BLOWUPS IN GAUGE AND CONSTRAINT MODES. Bernd Reimann, AEI in collaboration with M. Alcubierre, ICN (Mexico)

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1 BLOWUPS IN GAUGE AND CONSTRAINT MODES Bernd Remnn, AEI n ollboron M. Aluberre, ICN (Mexo) Jen, Jnury 30, 006 1

2 Tops Pologes ( soks nd bloups ) n sysems of PDEs Te soure rer for vodng bloups Evoluon Sysem: Guge: Generlzed rmon lpse nd sf Evoluon equons: Adjused ADM sysem Ensen equons n sperl symmery nd smplfed 3+1 se Anlyss Numerl resuls: Bloups n e lpse nd sf modes Bloups n e onsrn modes

3 Pologes for Sysems of PDEs We sudy (n 1+1 dmensons) e Cuy problem for srongly yperbol sysems of PDEs su s u & u p( u, v), A( u) v q( u, v). v x Here nd re veor-vlued funons, nd e prnpl pr A s srongly yperbol mrx n be dgonlzed o v R ere re s rel egenvlues. We en nrodue e egenfelds For sysems of PDEs velke rer, o knd of pologes n our: (S. Aln: Bloup for Nonlner Hyperbol Equons, 95) 1 AR dg[ 1 m], Grden srope / Sok : Due o rossng of rerss, dervves of evoluon vrbles beome nfne. (P. Lx: Hyperbol Sysems of Conservon Ls nd e Meml Teory of Sok Wves, 73; M. Aluberre: Te Apperne of Coordne Soks n Hyperbol Formulons of GR, 97) ODE-Mensm / Bloup n fne me : Here egenfelds blo up long er rerss due o qudr soure erms. (F. Jon: Nonlner Wve Equons, Formon of Sngulres, 79; Remnn e l.: Guge nd onsrn soks n one-dmensonl numerl relvy, 05) R 1 v. 3

4 ODE-Mensm nd Soure Crer Vrbles blo up n fne me due o self-nrese n er domn of nfluene. Exmple: Te soluon o e ODE, blos up e me unless Generlzng s de o e prevous sysem of PDEs, one fnds en rng e evoluon of e mn sysem n erms of egenfelds. In prulr, for e dervve long rers follos For lolzed perurbons rvelng dfferen speeds, our soure rer yelds: If lso mxed erms ( ) 1 0 jk x x j k j, k jk 0., q( u, ) j Ο( ) re mporn, e ry n ddon o se k jk 4. 0.

5 Guge: Generlzed Hrmon Coordnes Hrmon oordnes re defned by skng for n dped oordnes yelds e yperbol evoluon equons for e lpse nd e resled sf A generlzon of su oordnes, so-lled generlzed rmon oordnes, s proposed by Bon e l. for e me oordne ( Bon-Mssó-Lpse ) nd n M. Aluberre, A. Cor, J.A. González, D. Nuñez, B. Remnn nd M. Slgdo: Generlzed rmon oordnes nd yperbol sf ondons, 05 for e spl oordnes: Demndng one obns, x g An n 0 f() K K g K /. 1/ f() 1 A 1/ () 1 () (3) f (3) K. 0, 0 {1,,3} 5

6 Evoluon Equons: Adjused ADM sysem For e evoluon sysem e use e ADM-sysem n e verson of J. York, yelds for e 3+1 spl e onsrn equons H M R K ogeer e evoluon equons ( ( L L ) j ) K j K j j j K for e 3-mer j nd e exrns urvure Ts se of equons, oever, s no srongly yperbol. For s reson e djus e evoluon equons for e K s by ddng e Hmlonn onsrn Here e s re rbrry funons (noe e do no demnd ). j j K j ( R j K j j K KK. K j ( L ) K H... j j 0 j 0 K k K k j ) j j 6

7 f f For e lpse egenfeld rvelng egenspeed f e fnd f ons f 1 f f ' "mxed erms", ons / "mxed erms". : 0 Demndng ese oeffens vns, e obn (). nd In order o elmne n ddon e mxed soure erms, f urns ou s needed. Wen pplyng e ndre lner degenery rer, M. Aluberre n 96 obned e sme ondon (*) for e vodne of soks. Furermore, e soed 1+log slng f / s n pproxme. soluon o (*) f 1. Guge Bloups : 0 (*) nd smlrly for e sf egenfeld egenspeed e nfer f 1 ons / f 7

8 Numerl Resuls: Guge Bloup Inl d: Mnkosk mer n 1+1 dmensons n non-rvl oordnes Evoluon: Generlzed rmon lpse nd sf f = f 0 = ons nd = 0 = ons f =1 s preferred f =1 = f =1 s preferred mxed erms f ± ± re smll sne egenspeeds λf ± nd λ ± re que dfferen 8

9 In ddon o guge nd pysl modes, ere re modes reled o e volon of e onsrns. Bloups n su onsrn modes n be voded by modfyng e evoluon sysem. Wen rryng ou e nlyss n sperl symmery, e onsrn modes ve egenvlues ( (1/ ) g ), nd e sysem s srongly rr yperbol f 1/, 0. Seng n ons e qudr soure erm o 0, e obn evoluon sysems vod bloups f rk Smlrly, for smplfed 3+1 se (only dependeny on x, negleng off-dgonl mer omponens, no sf), e obned e onsrn modes rvel egenspeeds 1 ( yy zz ) g xx, ere yy zz 0,( f 1) / mus old. From ons follos for e vodne of bloups: Consrn Bloups 1 3 / "mxed erms" rk f 1 "mxed erms" x xx xx yy. zz 1 yy yy zz zz. 9

10 Numerl Resuls: Consrn Bloup (sperl symmery) Consrn-volng nl d: Perurbed Mnkosk mer n sperl symmery Evoluon: djused ADM + rmon slng (f =1) + vnsng sf sndrd ADM bloup-vodng fmly of djusmens rk 3 ere 1/,

11 Numerl Resuls: Consrn Bloup (smplfed 3+1 se) Consrn-volng nl d: Perurbed Mnkosk mer, y=z, only dependeny on x, negleng off-dgonl mer omponens Evoluon: djused ADM + rmon slng + zero sf bloup-vodng fmly of djusmens sndrd ADM / ere yy 1/ 4, 0 xx 1 yy yy 11

12 Conlusons In srongly yperbol sysems of PDEs, pologes n nd do rse: grden srope => soks ODE-mensm => bloups n fne me In order o vod ose, e proposed nd dsussed ondons: ndre lner degenery soure rer For Ensen s equons usng generlzed rmon guge, lredy n 1+1dmensons guge pologes our. In more spl dmensons, n ddon onsrn pologes rse. Hoever, usng good guge oes nd ddng onsrns n suble y o evoluon equons, one n obn velke evoluons ou soks nd bloups. Te nlyss s no ye been rred ou n full 3+1 dmensons, bu for evoluon sysems n numerl relvy s ler one sould use: srongly yperbol prnpl pr + bloup-vodng soure erms 1

Stability Analysis for VAR systems. )', a VAR model of order p (VAR(p)) can be written as:

Stability Analysis for VAR systems. )', a VAR model of order p (VAR(p)) can be written as: Sbl Anlss for VAR ssems For se of n me seres vrbles (,,, n ', VAR model of order p (VAR(p n be wren s: ( A + A + + Ap p + u where he A s re (nxn oeffen mres nd u ( u, u,, un ' s n unobservble d zero men