DARCY S LAW AND DIFFUSION FOR A TWO-FLUID EULER-MAXWELL SYSTEM WITH DISSIPATION

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1 Mathmatical Modls and Mthods in Applid Scincs c World Scintific Publishing Company DARCY S LAW AND DIFFUSION FOR A TWO-FLUID EULER-MAXWELL SYSTEM WITH DISSIPATION RENJUN DUAN Dpartmnt of Mathmatics, Th Chins Univrsity of Hong Kong, Shatin, Hongkong rjduan@math.cuhk.du.hk QINGQING LIU School of Mathmatics, South China Univrsity of Tchology, Guangzhou 564, P.R. China shuxuliuqingqing@6.com CHANGJIANG ZHU School of Mathmatics, South China Univrsity of Tchology, Guangzhou 564, P.R. China cjzhu@mail.ccnu.du.cn Rcivd Day Month Yar Rvisd Day Month Yar Communicatd by xxxxxxxxxx This papr is concrnd with th larg-tim bhavior of solutions to th Cauchy problm on th two-fluid Eulr-Maxwll systm with dissipation whn initial data ar around a constant quilibrium stat. Th main goal is th rigorous justification of diffusion phnomna in fluid plasma at th linar lvl. Prcisly, motivatd by th classical Darcy s law for th nonconductiv fluid, w first giv a huristic drivation of th asymptotic quations of th Eulr-Maxwll systn larg tim. It turns out that both th dnsity and th magntic fild tnd tim-asymptotically to th diffusion quations with diffusiv cofficints xplicitly dtrmind by givn physical paramtrs. Thn, in trms of th Fourir nrgy mthod, w analyz th linar dissipativ structur of th systm, which implis th almost xponntial tim-dcay proprty of solutions ovr th high-frquncy domain. Th ky part of th papr is th spctral analysis of th linarizd systm, xactly capturing th diffusiv fatur of solutions ovr th low-frquncy domain. Finally, undr som conditions on initial data, w show th convrgnc of th dnsitis and th magntic fild to th corrsponding linar diffusion wavs with th rat + t 5/4 in L norm and also th convrgnc of th vlocitis and th lctric fild to th corrsponding asymptotic profils givn in th sns of th gnnralizd Darcy s law with th fastr Corrsponding author

2 R. Duan, Q. Liu, C. Zhu rat + t 7/4 in L norm. Thus, this work can b also rgardd as th mathmatical proof of th Darcy s law in th contxt of collisional fluid plasma. Kywords: Eulr-Maxwll systm; dissipation; diffusion wavs; larg-tim bhavior. AMS Subjct Classification: 35B4,35Q35, 35P. Introduction It is gnrally blivd that th Darcy s law govrns th motion of th inviscid flow with frictional damping 7 or th slow viscous flow 5 in larg tim. It is quit nontrivial to mathmatically justify th larg-tim bhavior of solutions to thos rlativ physical systms, particularly in th cas whn vacuum appars, cf. 4,9,8,9. Bsids, thr ar also som rsults, for instanc, s and rfrncs thrin, to discuss th modifid Darcy s law for conducting porous mdia. In th papr, w attmpt to giv a rigorous proof of Darcy s laws and diffusion phnomna in th contxt of collisional fluid plasma whnvr th dnsitis of fluids ar clos to nonvacuum stats. In a wakly ionisd gas with a small nough ionisation fraction, chargd particls will intract primarily by mans of lastic collisions with nutral atoms rathr than with othr chargd particls, cf. 3. In such situation, th motion of fluid plasmas consisting of ions α = i, lctrons α = and nutral atoms is gnrally govrnd by th two-fluid Eulr-Maxwll systn thr spac dimnsions t n α + n α u α =, m α n α t u α + u α u α + p α n α = q α n α E + u α c B ν α m α n α u α, t E c B = 4π q α n α u α, t B + c E =, E = 4π q α n α, B =.. Hr th unknowns ar n α = n α t, x and u α = u α t, x R 3 with α = i,, dnoting th dnsitis and vlocitis of th α-spcis rspctivly, and also E = Et, x R 3 and B = Bt, x R 3, dnoting th slf-consistnt lctron and magntic filds rspctivly, for t > and x R 3. For th α-spcis, p α dpnding only on th dnsity is th prssur function which is smooth and satisfis p αn > for n >, and for simplicity w assum in th papr that th fluid is isothrmal and hnc p α n = T α n for th constant tmpratur T α >. Constants m α >, q α, ν α >, c > stand for th mass, charg and collision frquncy of α-spcis and th spd of light, rspctivly. Th constant 4π apparing in th systs rlatd to th spatial dimnsion. Notic q = and q i = Z in th gnral physical situation, whr > is th lctronic charg and Z is an positiv intgr. Without loss

