(·, ξ) =</sub>) J∈I (<sub>−1)</sub>JwJ, i∂twJ− h∆wJ = 0, wJ = wJ′ bm` R3\Θ_jn, k3

RXeX _ûbmHii , 1biBKiBQMb /2 ai`B+?`ix 2M /2?Q`b /2 /2mt Q#bi+H2b bi`B+i2K2Mi +QMp2t2b Qɍ w_∅+Q``2bTQM/ ¨ mM2 QM/2 HB#`2 2i wJ ¨ mM2 QM/2 `û~û+?B2 b2HQM HǶ?BbiQB`2 J TmBb B;MQ`Mi

H2b Q#bi+H2bX G2b wJ bQMi Q#i2Mmb b2HQM H Kûi?Q/2b "Eq , QM +?2`+?2 wJ bQmb H 7Q`K2

wJ(x, t) = ) k≥0 wJ_k(x, t)hke−i(ϕJ(x,ξ)|ξ|−tξ2)/h Qɍ ϕ_∅(x, ξ) = (x_{− y) · ξ/|ξ|- H2b T?b2 ϕ}J- |J| ≥ 1 pû`B}2Mi HǶû[miBQM 1BFQMH2 |<sub>∇ϕ</sub>J| = 1, ϕJ = ϕJ′ bm` R3\Θ_jn, 2i wJ k mM i`MbTQ`i b2HQM H T?b2 , (∂t+ 2|ξ|∇ϕJ ·∇ + |ξ|∆ϕJ)wJk =−i∆wJk₋₁, wJ_k = w_kJ′ bm` R3\Θjn, Qɍ QM  miBHBbû H +QMp2MiBQM wJ

−1= 0X lM2 bQHmiBQM mt û[miBQMb 1BFQMH2b 2bi +QMbi`mBi2

T` H Kûi?Q/2 /2 `û~2tBQM /2b T?b2b /ǶAFr , ∇ϕJ(x, ξ) +Q``2bTQM/Mi ¨ H /B`2+iBQM

/ǶmM `vQM ``Bpû 2M x T`ĕb āi`2 T`iB bmBpMi H /B`2+iBQM ξ

|ξ| 2i pQB` ûiû `û~û+?B bmBpMi

HǶ?BbiQB`2 J- 2M /Ƕmi`2b i2`K2b- H2 ;`/B2Mi /2 H T?b2 bmBi H2b `vQMbX

PM T2mi HQ`b `ûbQm/`2 H2b û[miBQMb /2 i`MbTQ`i bmBpMi H2b `vQMbX SQm` +2H¨- QM `2K`[m2 [m2 w(τ) = v(x + 2τ|ξ|∇ψ, t + τ) `ûbQmi HǶû[miBQM /Bzû`2MiB2HH2 Q`/BMB`2

∂τw=−|ξ|∆ψ(x + 2τ∇ψ(x))w, KBb QM T2mi BMiû;`2` U(GmM99)V , 2tT(−|ξ|" τ τ0 ∆ψ(x + 2s_{∇ψ(x))ds) =}+ Gψ(x + 2τ|ξ|∇ψ(x)) Gψ(x + 2τ0|ξ|∇ψ(x)) ,1/2 , Qɍ Gψ /ûbB;M2 H +Qm`#m`2 ;mbbB2MM2 /2 H bm`7+2 /2 MBp2m /2 ψX PM  /QM+- T` 2t2KTH2 TQm` wJ 0 wJ<sub>0</sub>(x + 2τ |ξ|<sub>∇ϕ</sub>J(x), t + τ ) = + GϕJ(x + 2τ |ξ|∇ϕJ(x)) GϕJ(x) ,1/2 wJ<sub>0</sub>(x, t) TQm` x ∈ ∂ΘjnX JBb w J 0 = wJ ′

