The Ginzburg-Landau approximation for pattern forming systems. Guido Schneider

Transcription

1 Gzburg-u pproimion pern mg yem Guido Shneider

2 We re ereed pern mg yem loe fir biliy wih bounded erizion round homogeneou ground e wr olved by wih, k, n N, y f n (k, ye ik e n (k, ble ground e unble ground e e n (k k " e n (k k > Perurbion ly

3 Deripion pilly eended pern mg yem by Gzburg-u equion perurbion z origl yem Gzburg-u A (iµ@ XA A ( i A A?

4 mple: Swif-Hohenberg u u " u u 3 z derivion G equion " G (, "A(X, e i "A(X, e i where X ", " Inerg SH equion give reidul u u " u u 3 " 3 e A 4@ XA A 3A A " 3 e 3i A 3 " 4 e i (4i@ 3 XA " 5 e i (@ 4 XA

5 Approimion orem e A C([, ],X A be oluion G equion n re e " > C>uh ll " (, " re re oluion u SH equion whih be pproimed by " G uh up [, /" ] ku( " G (k Xu pple C" 3/ u(, "A fron ("( " o ho

6 Approimion orem e A C([, ],X A be oluion G equion n re e " > C>uh ll " (, " re re oluion u SH equion whih be pproimed by " G uh up [, /" ] ku( " G (k Xu pple C" 3/ u(, "A fron ("( " o ho

7 Approimion orem e A C([, ],X A be oluion G equion n re e " > C>uh ll " (, " re re oluion u SH equion whih be pproimed by " G uh up [, /" ] ku( " G (k Xu pple C" 3/ Poible funion pe X A X u Sobolev pe H b unimly onuou funion C b,unif unimly lol Sobolev pe H l,u equipped wih rm up kuk H (,

8 Approimion orem e A C([, ],X A be oluion G equion n re e " > C>uh ll " (, " re re oluion u SH equion whih be pproimed by " G uh up [, /" ] ku( " G (k Xu pple C" 3/ emrk eul n-rivil e qudri erm origl yem eene oluion order O(" h be eblhed on O(/" -ime le

9 Ariviy very mll oluion origl yem develop uh wy i be deribed fer O(/" -ime le by oluion oied Gzburg-u equion Upper emionuiy G rr reled origl yem rr onverge G rr "! Peudo-orbi pproimion oluion origl yem be pproimed ll ime by pproimion bed on peudo-orbi oied G equion A peudo-orbi on piee- oluion wih jump! "! Globl eene Globl eene oluion G equion rfer globl eene oluion O(-neighborhood unble orig origl yem (Ariviy AND A A G AND pproimion

10 Some referene P Colle J-P kmn, ime dependen mpliude equion Swif-Hohenberg problem, Comm Mh Phy 3 (99, 39 5 W khu, Gzburg-u equion rr, J Nonler Siene 3 (993, A Mielke G Shneider, Arr modulion equion on unbounded dom - eene ompron, Nonleriy 8 (995, 5,

11 G Shneider, Globl eene reul pern mg yem - Appliion 3D Nvier-Ske problem -, J Mh Pure Appl, IX 78 (999, 65 3 orem Globl eene oluion Couee- ylor problem O(-neighborhood kly unble Couee flow Hl,u wih >3/ O(- neighborhood mpped O("-neighborhood by flow

12 HA S I?

13 NO! re re ill my relev open problem bou Gzburg- u mlm whih re overed by ory bee ime-periodi yem: G Shneider H Ueker, mpliude equion fir biliy elero-onveion nemi liquid ryl e wo unbounded pe direion, Nonleriy (7, 6, (removed ondiion ued phy lierure frequeny ime-periodi g h be u ienly big Pern mg yem wih rml fluuion: uigi Amedeo Bihi, Dirk Blömker, Guido Shneider equion SPD on unbounded dom rxiv:76558 Modulion