3 Two fluid Eulr-Maxwll systm 3 of gnrality, w may assum Z = through th papr, sinc it can b normalizd to b unit undr th transformation ñ i = Zn i, q i =, = m i n Z, p in = p i. Z Initial data ar givn by with th compatibility condition [n α, u α, E, B] t= = [n α, u α, E, B ],. E = 4π q α n α, B =..3 Th papr is mainly concrnd with th larg-tim asymptotic bhavior of solutions to th Cauchy problm on th two-fluid Eulr-Maxwll systm with collisions whnvr initial data ar clos to a constant quilibrium stat [n α =, u α =, E =, B = ]. Notic that collision trms play a ky rol in th analysis of th problm; s 33, for instanc. For that purpos, w first stat th global-in-tim xistnc in th following Thorm.. Lt th intgr N 3. If [n α, u α ] H N + [E, B ] H N is sufficintly small thn th Cauchy problm.,.,.3 admits a uniqu global solution n α, u α, E, B C[, ; H N R 3 Lip[, ; H N R 3. Th mor prcis statmnt of Thorm. will b givn in Sction 5., and its proof is basd on th dirct nrgy stimats, cf. 6,8. To furthr study th asymptotic bhaviour of solutions, w introduc th larg-tim asymptotic profil as follows. Lt G µ t, x = 4πµt 3/ xp{ x /4µt} b th hat krnl with th diffusion cofficint µ >. Lt us dfin th ambipolar diffusiv cofficint µ > and th magntic diffusiv cofficint µ > by µ = T i + T c ν i ν, µ = ν i + ν 4π ν i + ν,.4 rspctivly. Corrsponding to givn initial data., w dfin th asymptotic profil [n, u α, E, B] by n = m α ν α G µ t, n α,.5 ν i + ν B = G µ t, B,.6

4 4 R. Duan, Q. Liu, C. Zhu and u α t, x = T i + T nt, x + c ν Bt, x,.7 ν i + ν 4πq α ν i + ν Et, x = T i ν T ν i nt, x + c ν i ν ν i + ν 4π Bt, x,.8 ν i + ν for α = i,. With th abov notations, th main rsult of th papr rgarding th asymptotic bhaviour of solutions is statd as follows. Thorm.. Thr ar constants ɛ > and C > such that if [n α, u α ] H L + [E, B ] H L < ɛ,.9 thn th solution to th Cauchy problm.,.,.3 satisfis n α n L + B B L C + t 5 4,. and for all t. u α u α L + E E L C + t 7 4,. W giv a fw rmarks on Thorm.. First of all, from th proof latr on, undr th assumption.9 th solution to th Cauchy problm.,.,.3 around th constant quilibrium stat njoys th tim-dcay proprty [n α, u α ] L + [E, B] L C + t 3 4, whr th tim-dcay rat must b optimal for gnral initial data with B du to thos rsults from th spctral analysis givn in Sction 4; s Corollary 4. for instanc. On th othr hand th larg-tim asymptotic profil also satisfis n L + B L C + t 3 4, u α L + E L C + t 5 4, which ar also optimal in trms of th dfinition.5,.6,.7,.8 of [n, u α, E, B]. Thrfor it is nontrivial to obtain th fastr tim-dcay rats. and., and this also assurs that [ + n, u α, E, B] indd can b rgardd as th mor accurat larg-tim asymptotic profil for solutions to th Cauchy problm undr considration, compard to th trivial constant quilibrium stat. Notic that n and B ar diffusion wavs by.5 and.6 as wll as.4, and u α and E ar dfind in trms of thos two diffusion wavs by.7 and.8. From th huristic drivation of th larg-tim asymptotic profils in th nxt sction, w s that th asymptotic profils can b solvd from th asymptotic quations obtaind in a formal way in th sns of th Darcy s law. In th cas without any lctromagntic fild, thr hav bn xtnsiv mathmatical studis of th larg-tim bhavior for