0 bm` ∂Θjn 2i QM T2mi `ûBiû`2` H2 `BbQMM2K2Mi 2M `2KQMiMi

H2b `vQMb Dmb[mǶ¨ w∅

0 TQm` H2[m2H `ûbQm/`2 H2b û[miBQMb /2 i`MbTQ`i 2bi BKKû/Bi @ H

HQ+HBbiBQM /2 q MQmb bbm`Mi 2M T`iB+mHB2` [m2 HǶQM TmBbb2 2M 2z2i `2KQMi2` H2b `vQMbX PM Q#iB2Mi w₀J(x, t) = ΛϕJ(x, ξ)q(X−2t(x, |ξ|∇ϕJ), ξ), ΛϕJ(x, ξ) = + GϕJ(x) GϕJ(X−1(x, |ξ|∇ϕJ)) ,1/2 × · · · × -Gϕ(X−|J|−1(x, |ξ|_∇ϕJ)) Gϕ(X−|J|_{(x, |ξ|∇ϕJ))} .1/2 , Qɍ Xt /ûbB;M2 H +QKTQbMi2 bTiBH2 /m ~Qi Xi H2b TQBMib /m #Q`/ ii2BMib T`ĕb i `û@

~2tBQMb- 2i /2b 2tT`2bbBQMb bBKBHB`2b bQMi pH#H2b TQm` k ≥ 1X

RX 1biBKiBQMb /2 ai`B+?`ix ¨ HǶ2tiû`B2m` /2 THmbB2m`b Q#bi+H2b +QMp2t2b

LQiQMb [m2 H `+BM2 +``û2 /m T`Q/mBi /2b `TTQ`ib /2b +Qm`#m`2b :mbbB2MM2b ΛϕJ

2bi HǶMHQ;m2 /2 H `+BM2 +``û2 /m T`Q/mBi /2b C+Q#B2MM2b TT`BbbMi /Mb (":>Ry)X *Ƕ2bi +2 i2`K2 [mB 2bi `2bTQMb#H2 /2 H /û+`QBbbM+2 i2KTQ`2HH2 /m ~Qi , /ǶT`ĕb AFr (AF3k-AF33

)-|ΛϕJ| ! λ|J|

Qɍ 0 < λ < 1 2bi H `+BM2 +``û2 /m T`Q/mBi /2b /2mt pH2m`b T`QT`2b /2 KQ/mH2 THmb T2iBi [m2 1 /2 H /Bzû`2MiB2HH2 /2 HǶTTHB+iBQM /2 SQBM+`û bbQ+Bû2 ¨ H i`D2+iQB`2 +TiBp2X RXeXjX G2b /û`Bpû2b /B`2+iBQMM2HH2b /2 H T?b2 PM  /QM+ eiht∆δy ≃ 1 (2πh)3 ) J∈I ) k≥0 " wJ_k(x, t, ξ)hke−i(ϕJ(x,ξ)|ξ|−tξ2)/h_dξ, Qɍ H2b wJ

k 2i ϕJ QMi ûiû +QMbi`mBib T`û+û/2KK2MiX aB mM `ûbmHii /ǶAFr (AF3k- AF33)

+QM+2`MMi H2 T`Q/mBi /m `TTQ`i /2b +Qm`#m`2b :mbbB2MM2b MQmb bbm`2` H /û+`QBbbM+2 i2KTQ`2HH2 /m ~Qi- BH 2bi Mû+2bbB`2 TQm` +QM+Hm`2 /2 ;;M2` i`QBb /2KB2b TmBbbM+2b /2 hX LQmb miBHBbQMb TQm` +2H mM `;mK2Mi /2 T?b2 biiBQMMB`2X AH 2bi /QM+ Mû+2bbB`2- /ǶmM2 T`i- /2 TQmpQB` +QMi`ƬH2` H2b /û`Bpû2b /B`2+iBQMM2HH2b [mB bm`;B`QMi- BMbB [m2 /Ƕûim/B2` H2b TQBMib biiBQMMB`2b /2b T?b2b SJ(x, ξ, t) = ϕJ(x, ξ)|ξ|− tξ2 2i /2 KQMi`2` [m2 b