14 Pern mg yem wih rvion lw - Flow down led flow - Mrgoni problem - Frdy problem

15 Clil iuion v rvion lw iuion Clil iuion: qudri erion riil mode give nriil mode Crvion lw iuion: h longer rue

16 Approimion orem (Ce Häker, G Shneider, D Zimmermn, Juifiion Gzburg- u pproimion e mrglly ble long wve, J Nonler Si (,, 93 3 Approimion orem (Ce G Shneider D Zimmermn, Juifiion Gzburg-u pproimion biliy i pper Mrgoni onveion, Mh Mehod Appl Si 36 (3, 9, 3 3 D Zimmermn, Juifiion Gzburg-u pproimion quiler problem - ludg ppliion Bénrd-Mrgoni onveion, Prepr Univeriä Sugr (6, 89 pge Ariviy orem (Ce W-P Düll, Kouroh Sei Khi, G Shneider, D Zimmermn, Ariviy Gzburg-u mode dribuion pern mg yem wih mrglly ble long mode, J Di er quion 6 (6,,

17 Pro pproimion orem (ubi e z G A(X, e i A(X, e i reidul e(u u ( u u u 3 improved z G 3 A 3 (, 3 improved reidul e( 3 3 ( A 3 64A 3 O( 4

18 error u, equion error ( e( Apply n Gronwll equliy prove up [, /" ] k(k X O(

19 ee ypuv Shmid reduion; (99, Melbourne (9 he mode-filer We ome bk hlg erm In Mielke (bk e(u u ( u (u mode-filer We ome hlg o (u e(u u ( u Sh94b, re ereed pproimion reul (Colle kmn [vh9, Sh94, il iuion, f SU7], e neurl lg llerm whih In el lil iuion, f [vh9, Sh94, Sh94b, SU7 ( improved pproimion o y given n fd on erm do fer erg z u ( 99; Kirrmn e l 99; Shneider 994, 994; ká e l 9 on ll erm whih do el fer erg z u We ome bk hlg erm ( In mode-filer wve number k, where ( ( uh orem, where e pern-mg SU7], e neurl mode wve number k ( ( Pro pproimion (qudri re re oluion yem whih behve A dire eime ll [, / ] wih help Gronwll equliy proimion uh il iuion, f [vh9, Sh94b, SU7], help e neurl e, lled mode-filer ll been rodued y re defed Aodire eime [, / ] wih Gronwll equliy i Sh94, hve ( uh Oule pro emple, o lled mode-filer hve been ro r Gzburg u equion Moreover, re re riviy reul erm ( poible due whih ould led r growh re O(e n wve number k, where ( ( uh ipliion operr Fourier pe hrough whih poible duey erm ( Shneider ould led growh re 995, howg every mll oluion beo(e derib e( O( O( rodued re defed mulipliion operr Fourier pe hrough r> mode-filer h beome unbounded rodued Gzburg u / defed Bee, epl error u fie o lled hve been y re ime by equion Combg pproim r > h beome unbounded / Bee, e u F (k (ku (k, j ±, j j prove F (kupper (ku (k, j ± how overome problem, how how mke reidul j u jmll A mer h f n n behoen o lrge diuly iviy reul llow emi-onuiy r liion operr Fourier pe hrough how overome h problem, (how how mke reidul mll ( ( e(, ±, Shneider 995; Shneider 999, hdowg by peudo-orbi, jour : {,boundg }, wih ± (k k /3, erm ( e( ll [, / ] (k k /3, where : {, }, wih j ± u F (k (ku (k, j ±, improved pproimion oy given n fd 999b pj j reul pern-mg yem (Shneider 994b, er f hve O( O( error pli f where reidul Appendi