5 Two fluid Eulr-Maxwll systm 5 th dampd Eulr systm basing on th Darcy s law; s,8,8,9 and rfrnc thrin. Howvr fw rigorous rsults ar known for such physical law in th contxt of two-fluid plasma with collisions. This work can b rgardd to som xtnt as th gnralisation of th Darcy s law for th classical non-conductiv fluid to th plasma fluid undr th influnc of th slf-consistnt lctromagntic fild. Th scond issu is concrnd with th condition.9 rgarding th H rgularity of initial data. In fact, as sn from Thorm., th global xistnc of solutions can b assurd for small initial data in H 3 only. Not that th dampd Eulr-Maxwll is of th rgularity-loss typ, corrsponding to th fact that ignvalus of th linarizd systm may tnd asymptotically to th imaginary axis as th frquncy gos to infinity; s 6 in th on-fluid cas. Thr has bn a gnral thory dvlopd in 36 in trms of th Fourir nrgy mthod to study th dcay structur of gnral symmtric hyprbolic systms with partial rlaxations of th rgularity-loss typ. Th main fatur of tim-dcay proprtis for such rgularityloss systs that solutions ovr th high-frquncy domain can still gain th nough tim-dcay rat by compnsating nough rgularity of initial data. Thrfor, th highr ordr Sobolv rgularity is ssntially usd to complt th proof of Thorm. for th larg-tim bhaviour of solutions, particularly for obtaining th xplicit tim-dcay rat ovr th high-frquncy domain. Third, th ky point of Thorm. is to prsnt th convrgnc in L norm of th solution [n α, u α, E, B] to th profil [ + n, u α, E, B] if initial data approach th constant stady stat in th sns of.9. Thr could b svral dirct gnralisations of th currnt rsult. As in 37, it can b xpctd to obtain th convrgnc rats for th drivativs up to to som ordr. In gnral, th highr th ordr of drivativs is, th fastr thy dcay in tim. Anothr possibl approach for obtaining th global xistnc and convrgnc of solutions to th constant stady stat is to introduc as in 5 th ngativ Sobolv spac basing on th pur nrgy mthod togthr with th functional intrpolation inqualitis, whr th advantag is that both L norms of th highr drivativs and L norms of th zro-ordr ar not ncssarily small. Howvr, it sms still nontrivial to apply such mthod to obtain th larg-tim asymptotic bhaviour. and.. Th final rmark is concrnd with th nonlinar diffusion of th two fluid Eulr- Maxwll systm with collisions. In fact, th currnt work is don at th linarizd lvl. Evn for th gnral prssur functions P α α = i,, by using th sam formal drivation as in Sction, th dnsity satisfis th nonlinar hat quation t n P n =, whr P is in connction with P α α = i, as wll as othr physical paramtrs. Th nonlinar hat quation abov is also a typ of th porous mdiuquation. Thus, it would b intrsting and challnging to furthr invstigat th asymptotic stability of th nonlinar diffusion wavs, cf. 8. W hop to rport it in th futur study. To prov Thorm. w nd to carry out th spctral analysis of th linarizd

6 6 R. Duan, Q. Liu, C. Zhu systm around th constant stady stat; s 6, for instanc. In fact, th solution can b writtn as th sum of th fluid part and th lctromagntic part in th form of ρ α t, x ρ α t, x u α t, x Et, x Bt, x = u α, t, x E t, x + u α, t, x E t, x Bt, x Howvr it sms difficult to giv an xplicit rprsntation of solutions to two ignvalu problms du to th high phas dimnsions undr considration. Th main ida is to obtain th asymptotic xpansions of solutions to th linarizd systm as th frquncy k cf. 7, ; s Sction 4. On trick to dal with th lctromagntic part is to first rduc th systm to th high-ordr ODE of th magntic fild B only, thn study th asymptotic xpansion of B as k, and finally apply th Fourir nrgy mthod to stimat th othr two componnts u α, and E cf. and rfrnc thrin; s Lmma 4.4. For k, it can b dirctly tratd by th Fourir nrgy mthod sinc th linarizd solution oprator in th Fourir spac bhavs lik } xp { λ k + k t, which lads to th almost xponntial tim-dcay dpnding on rgularity of initial data; s Sction 3. In th man tim, w find that th larg-tim bhavior of solutions to th two-fluid Eulr-Maxwll systm. is govrnd by th following two subsystms t n + nu =, and P α n = q α ne ν α m α nu, α = i,, q α E ν α m α u α, =, α = i,, c B = 4πn q α u α,, t B + c E =. For mor dtails s Sction and Sction 4. Finally w would mntion th following works rlatd to th papr: som drivations and numrical computations of th rlativ modls,,5,35, global xistnc and larg-tim bhavior for th dampd Eulr-Maxwll systm 3,6,8,3,3,34,37,38,39,4, global xistnc in th non-damping cas,4,, and asymptotic limits undr small paramtrs 6,3,3. Th rst of th papr is organisd as follows. In Sction, w giv th huristic drivation of diffusion wavs motivatd by th classical Darcy s law. In Sction 3 w rformulat th Cauchy problm on th Eulr-Maxwll systm around th constant.