>2bbB2MM2 v 2bi MQM /û;ûMû`û2X

SQm` +2 7B`2- MQmb /Bzû`2M+BQMb HǶû[miBQM 1BFQMH2 T` `TTQ`i ¨ ξ 2i MQmb `ûbQHpQMb bmB@ pMi H2b `vQMb HǶû[miBQM /2 i`MbTQ`i [mB TT`ŗiX S` 2t2KTH2- /û`BpMi |∇(ϕJ|ξ|)|2=

|ξ|2 QM Q#iB2Mi

|ξ|_∇(∂ξiϕJ|ξ|) ·∇ϕJ =

ξi

|ξ|,

H [mMiBiû ∂ξiϕJ|ξ|[mB MQmb BMiû`2bb2 2bi /QM+ i`MbTQ`iû2 bmBpMi ∇ϕJ- +Ƕ2bi ¨ /B`2 bmBpMi

H2b `vQMbX 1M BMiû;`Mi HǶû[miBQM /Bzû`2MiB2HH2 Q`/BMB`2 [m2 pû`B}2 τ −→ ∂ξi(ϕJ|ξ|)(x +

τ_∇ϕJ(x, ξ))- 2i 2M miBHBbMi H +QM/BiBQM m #Q`/ TQm` H2b T?b2b ϕJ = ϕJ′ bm` ∂Θ<sub>j</sub> n- QM Q#iB2Mi TQm` x ∈ ∂Θjn ∂ξi(ϕJ|ξ|)(x + τ∇ϕJ(x, ξ), ξ) = ∂ξi(ϕJ′|ξ|)(x, ξ) + τ ξi |ξ|, 2i QM Biĕ`2 H2 `BbQMM2K2Mi Dmb[mǶ¨ H T`2KBĕ`2 T?b2X PM Q#iB2Mi

DξSJ(x, ξ, t) = ˆX_−2t(x, |ξ|∇ϕJ(x, ξ))− y

2i H T?b2 biiBQMM2 2M H2 p2+i2m` ξ = sJ(x, t)T2`K2iiMi /Ƕii2BM/`2 x /2TmBb y 2M i2KTb

2t 2M +QKK2MÏMi T` mM2 `û~2tBQM bm` Θj1X

.2 H KāK2 KMBĕ`2- MQmb `ûbQHpQMb 2M `2KQMiMi H2b `vQMb HǶû[miBQM /2 i`MbTQ`i pû`B}û2 T` H T?b2 ∇/∂_ξ2 iξjψJ 0 ·∇ψJ = δij− ∂ξi∇ψJ· ∂ξj∇ψJ jy

RXeX _ûbmHii , 1biBKiBQMb /2 ai`B+?`ix 2M /2?Q`b /2 /2mt Q#bi+H2b bi`B+i2K2Mi +QMp2t2b TQm` KQMi`2` b MQM /û;ûMû`2b+2M+2 2M H2b TQBMib biiBQMMB`2b TQm` t ≥ t0>0X

6BMH2K2Mi- TQm` +QMi`ƬH2` H2b /û`Bpû2b /B`2+iBQMM2HH2b [mB TT`ŗi`QMi- BH bǶ;Bi /2 +QMi`ƬH2` H2b /û`Bpû2b bTiBQ@/B`2+iBQM2HH2b /2b T?b2bX SQm` +2 7B`2- MQmb bmBpQMb H KāK2 B/û2 [m2 T`û+û/2KK2Mi , Dα

ξϕJ pû`B}2

∇(DξαϕJ) ·∇ϕJ = RJ,

Qɍ RJ M2 +QMiB2Mi [m2 /2b /û`Bpû2b /ǶQ`/`2 BM7û`B2m`X PM `ûbQm/ +2ii2 û[miBQM /2 i`MbTQ`i

2M `2KQMiMi H2b `vQMb- QM T`2M/ H2b /û`Bpû2b bTiBH2b /2 H bQHmiBQM 2i QM Q#iB2Mi T` BM/m+iBQM

|Dα_ξD_xβ_∇ϕJ|≤ Dα,β|J|,

mM `ûbmHii /ǶAFr MQmb bbm`Mi HǶBMBiBHBbiBQM α = 0X aB mM i2H `ûbmHii b2K#H2 #`miH- BH 2bi 2M 7Bi bm{bMi TQm` Q#i2MB` H2 +QMi`ƬH2 [mB MQmb BMiû`2bb2 2M /2b i2KTb HQ;`Bi?KB[m2b , 2M 2z2i- H2b /û`Bpû2b /B`2+iBQMM2HH2b [mB TT`ŗi`QMi bQMi +QMi`2 /2b i2`K2b /ǶQ`/`2 m KQBMb h- Q`- TQm` wJ