pproimion y onrued eime /3, improved modified pproimion A mer hve O( O A o given n fd ll e yem m kehed Fig proimion uh }, :reidul {, wih ±From (k w kon number eime /3, ie, orem riil n-riil pr led dierenly, dierene hve O( error pli riil n-riil pr led re given uprovendieren A pproimion been ligh e(u u ( u (u proimion uh n er hve O( O( error pli ierenly, f y fy yem, where ± ± ± uion degenered iuion oeffiien O( O( ie, e(, where± ± ± 3 vhe n on ll erm whih do el fer erg z u (7 il n-riil pr led dierenly, ie, m ion po h been dued Bizer Shneider Me e( O( O( y fy yem m dire mer ± f n n be hoen o lrge diuly A eime ll [, / ] wih help Gronwll equliy will Gzburg u equion mpliude equion hve been h A, where y fy yem ± ± mode-filer We ome bk hlg erm ( boundg f ( O( hoen O(, ourrene r ( O( (997 Gzburg u equion In our erm ( e( ll [, / ] Apoible mer n n be o lrge diuly mp due erm ( whih ould led growh re O(e lil iuion, f [vh9, Sh94, Sh94b, SU7], e neu e Hopf bifurion Fourier wve number k h O(, ( O( O( r Appendi A modified pproimion onrued eime our boundg erm ( e( ll [, / ( O( O beome unbounded / Bee, ep > h wve ( O( O(, mode number k, where ( ( uh reidul re given From w on qudri eime dierene reidul u eime mll how O( Appendi A modified pproimion onrued fo f on erion riil mode overome h problem, how how mke pplied due f pplied on qudri er emple, hve been rodued y re defi o (mode-filer O( O( lled dierene,reidul ie, ( (( w Se eime epnilly dmped hrough re given on u From ie, erion riil mode vhe, ( (( Se ep hrough mulipliion operr Fourier pe improved pproimion o y given n fd e he f pplied on qudri erion riil mode igroup genered ey obgenered eime ( O( by hrough ( i ey pnilly dmped emigroup by i ob mode-filer We ome bk hlg erm ( In ie, ( (( Se epnilly dmped hrough nerg O( fir equion yield proimion h uh u F (k (ku (k, j ±, yield n eime O( Inerg h fir equion j j Sh94b, SU7], e f lil iuion, [vh9, Sh94, neurl genered mode-filer bk hlg erm ( roup by ( i We eyome ob eime O( n uh ( O( O( e( O( O( mode wve number k, where ( ( ( O( where j : {, }, wih k /3, e erg h fir equion yield ± (k O( lil iuion, f [vh9, Sh94, Sh94b, SU7], ne 4 Fig elev perum derivion Gzburg u equion 4 re defed emple, o lled mode-filer hve been rodued y O( mer O( A f hve O( error rel o A mer f n n be hoen lrge diuly mode wve number k, where ( ( u A dire eime ll [, / ] wih help Gronwll equliy w pr eigenvlue λ(k ploed funion Fourier wve ( O( O( operr Fourier pe hrough mulipliion pr perum lie rily below le e λ mgniude bi riil n-riil led dierenly, ie, poible o ob O( O( ll [, / ] 4 our boundg erm ( e( ll [, / emple, o lled mode-filer hve been rodued y re de] prmeer perurbion <ε, where hrough y fy yem u F (kfourier ± operr ±pproimion ±pe j±, j j (ku (k, A modified onrued eime Appendi mulipliion