7 Two fluid Eulr-Maxwll systm 7 stady stat, and study th dcay structur of th linarizd homognous systm by th Fourir nrgy mthod. In Sction 4, w prsnt th spctral analysis of th linarizd systm by thr parts. Th fist part is for th fluid, th scond on for th lctromagntic fild, and th third on for th xtra tim-dcay of solutions with spcial initial data. Th rsult in th third part accounts for stimating th inhomognous sourc trms. In Sction 5, w first prov th global xistnc of solutions by th nrgy mthod, show th tim asymptotic rat of solutions around th constant stats and thn obtain th main rsult concrning th tim asymptotic rat around linar diffusion wavs. Notations. Lt us introduc som notations for th us throughout this papr. C dnots som positiv gnrally larg constant and λ dnots som positiv gnrally small constant, whr both C and λ may tak diffrnt valus in diffrnt placs. For two quantitis a and b, a b mans λa b λa for a gnric constant < λ <. For any intgrm, w us H m, Ḣ m to dnot th usual Sobolv spac H m R 3 and th corrsponding m-ordr homognous Sobolv spac, rspctivly. St L = H m whn m =. For simplicity, th norm of H s dnotd by m with =. W us, to dnot th innr product ovr th Hilbrt spac L R 3, i.. f, g = fxgxdx, f = fx, g = gx L R 3. R 3 For a multi-indx α = [α, α, α 3 ], w dnot α = x α x α x α3 3. Th lngth of α is α = α + α + α 3. For simplicity, w also st j = xj for j =,, 3.. Huristic drivation of diffusion wavs In this sction w would provid a huristic drivation of th larg-tim asymptotic quations of th dnsitis, vlocitis and th lctromagntic fild. Indd, both th dnsitis and th magntic fild satisfy th diffusion quations with diffrnt diffusion cofficints in trms of thos physical paramtrs apparing in th systm, and th vlocitis and th lctric fild ar dfind by th dnsitis and th magntic fild according to th Darcy s law... Diffusion of dnsitis W first giv a formal drivation of th larg-tim asymptotic quations of dnsitis and vlocitis. Assum th quasinutral condition n i = n = nt, x, u i = u = ut, x,. and also assum that th background magntic fild is a constant vctor, for instanc, B =,, B is constant along x 3 -dirction. Not that B hr is not ncssarily assumd to b zro. W start from th asymptotic momntuquations for α = i and : p α n = q α n E + u c B ν α m α nu..

8 8 R. Duan, Q. Liu, C. Zhu Along B,. rducs to i.., p α n B = q α ne B ν α m α nu B. { pi n B = ne B ν i nu B, p n B = ne B ν nu B. It can b furthr writtn in th matrix form: pi B n νi m = i n E B. p B n ν n u B On can solv E B and u B as p i + p B u B =, ν + ν i n E B = pi ν i p ν B n ν i +. ν Notic that sinc B =,, B is along th x 3 -dirction, thn 3 p i n + p n u 3 =, ν + ν i n E 3 = ν i + ν pin 3 ν i pn n ν..3 Along th x x -plan normal to B, noticing u B = u B, u B,,. rducs to p α = q α n E + B c u ν α m α nu, p α = q α n E B c u ν α m α nu, i.., This implis u u mα ν α n qαn q αn c B c B m αν α n u u + q α n E q mα ν = α n αn c B [ qαn c B m q α n αν α n E p = α..4 p α E for α = i and. W dnot q mα ν A α := α n αn c B qαn c B m. αν α n E + p α p α ],.5

9 Two fluid Eulr-Maxwll systm 9 Thn ltting th right-hand trms of.5 b qual for α = i and furthr implis q i na E i + A p i i = q na E + A p. E p i E p Du to th isothrmal assumption p α n = T α n, on has q i na i q na E = T i A i Thrfor, u E E = q i na i Plugging.6 back into.5 givs u = [ q i na i q i na i E T A n. n q na T i A i T A n..6 n q na T i A i T A + T i A ] n i n Lt us giv an xplicit computation of th cofficint matrix in.7: G = q i na i q i na i q na T i A i Notic q i =, q =, and A mα ν α = α n qαn c B q dt A αn α c B m = A T αν α n dt A α, α.7. T A + T i A i..8 whr dt A α = n m ανα + q α c B. To cancl n in dt A α, w writ na α = n dt A α A T α = Thn on can comput.8 as m αν α m α ν α + c B qα c B m α ν α + c B qα c B m α ν α + c B m αν α m α ν α + c B := K α. G = q i na i q i na i q na T i A i T A + T i A i = n [ n A T i A T i + n ] [ A T Ti A T i T ] A T + T i A T i dt A i dt A i dt A dt A i dt A dt A i = n A T i M T i A T i + n A T i M T A T + M M T i A T i dt A i dt A i dt A i dt A dt A i = n A T M T i A T i + n A T i M T A T dt A dt A i dt A i dt A = K K i + K T i n K i + K i K i + K T n K = T i n K K i + K K i T n K ik i + K K,.9 whr M = K i + K. Dnoting C i = m i ν i + c B, C = m ν + c B,

10 R. Duan, Q. Liu, C. Zhu on has K i + K = ν i C i Hnc, = = B c B c B c ν i + ν C B c ν i C i + mν B C c C i ν i C C i ν i+ν ν iν + B c C ic + mν C B c ν C i C B ν i+ν ν i+ν ν ν i c C ic = ν i + ν C i C B ν i+ν ν ν i c C ic ν iν + B c C ic m iν i ν + B B c c ν ν i. B c ν ν i ν i ν + B c dtk i + K = ν i + ν C i C, whr w hav usd th idntity C i C = m i νi m ν + B c m i νi + m ν B + c. It is thrfor straightforward to s K i + K = m iν i ν + B c B c ν ν i. ν i + ν B c ν ν i ν i ν + B c Aftr strnuous computations, on can vrify that K K i + K K i = K i K i + K K =. ν i + ν Hr w hav omittd th proof of th abov idntity for brvity. Plugging this idntity into.9 yilds that th cofficint matrix G in.7 is givn by G = Ti+T ν i+ν.. n Ti+T ν i+ν This togthr with.3 imply that u Ti+T ν i+ν n n u = Ti+T m iν i+ν n.. u 3 Ti+T ν i+ν 3 n