k ̸= 0- QM  |J| ≈ t ≤ .| HQ; h|- 2i H T2`i2 /Ƕ. TmBbbM+2 /2 h [mB

TT`ŗi HQ`b MǶBM/mBi m+mM2 T2`i2X

RXeX9X S?b2 biiBQMMB`2 2i +QM+HmbBQM

BMbB- T`ĕb mM `;mK2Mi /2 T?b2 biiBQMMB`2- QM Q#iB2Mi TQm` t ≥ t0

eiht∆δy≃ 1 (2π)3h−3/2 ) J∈I ) k_≥0 ˜ w_kJ(x, t, ξ)hke−isJ(x,t)/h_, Qɍ ˜wJ

0 = wJ0 2i- TQm` k ≥ 1- ˜wkJ 2bi mM2 +QK#BMBbQM HBMûB`2b /2 /û`Bpû2b /B`2+iBQMM2HH2b /2

w_kJX .ǶT`ĕb H2 `ûbmHii /ǶAFr bm` H2b T?b2b T`û+û/2KK2Mi +Biû 2i +QKK2 TQm` w_kJ _̸= 0-QM  |J| ≈ t ≤ .| HQ; h|-) J∈I |wJ₀(x, t, ξ)| ! ) J | wJ k̸=0 λ|J|!) r"t λr_{≤ e}−µt URXeXRV

TQm` mM +2`iBM µ > 0 M2 /ûT2M/Mi [m2 /2 H ;ûQKûi`B2 /2b Q#bi+H2bX .2 H KāK2 KMBĕ`2- H2b i2`K2b ˜wJ k(x, t, ξ)hk /K2ii2Mi mM2 #Q`M2 bBKBHB`2 TmBb[m2 +QKK2 ûpQ[mû T`û+û/2KK2Mi- T`2M/`2 mM2 /û`Bpû2 /B`2+iBQMM2HH2 /2 wJ k BM/mBi m TB`2 mM2 T2`i2 /2 h−% TQm` /2b i2KTb HQ;`Bi?KB[m2bX BMbB /QM+- TQm` t ∈ (t0, .|HQ; h|) , |eith∆δy| ! h−3/2e−µt

2i TQm` /2b i2KTb T2iBib- eith∆_δ

y 2bi mM2 QM/2 HB#`2 UHǶ`;mK2Mi /2 i`MbHiBQM /2 (ApRy)

bbm`Mi [m2 HǶQM TmBbb2 T`2M/`2 mM2 /QMMû2 bmTTQ`iû2 HQBM /m #Q`/V 2i QM T2mi miBHBb2` HǶ2biBKiBQM /2 /BbT2`bBQM TQM+im2HH2 /2 R3<sub>- +2 [mB +HƬi H T`2mp2X</sub>

RX 1biBKiBQMb /2 ai`B+?`ix ¨ HǶ2tiû`B2m` /2 THmbB2m`b Q#bi+H2b +QMp2t2b RXeX8X GǶû[miBQM /2b QM/2b

.Mb H2 +b /2 HǶû[miBQM /2b QM/2b- H `û/m+iBQM /m T`Q#HĕK2 2bi 2t+i2K2Mi bBKBHB`2-2M miBHBbMi H /û+`QBbbM+2 HQ+H2 /2 HǶûM2`;B2 THmiƬi [m2 HǶ2biBKiBQM `û;mH`BbMi2- 2i H2 `ûbmHii /2 aKBi? 2i aQ;;2 (aaN8) ¨ HǶ2tiû`B2m` /ǶmM +QMp2t2 bi`B+i THmiƬi [m2 H2 `ûbmHii /2 (ApRy)X