20 Pro pproimion orem (rvion lw e II y fy yem m ( O( O(, ( O( O(, ( O( O( eive equion due rvion lw ruure (( (,

21 h eion on ummry urfe new ideg Hene, ug rved n film bee ued hle Mrgoni problem e h fluid In hi e friion o rong long wve perurbion re longer ble friion o rong long wve urfe perurbion re longe h long wve perurbion re longer ble ree lefmrgoni urfe righ movg wve perurbion imply vh diuively ndle problem e h fluid In h problem wve ree lef righ movg wve perurbion imply vh diuively riion o rong long wve urfe perurbion re longer bl ree lef righ movg wve perurbion imply vh diu ng perurbion imply vh diuively e, ug g rved quiie refleion ymmery ng long wve urfe perurbion re longer ble Hene, ug g refleion rved quiie imply refleion ymmery o lef righ movg wve perurbion vh diuively erved quiie ymmery Hene, ug g rved quiie refleion ymmery blem eive equion due rvion lw ruure ovg wve perurbion imply vh diuively problem gproblem g rved quiie refleion ymmery h rved quiie refleion ymmery (( (, good model behvior (( (, (( (, (( (, model ( Se w good model behvior neurl mode lo order, f our mode lo model behvior neurl mode lo order, f our hvior neurl order, f our good (( (, model ( led mode mul problem on model ((, fir ide [HSZ] doe good behvior neurl lo order, f Vriion del ( Se w ner ideniy hge oorde poible Sehge w oorde ner ideniy oorde poible ner( ideniy poiblehge model ( Se w doe ner ideniy hge oorde po fir ide [HSZ] doe pply Hover, eond ide eem ] doe pply Hover, eond ide eem fir ide [HSZ] pply Hover, eond ide eem be uible ol Hover, w d behvior model neurl behvior neurl mode lo order, f ou mode lo order, f our / ide Hover, eond ide fir / / pply [HSZ] doe ver, hve e ( d O( ln whih O( doe Se uible hve unbounded (d oorde d O( whih be uible Hover, hve e e/poible lnln whih ( w ol Hover, ner ideniy hge poibl be ner ideniy hge oorde ol be uible ol Hover, hve e ( d O( ln wh oe llow onrol nler erm order unbounded doe llow onrol nler erm order O( ly up h po ounded doe llow onrol nler erm order SZ] doe [HSZ] pply Hover, eond ide eem fir ide doe pply Hover, eond ide eem po ly epled 3 ly unbounded doe llow onrol nler erm O( ly up h po epled deil Seion 3 ly Seion deil up h po epled deil Seion 3 ly / Soluion void h und growh / ( d O( ln whih over, hve e wh n epled deil Seion 4 ide uible ol Hover, hve e ( d O( ln whih i O( ly up h po epled deil Seion 3 void h und growh n epled deil Seion 4 ide void h und growh n epled deil Seion 4 idei fir onider ± d doe llow onrol nler erm order ime dierenible funion llow Xime, eondly ed doe onrol nler erm orde void h und growh n epled deil Seion 4 fir onider dierenible funion X, eondly ± r onider ime dierenible funion X, eondly ± pli ue (( ( O(,, hirdly ue n h po epled deil Seion 3 ly pli up, h ue n deil Seion ( 3 lyi O(, fir onider ime dierenible funion X, hirdly, hirdly (( po epled e ± pli ue n (( ( O(, e ly / / epled epli ( n d O(/ h llow bound growh deil Seion 4 ide flly ue e ue e ue, hirdly n (( ( / flly ( d O(/ h llow bound h und growh n epled deil Seion 4 ide ( d e flly ue O(/ h llow bound -erm by O(-bound on O(/ ime le fluene ime dierenible funion X, eondly / fluene nler O(-er by nler O(-erm O(-bound on O(/ ime le flly ue e ( d O(/ h llow boun onider ime dierenible funion X, eondl ± o how full yem repe f uene nler O(-erm by O(-bound on O(/ ime le ± dly ue n (( ( yem prie O( repe, ± h how prie h how full f fluene nler O(-erm by O(-bound on O(/ ime nible funion X A lredy id h will, hirdly ue n (( ( O( yem repe h how full f ± /prie hoen ime dierenible funion X A lredy id h will e ( d O(/ h llow bound hoen ime dierenible prie h how full yem repe f / ime dierenible en funion X A lredy id h will h llow bound h lyo(-erm ue e ( d O(/ be d Seion 4 by O(-bound on O(/ ime le be 4 ime dierenible funiond X Seion A 4 lredy id hi dhoen Seion nler O(-erm O(-bound full f on O(/ ime l ± be how d Seion 4 yembyrepe 3rie bi eime h how full id yem f ± i erenible funion X A lredy h repe will

22 An open problem: Globl eene pern mg yem wih rvion lw Approimion orem: Ariviy orem: Di uly: Syem modulion equion h epnilly rg XA A AB A X( A mple: Mrgoni problem wih kly unble orig