11 Two fluid Eulr-Maxwll systm Thrfor, using th first quation of. for th consrvation of mass undr th quasinutral assumption., w obtain that n satisfis th diffusion quation t n T i + T n =.. ν i + ν It rmains to dtrmin th componnts of E normal to B, namly E and E. In fact,.4 implis that E q α n α E u n = A α T α, u n for α = i and. Taking α = i for instanc and thn using.7,.8 and., on has E = n mi ν i n c B n E n n c B m G + T i n iν i n n n n = n mi ν i n c B n n c B m G + T i n iν i n n n = T iν T ν i B T i+t m iν i+ν c ν i+ν n. n T i+t n B c ν i+ν This togthr with.3 imply that T iν T ν i B E m iν i+ν c n E = B T i+t c m iν i+ν E 3 T iν T ν i ν i+ν T i+t ν i+ν T iν T ν i ν i+ν T iν T ν i ν i+ν n n.3. 3 n W point out that in th cofficint matrix on th right of.3, th diagonal ntris ar qual and indpndnt of B, and th non-diagonal ntris ar skwsymmtric and linar in B... Diffusion of th magntic fild Notic that th larg-tim asymptotic profils of u and E givn by. and.3 ar along th gravitational dirction of th diffusiv dnsity n dtrmind by.. Lt u α, α = i, and E b th asymptotic profils along th dirction normal to th gravitation of th dnsity. Thn by taking th background dnsitis as n i = n =, w xpct that th larg-tim profil of th magntic fild B is govrnd by th following systm E + ν i u i, =, E + ν u, =, c B + 4πu i, u, =, t B + c E =.

12 R. Duan, Q. Liu, C. Zhu It is asy to obtain that B satisfis th diffusion quation t B c ν i ν 4π B =, ν i + ν and u α, α = i, and E ar givn by u i, = E = c ν B, m i ν i 4π ν i + ν u, = E = c ν i B, ν 4π ν i + ν E = c ν i ν 4π B. ν i + ν 3. Dcay proprty of linarizd systm In this sction, w study th tim-dcay proprty of solutions to th linarizd systm basing on th Fourir nrgy mthod. Th rsult of this part is similar to th cas of on-fluid in 6, and also similar to th study of two-spcis kintic Vlasov-Maxwll-Boltzmann systn 9. Th main motivation to prsnt this part is to undrstand th linar dissipativ structur of such complx systm as in 36 in trms of th dirct nrgy mthod and also provid a clu to th mor dlicat spctral analysis to b givn latr on. Notic that th ky stimat 3.7 in this sction will b usd to dal with th tim-dcay proprty of solutions ovr th high-frquncy domain in th nxt sctions. 3.. Rformulation of th problm W assum that th stady stat of th Eulr-Maxwll systm. is trivial, taking th form of n α =, u α =, E = B =. Bfor constructing th mor accurat larg-tim asymptotic profil around th trivial stady stat, w first considr th linarizd systm around th abov constant stat. For that, lt us st ρ α = n α for α = i and. Thn U := [ρ α, u α, E, B] satisfis t ρ α + u α = g α, Initial data ar givn by m α t u α + T α ρ α q α E + m α ν α u α = g α, t E c B + 4π q α u α = g 3, t B + c E =, E = 4π q α ρ α, B =. 3. [ρ α, u α, E, B] t= = [n α, u α, E, B ], 3.

13 Two fluid Eulr-Maxwll systm 3 with th compatibility condition E = 4π Hr th inhomgnous sourc trms ar g α = ρ α u α := f α, p g α = m α u α u α α ρ α + ρ α + g 3 = 4π q α ρ α u α. q α ρ α, B =. 3.3 p α u α ρ α + q α c B, Notic in th isothrmal cas that p αn = T α for any n > Linar dcay structur In this sction, for brvity of prsntation w still us U = [ρ α, u α, E, B] to dnot th solution to th linarizd homognous systm t ρ α + u α =, with givn initial data m α t u α + T α ρ α q α E + m α ν α u α =, t E c B + 4π q α u α =, t B + c E =, E = 4π q α ρ α, B =, satisfying th compatibility condition E = 4π 3.5 [ρ α, u α, E, B] t= = [ρ α, u α, E, B ], 3.6 q α ρ α, B =. 3.7 Th goal of this sction is to apply th Fourir nrgy mthod to th Cauchy problm 3.5, 3.6, 3.7 to show that thr xists a tim-frquncy Lyapunov functional which is quivalnt with Ût, k and morovr its dissipation rat can also b charactrizd by th functional itslf. Lt us stat th main rsult of this sction as follows. Thorm 3.. Lt Ut, x, t >, x R 3, b a wll-dfind solution to th systm Thr is a tim-frquncy Lyapunov functional EÛt, k with EÛt, k Û := [ˆρ α, û α ] + Ê + ˆB, 3.8