G +QMbi`m+iBQM /2 T`Kûi`Bt- bBKBHB`2- b2 7Bi KBMi2MMi TQm` HǶû[miBQM /2b QM/2bX G /Bzû`2M+2 7QM/K2MiH2 ûiMi [m2 H2b `vQMb bQMi KBMi2MMi bmBpBb ¨ pBi2bb2 1- +2H b2 i`/mBbMi- m MBp2m /2 H T?b2- T` H2 7Bi [m2 +2ii2 /2`MBĕ`2 biiBQMM2 KBMi2MMi bm` /2b /B`2+iBQMb TH2BM2b- 2i QM miBHBb2 HQ`b mM `;mK2Mi /2 T?b2 biiBQMMB`2 bm` +?[m2 bT?ĕ`2 Sd−1_{(0, s)X PM M2 ;;M2 THmb KBMi2MMi [m2} d₋₁

2 TmBbbM+2b /2 h U+2 [mB i`/mBi H2

+`+iĕ`2 KQBMb /BbT2`bB7 /2 HǶû[miBQM /2b QM/2bV- 2i QM Q#iB2Mi /2 H KāK2 KMBĕ`2 [m2 TQm` a+?`ƺ/BM;2` HǶ2biBKiBQM /BbT2`bBp2 T`ĕb /2 H i`D2+iQB`2 +Tiû2X

RXdX _ûbmHii , GǶ2tiû`B2m` /2 N ≥ 3 Q#bi+H2b bi`B+i2K2Mi

+QMp2t2b pû`B}Mi H +QM/BiBQM /ǶAFr

LQmb ;ûMû`HBbQMb H2 `ûbmHii T`û+û/2Mi ¨ HǶ2tiû`B2m` /2 N ≥ 3 Q#bi+H2b pû`B}Mi H +QM/BiBQM /ǶAFr ,

h?ûQ`ĕK2 RX8 U(G7R3#)VX aQBi (Θi)1≤i≤N mM2 7KBHH2 /2 +QMp2t2b bi`B+ib /2 R3- pû`B}Mi

H2b +QM/BiBQMb /ǶAFr URX9XRV 2i URX9XkV- 2i Ω = R3_{\ ∪} 1≤i≤N

Θ_iX HQ`b- bQmb H +QM/BiBQM

/Ƕ/KBbbB#BHBiû ?Q`b TQBMi HBKBi2- /2b 2biBKiBQMb /2 ai`B+?`ix ;HQ#H2b 2i bMb T2`i2b bQMi pH#H2b /Mb Ω TQm` H2b û[miBQMb /2 a+?`ƺ/BM;2` 2i /2b QM/2b ,

∥u∥Lq_(R,Lr_(Ω))!∥u₀∥_H˙s+∥u1∥H˙s−1, UPM/2bV

∥u∥Lq_(R,Lr_(Ω))!∥u₀∥_L2. Ua+?`ƺ/BM;2`V

*2 `ûbmHii 2bi KQMi`û /Mb (G7R3#)- BH 2bi B+B T`ûb2Miû 2M T`iB2 AAXjX .QMMQMb mM T2`Ïm /2 H T`2mp2 /Mb H2 +b /2 HǶû[miBQM /2 a+?`ƺ/BM;2`X

1biBKiBQM `û;mH`BbMi2 bMb T2`i2 /Mb H xQM2 MQM@+Tiû2

G2 THM /2 H T`2mp2 2bi H2 KāK2 [m2 TQm` H2 +b H2 THmb bBKTH2 /2 /2mt Q#bi+H2bX SQm`pm [m2 HǶQM /BbTQb2 /ǶmM2 2biBKiBQM `û;mH`BbMi2 bMb T2`i2 ¨ HǶ2tiû`B2m` /2 HǶ2Mb2K#H2 +Tiû-HǶ`;mK2Mi T`û+û/2Mi `2bi2 pHB/2 TQm` KQMi`2` [m2 HǶQM T2mi b2 `û/mB`2 ¨ KQMi`2` /2b 2biBKiBQMb T`ĕb /2 HǶ2Mb2K#H2 +Tiû 2M i2KTb HQ;`Bi?KB[m2X *2T2M/Mi- QM M2 /BbTQb2 THmb /ǶmM2 ;ûQKûi`B2 T`iB+mHBĕ`2 MQmb T2`K2iiMi /ǶmiBHBb2` mM2 2biBKiBQM /2 `ûbQHpMi2 2M /2?Q`b /ǶmM Q#bi+H2 MQM@+TiMi- 2i BH MQmb 7mi /QM+ /QMM2` mM2 T`2mp2 ;ûMû`H2X LQiQMb- /2 THmb- [m2 MQmb KQMi`QMb B+B mM2 2biBKiBQM ¨ HǶ2tiû`B2m` /2 HǶ2Mb2K#H2 +Tiû

/Mb HǶ2bT+2 /2b T?b2b 2i MQM THmb bBKTH2K2Mi /2 b T`QD2+iBQM bTiBH2- +Ƕ2bi ¨ /B`2 [m2

H 7QM+iBQM /2 +QmTm`2 2M 2bT+2 2bi `2KTH+û2 T` mM QTû`i2m` /QMi H2 KB+`Q@bmTTQ`i 2bi /BbDQBMi /2 HǶ2Mb2K#H2 +Tiû pm +QKK2 bQmb@2Mb2K#H2 /2 HǶ2bT+2 /2b T?b2bX

RXdX _ûbmHii , GǶ2tiû`B2m` /2 N ≥ 3 Q#bi+H2b bi`B+i2K2Mi +QMp2t2b pû`B}Mi H +QM/BiBQM /ǶAFr LQmb miBHBbQMb TQm` +2H mM `;mK2Mi /2 +QKKmii2m`- bBKBHB`2 ¨ +2HmB TT`BbbMi /Mb (":>Ry)- /Mb HǶ2bT`Bi /2 JQ`r2ix- _HibQM 2i ai`mbb (J_add), TQm` iQmi2 bQHmiBQM u /2 HǶû[miBQM /2 a+?`ƺ/BM;2` 2i iQmi QTû`i2m` A QM  ,

⟨Au, u⟩(T ) − ⟨Au, u⟩(0) = " T

0

"

Ω⟨[i∆, A]u, u⟩ +

" T

0

"

∂Ω⟨Au, ∂ nu⟩.

aB HǶQM +?QBbBi A /ǶQ`/`2 xû`Q vMi mM bvK#QH2 TQbBiB7- 2i 2HHBTiB[m2 bm` HǶ2Mb2K#H2 [mB MQmb BMiû`2bb2- 2i i2H [m2 H2 i2`K2 /2 #Q`/ TT`BbbMi  H2 #QM bB;M2 ¨ /2b i2`K2b /ǶQ`/`2 BM7û`B2m` T`ĕb- QM Q#iB2Mi /QM+ HǶ2biBKiBQM pQmHm2X G Kûi?Q/2 /2 (J_add) bǶ/Ti2 ¨ HǶû[miBQM /2 a+?`ƺ/BM;2` TQm` +H+mH2` H2 bvK#QH2 /2 A m #Q`/- 2i MQmb `2T`2MQMb H2m` +QMbi`m+iBQM /ǶmM2 7QM+iBQM /Ƕû+?TT2K2Mi Ĝ pH#H2 TQm` mM Q#bi+H2 MQM@+TiMi ;ûMû@ `B[m2 Ĝ ¨ HǶ2tiû`B2m` /2 HǶ2Mb2K#H2 +Tiû TQm` +QMbi`mB`2 H2 bvK#QH2 /2 AX LQiQMb [m2 HǶQM M2 /BbTQb2 /ǶmM2 2tT`2bbBQM /m bvK#QH2 m #Q`/ [m2 /Mb mM2 xQM2 {τ − η2_{>0}- 2i TQm`}

+2ii2 `BbQM- +2ii2 Kûi?Q/2 M2 T2`K2i Tb /2 KQMi`2` [m2 H2 i2`K2 /2 #Q`/ 2bi TQbBiB7 ¨ /2b i2`K2b /ǶQ`/`2 BM7û`B2m` T`ĕb- KBb b2mH2K2Mi [mǶBH 2bi T`2b[m2 TQbBiB7 ¨ /2b i2`K2b /ǶQ`/`2 BM7û`B2m` T`ĕb- 2i MQmb Q#i2MQMb mM2 2biBKiBQM /2 H 7Q`K2