14 4 R. Duan, Q. Liu, C. Zhu such that, for som λ >, th Lyapunov inquality holds for any t > and k R 3. d EÛt, k + λ k dt + k EÛt, k 3.9 Proof. As in 6, w us th following notations. For an intgrabl function f : R 3 R, its Fourir transfors dfind by ˆfk = xp ix kfxdx, R 3 x k := 3 x j k j, k R 3, whr i = C is th imaginary unit. For two complx numbrs or vctors a and b, a b dnots th dot product of a with th complx conjugat of b. Taking th Fourir transforn x for 3.5, Û = [ˆρ α, û α, Ê, ˆB] satisfis t ˆρ α + ik û α =, j= m α t û α + T α ikˆρ α q α Ê + m α ν α û α =, t Ê cik ˆB + 4π q α û α =, t ˆB + cik Ê =, ik Ê = 4π q α ˆρ α, ik ˆB =, t >, k R First of all, it is straightforward to obtain from th first four quations of 3. that d [ Tα dt ˆρ ] α, m d ] αû α + [Ê, ˆB + m αν α û α =. 3. 4π dt By taking th complx dot product of th scond quation of 3. with ikˆρ α, rplacing t ˆρ α by th first quation of 3., taking th ral part, and taking summation for α = i,, on has t Rm α û α ikˆρ α + T α k ˆρ α + 4π q α ˆρ α = m α k û α m α ν α Rû α ikˆρ α, which by using th Cauchy-Schwarz inquality, implis t Rm α û α ikˆρ α + λ k ˆρ α + 4π q α ˆρ α C + k û α.

15 Two fluid Eulr-Maxwll systm 5 Dividing it by + k givs Rm α û α ikˆρ α t + k +λ k + k ˆρ α + 4π + k q α ˆρ α C û α. 3. In a similar way, by taking th complx dot product of th scond quation of 3. with 4πq α Ê/T α, rplacing t Ê by th third quation of 3., and taking summation for α = i,, on has t 4πm α q α û α T Ê+ k Ê + 4πqα α T α + 4πm α q α û α 4π q α û α + T α whr w hav usd ik Ê = 4π th Cauchy-Schwarz inquality imply t Ê = 4πm α q α û α cik T ˆB α 4πq α m α ν α û α Ê, 3.3 T α q α ˆρ α. Taking th ral part of 3.3 and using 4πm α q α Rû α T Ê + k Ê + 4πqα α T Ê α 4πm α q α Rû α cik T ˆB + C û α, α which furthr multiplying it by k / + k givs t 4πm α q α T α k Rû α Ê + k + k k Ê + k + 4πm α q α T α 4πq α T α k Ê + k k Rû α cik ˆB + k + C û α. 3.4 Similarly, it follows froquations of th lctromagntic fild in 3. that t Ê ik ˆB + c k ˆB = c k Ê + 4π q α û α ik ˆB, which aftr using Cauchy-Schwarz and dividing it by + k, implis Finally, lt s dfin EÛt, k = RÊ ik ˆB λ k ˆB c k Ê t + k + + k + k + C û α. 3.5 κ [ T α ˆρ α, ] m α û α + [ Ê, ˆB] + κ 4πm α q α T α Rm α û α ikˆρ α + k k Rû α Ê RÊ ik ˆB + k + κ κ + k,

16 6 R. Duan, Q. Liu, C. Zhu for constants < κ, κ to b dtrmind. Notic that as long as < κ i is small nough for i =,, thn EÛt, k Ût holds tru and 3.8 is provd. Th sum of 3., 3. κ, 3.4 κ and 3.5 κ κ givs t EÛt, k + λ û α + λ k + k ˆρ α + λ k + k [Ê, ˆB],3.6 whr w hav usd th idntity k ˆB = k ˆB du to k ˆB = and also usd th following Cauchy-Schwarz inquality κ 4πm α q α T α k Rû α ik ˆB + k 4πm α q α T α κ k 4 û α 4ɛκ + k + 4πm α q α T α ɛκ κ k ˆB + k. First, w chos ɛ > such that ɛ4π m α q α T α λ for λ apparing on th lft of 3.5, and thn lt κ > b fixd and lt κ > b furthr chosn small nough. Thrfor, 3.9 follows from 3.6 by noticing û α + λ k + k This complts th proof of Thorm 3.. ρ α + λ k + k [Ê, ˆB] λ k + k Û. Thorm 3. dirctly lads to th pointwis tim-frquncy stimat on th modular Ût, k in trms of initial data modular Ûk, which is th sam as 6. Corollary 3.. Lt Ut, x, t, x R 3 b a wll-dfind solution to th systm Thn, thr ar λ >, C > such that Ût, k Cxp λ k t + k Ûk 3.7 holds for any t and k R 3. Basd on th pointwis tim-frquncy stimat 3.7, it is also straightforward to obtain th L p -L q tim-dcay proprty to th Cauchy problm Formally, th solution to th Cauchy problm is dnotd by Ut = [ρ α, u α, E, B] = tl U, whr tl for t is said to b th linarizd solution oprator corrsponding to th linarizd Eulr-Maxwll systm.