∥PTh(φ)u∥L2_H1/2 ≤ C(∥u0∥L2 + δ∥PT_h( ˜φ)u∥_L2_H1/2) + CδO(h∞),

Qɍ ˜φ = 1 bm` H2 bmTTQ`i /2 φX PM Biĕ`2 +2 `BbQMM2K2Mi T`ĕb pQB` +?QBbBi δ > 0 bm{bK@ K2Mi T2iBi }M /ǶQ#i2MB` H2 `ûbmHiiX G2b i2`K2b /ǶQ`/`2 BM7û`B2m` TT`BbbMi bQMi ;û`ûb 2M miBHBbMi HǶ2biBKiBQM p2+ T2`i2X

LQiQMb [m2- /Mb HǶ2bT`Bi /2 (.oRk)- QM T2mi Biû`2` H T`2mp2 2i +QMi`ƬH2` ¨ +?[m2 ûiT2 H2b i2`K2b /ǶQ`/`2 BM7û`B2m` T` HǶ?vTQi?ĕb2 /2 `û+m``2M+2- TQm` KQMi`2` [m2 HǶ2z2i `û;mH`BbMi bMb T2`i2 /Mb H xQM2 MQM +Tiû2 2bi 2M 7Bi pH#H2 /ĕb [m2 HǶQM /BbTQb2 /ǶmM 2z2i `û;mH`BbMi p2+ T2`i2 TQHvMQKBH2- bbm`Mi HǶBMBiBHBbiBQM /m `BbQMM2K2MiX _û/m+iBQM m pQBbBM;2 /2b i`D2+iQB`2b Tû`BQ/B[m2b

1M miBHBbMi HǶ2biBKiBQM `û;mH`BbMi2 /Mb H xQM2 MQM +Tiû2- QM b2 `û/mBi /QM+ +QKK2 T`û+û/2KK2Mi ¨ KQMi`2` /2b 2biBKiBQMb /2 ai`B+?`ix bMb T2`i2 2M i2KTb HQ;`Bi?KB[m2-KB+`Q@HQ+H2K2Mi T`ĕb /2 HǶ2Mb2K#H2 +Tiû ,

∥PTh(φ)e−it∆Dψ(−h2∆)u0∥Lp(0,h|HQ; h|)Lq_(Ω) ≤ C∥u₀∥_L2,

Qɍ φ ∈ C∞_(R3_{× R}3_{) 2bi bmTTQ`iû2 T`ĕb /2 HǶ2Mb2K#H2 +Tiû K ⊂ T}⋆_{Ω∪ T}⋆∂Ω- 2i- pB /2b

`;mK2Mib /2 /mHBiû- BH bm{i /QM+ /2 KQMi`2` HǶ2biBKiBQM /2 /BbT2`bBQM ∥PTh(φ)eith∆ψ(−h2∆)PTh(φ)⋆∥L1_→L∞ !(ht)−3/2, ∀0 ≤ t ≤ .| HQ; h|.

SQm` MQmb `û/mB`2 ¨ H +QMbi`m+iBQM /ǶmM2 T`Kûi`Bt T`ĕb /ǶmM2 i`D2+iQB`2 +Tiû2- MQmb û+`BpQMb- BMbTB`ûb T` (":>Ry), eiT h∆ _≈ ) F=(k1,··· ,kL) eits0∆_Π kLe iτ h∆_Π kL−1· · · Πk1e iτ h∆_{+ R} 1(t) + R2(t), jj

RX 1biBKiBQMb /2 ai`B+?`ix ¨ HǶ2tiû`B2m` /2 THmbB2m`b Q#bi+H2b +QMp2t2b

Qɍ H2b QTû`i2m`b ΠkbQMi /2b QTû`i2m` /2 +QmTm`2 KB+`QHQ+mt HQ+HBbûb T`ĕb /2b /Bzû`2Mib

b2;K2Mib +QKTQbMi HǶ2Mb2K#H2 +TiûX PM +?QBbBi τ bb2x T2iBi- 2i H2b i2`K2b /2 +2ii2 bQKK2 +Q``2bTQM/2Mi mt T`iB2b /m ~Qi [mB `2bi2Mi T`ĕb /ǶmM2 i`D2+iQB`2 Tû`BQ/B[m2X G T`iB2 R1+Q``2bTQM/ m ~Qi [mB 2bi HQBM /2b Q#bi+H2b 2i v `2bi2 2i H T`iB2 R2 +Q``2bTQM/

¨ mM2 T`iB2 /m ~Qi [mB 2bi HQBM /2b Q#bi+H2b 2i v `2bi2- mM2 7QBb û+QmHû mM i2KTb T0 ≥ 0

M2 /ûT2M/Mi [m2 /2b ΠiX .QM+- 2M i2KTb t ≥ T0- BH bm{i /2 bǶBMiû`2bb2` mt T`iB2b /m

~Qi [mB `2bi2Mi T`ĕb /ǶmM2 i`D2+iQB`2 Tû`BQ/B[m2X

SQm` ;û`2` H2b i2KTb 0 ≤ t ≤ T0- QM +?QBbBi φ bmTTQ`iû2 bb2x T`ĕb /2 K TQm` [mǶ2M

i2KTb 0 ≤ t ≤ T0- H2b `vQMb Bbbmb /2b TQBMib /m bmTTQ`i /2 φ M2 bQMi DKBb iM;2Mib

m #Q`/ /m /QKBM2X HQ`b H +QMbi`m+iBQM /2 T`Kûi`Bt /ǶAFr T`û+2/2KK2Mi ûpQ[mû2 T2`K2i /ǶQ#i2MB` mM2 2biBKiBQM /BbT2`bBp2 2M i2KTb 0 ≤ t ≤ T0 , H2 ~Qi bǶû+`B` 2M 2z2i

+QKK2 mM2 bQKK2 U/ûT2M/Mi /2 T0V /ǶQM/2b `û~û+?B2b- +?+mM2 /Ƕ2Mi`2 2HH2 pû`B}Mi

HǶ2biBKiBQM /BbT2`bBp2X *QM+HmbBQM

lM2 7QBb HQ+HBbûb T`ĕb /ǶmM2 i`D2+iQB`2 Tû`BQ/B[m2- H +QMbi`m+iBQM T`û+û/2KK2Mi ûpQ@ [mû2 /Mb H2 +b /2 /2mt Q#bi+H2b bi`B+i2K2Mi +QMp2t2b- BMbB [m2 H2b `;mK2Mib /2 T?b2 biiBQMMB`2- bǶTTHB[m2Mi /2 H KāK2 KMBĕ`2 [m2 TQm` /2mt Q#bi+H2bX *2T2M/Mi- TQm` i2`KBM2` H T`2mp2- BH bǶ;Bi /2 +QMi`QH2` H2b bQKK2b MHQ;m2b ¨ URXeXRV- bQBi

)

J∈I

|wJk|,

Qɍ HǶ2Mb2K#H2 I /2b ?BbiQB`2b /2 `û~2tBQM TQbbB#H2b 2bi KBMi2MMi #2m+QmT THmb ;`M/-2i b iBHH2 2Mi`2 2M +QKTûiBiBQM p2+ H /û+`QBbbM+2 i2KTQ`2HH2 /2 +?+mM2 /2b QM/2b wJ , 2M /Ƕmi`2b i2`K2b- BH v  mM2 +QKTûiBiBQM 2Mi`2 H2 MQK#`2 /2 T`iB2b /2 HǶQM/2 [mB

`2bi2Mi TBû;û2b 2Mi`2 H2b Q#bi+H2b 2i H /û+`QBbbM+2 /2 +?+mM2 /2 +2b T`iB2bX GǶ?vTQi?ĕb2

/ǶAFr /Ƕ?vT2`#QHB+Biû 7Q`i2 URX9XRV MQmb bbm`2 [m2 +2ii2 +QKTûiBiBQM  HB2m /Mb mM b2Mb 7pQ`#H2- 2i HǶQM T2mi +HQ`2 H T`2mp2X

_2K`[m2X aB HǶ?vTQi?ĕb2 /Ƕ?vT2`#QHB+Biû 7Q`i2 URX9XRV TT`ŗi /2 KMBĕ`2 Mim`2HH2- MQmb

B;MQ`QMb bB 2HH2 2bi QTiBKH2 , 2bbv2` /ǶQ#i2MB` mM +QMi`2 2t2KTH2 /Mb mM +b Qɍ 2HH2 MǶ2bi Tb pû`B}û2 b2`Bi /B;M2 /ǶBMi2`āiX