17 Two fluid Eulr-Maxwll systm 7 Corollary 3. s 6 for instanc. Lt p, r q, l and lt m b an intgr. Dfin [ l + 3 r ] l, if l is intgr and r = q =, = q + [l + 3 r q ] +, othrwis, 3.8 whr [ ] dnots th intgr part of th argumnt. Suppos U satisfying 3.7. Thn tl satisfis th following tim-dcay proprty: Lt U L q C + t 3 p q m U L p for any t, whr C = Cm, p, r, q, l. +C + t l m+[l+3 r q ]+ U L r 4. Spctral rprsntation In ordr to study th mor accurat larg-tim asymptotic profil, w nd to carry out th spctral analysis of th linarizd systm. 4.. Prparations As in 6, th linarizd systm 3.5 can b writtn as two dcoupld subsystms which govrn th tim volution of ρ α, u α, E and u α, E and B rspctivly. W dcompos th solution to into two parts in th form of ρ α t, x ρ α t, x u α t, x Et, x Bt, x whr u α,, u α, ar dfind by = u α, t, x E t, x + u α, t, x E t, x Bt, x u α, = u α, u α, = u α,, 4. and likwis for E, E. For brvity, th first part on th right of 4. is calld th fluid part and th scond part is calld th lctromagntic part, and w also writ U = [ρ i, ρ, u i,, u, ], Notic that to th nd, E is always givn by U = [u i,, u,, E, B]. E = 4π ρ i ρ. W now driv th quations of U and U and thir asymptotic quations that on may xpct in th larg tim. Taking th divrgnc of th scond quation of 3.5, it follows that { t ρ α + u α =, 4. m α t u α q α E + T α ρ α + m α ν α u α =.

18 8 R. Duan, Q. Liu, C. Zhu Applying to th scond quation of 4. and noticing u α = u α,, w s that th fluid part U satisfis { t ρ α + u α, =, 4.3 m α t u α, q α E + T α ρ α + m α ν α u α, =. Initial data ar givn by [ρ α, u α, ] t= = [ρ α, u α, ]. 4.4 As sn latr on and also in th sns of th Darcy s law, th xpctd asymptotic profil of th fluid part satisfis { t ρ + ū =, with initial data ρ t= = ρ = T α ρ q α Ē + m α ν α ū =, ν i ν ρ i + ρ. ν i + ν ν i + ν Thrfor, ρ, ū and Ē ar dtrmind according to th following quations t ρ T i + T ρ =, ν i + ν ū = T i + T ρ, 4.5 ν i + ν Ē = T i ν T ν i ν i + ν ρ, whr initial data ū, and Ē, of ū and Ē ar dtrmind by ρ in trms of th last two quations of 4.5, rspctivly. For latr us, lt us dfin P ik, P ik, P 3 ik to b thr row vctors in R 8 by [ ] P ν i ν ik =:,,,, ν i + ν ν i + ν [ P ik =: T i + T ν i ik, T i + T ν i + ν ν i + ν ν i + ν [ P 3 Ti ν T ν i ν i ik =: ik, ν i + ν ν i + ν T i ν T ν i ν i + ν ] ν ik,,, ν i + ν ν ν i + ν ik,, Thn th larg-tim asymptotic profil can b xprssd in trms of th Fourir transform by ˆ ρ = xp T i + T k t P ikû T ν i + ν, 4.6 ˆū = xp T i + T k t P ikû T ν i + ν, 4.7 ˆĒ = xp T i + T k t P 3 ikû T ν i + ν. 4.8 ].

19 Two fluid Eulr-Maxwll systm 9 Th lctromagntic part satisfis th following quations: t u i, E + ν i u i, =, t u, + E + ν u, =, t E c B + 4πu i, u, =, t B + c E =, 4.9 with initial data [u α,, E, B] t= = [u α,, E,, B ]. 4. Th xpctd larg-tim asymptotic profil for th lctromagntic part is dtrmind by th following quations in th sns of Darcy s law again: Ē + ν i ū i, =, Ē + ν ū, =, c B πū i, ū, =, t B + c Ē =. As bfor, it is straightforward to obtain c t B ν i ν 4π ν i + ν B =, ū i, = Ē = c ν ν i 4π ν i + ν B, ū, = Ē = c ν i ν 4π ν i + ν B, Ē = c ν i ν 4π ν i + ν B, with initial data 4. B t= = B, whr initial data ū i,, ū,, Ē, ar givn from B according to th last thr quations of 4.. Notic that th asymptotic profil B of th magntic fild can b xprssd in trm of th Fourir transform by c ˆ Bt, ν i ν k = xp 4π ν i + ν k t ˆB k Spctral rprsntation for fluid part 4... Asymptotic xpansions and xprssions Aftr taking th Fourir transformation in x for 4.3, rplacing Ê by 4π ik k ˆρ i ˆρ, th fluid part Û = [ˆρ i, ˆρ, û i,, û, ] satisfis th following systm of st-ordr

20 R. Duan, Q. Liu, C. Zhu ODEs t ˆρ i + ik û i, =, t ˆρ + ik û, =, t û i, + T i ikˆρ i + 4π t û, + T ikˆρ + 4π ik k ˆρ i 4π ik k ˆρ 4π ik k ˆρ + ν i û i, =, ik k ˆρ i + ν û, =. 4.4 Initial data ar givn as Û t, k t= = Û k =: [ˆρ i, ˆρ, k k û i, k k û ]. 4.5 Thn th solution to 4.4, 4.5 can b writtn as Û t, k T = Aikt Û k T, with th matrix Aik dfind by ζ ζ Aik =: Ti ζ 4π ζ 4π ζ ζ ζ ν i, 4π ζ ζ T ζ 4π ζ ζ ν whr w hav dnotd ζ = ik on th right. In th squl, for brvity, with a littl abus of notation, for a positiv intgr l, w also us ζ l to dnot ζ l ζ if l is odd, and ζ l if l is vn. By dirct computation, w s that th charactristic polynomial of Aζ is dtλi Aik = λ 4 + ν i + ν λ 3 + Ti + It follows from 4.6 that ν i ν ν + T ν i Ti T + ζ 4 4π T i + T ζ 4 λ j = ν i + ν, i= i j 4 4 i= Ti + T ζ + 4π + λ ζ + 4π ν + ν i λ. 4.6 Ti λ i λ j = ν i ν + T ζ + 4π +, λ j = T it ζ 4 4π T i + T ζ.

21 Two fluid Eulr-Maxwll systm First, w analyz th roots of th abov charactristic quation 4.6 and thir asymptotic proprtis as ζ. Th prturbation thory s 7 or 4 for onparamtr family of matrix Aζ for ζ implis that λ j ζ has th following asymptotic xpansions: Notic that λ j with gλ = λ 3 + ν i + ν λ + For latr us w also st λ j ζ = + l= λ l j ζ l. ar th roots of th following quation: λgλ =, ν i ν + 4π + λ + 4π ν + ν i. gλ = λ 3 + c λ + c λ + c. 4.7 On can list som lmntary proprtis of th function gλ as follows: g = 4π ν + ν i > ; g ν i + ν = ν i ν ν i ν 4π g λ = 3λ + ν i + ν λ + 4π + > for λ ; ν i + ν < ; ν i ν + 4π ν i ν + 4π + + g λ = λ + λ + ν i + ν λ + 4π + >, for λ ν i + ν ; gλ is strictly incrasing ovr λ ν i + ν or λ. ν i ν + ν i ν + Th abov proprtis imply that th quation gλ = has at last on ral root dnotd by σ which satisfis ν i + ν < σ <. At this tim, although w hav known that thr is at last on ral root, it is not clar whthr ths roots ar distinct or not. W can distinguish svral possibl cass using th discriminant, = 8c c c 4c 3 c + c c 4c 3 7c. >, thn gλ = has thr distinct ral roots; <, thn gλ = has on ral root and two nonral complx conjugat roots; =, thn gλ = has a multipl root and all its roots ar ral. Through th papr, w only considr th first two cass. Not that th third cas is much hardr to study as Puisux xpansions of th ignvalus hav to b usd in that cas. Undr this assumption, in ordr to giv th asymptotic xprssions of

22 R. Duan, Q. Liu, C. Zhu Aikt as k, w s that th solution matrix Aikt has th spctral dcomposition 4 Aikt = xpλ j iktp j ik, j= whr λ j ζ ar th ignvalus of Aζ and P j ζ ar th corrsponding ignprojctions. Notic that P j ζ can b writtn as P j ζ = l j Aζ λ l ζi λ j ζ λ l ζ, whr w hav assumd that all λ j ζ ar distinct to ach othr for k small nough. In trms of th graph of gλ, on can s whn >, gλ = has thr distinct ngativ ral roots. Whn <, assuming that a + bi, a bi ar two conjugat complx roots and plugging a + bi into gλ =, on has th following two quations: R : a 3 3ab + ν i + ν a b + + 4π ν + ν i =, Im : 3a b b 3 + ν i + ν ab + Sinc b, substituting b = 3a + ν i + ν a + ν i ν + 4π + a ν i ν + 4π + b =. ν i ν + 4π + back into th quation of th ral part abov, w hav a 3 + a ν i + ν + a ν i ν + 4π + + ν i + ν + ν i ν + ν ν i + 4π ν i + 4π ν =. Thn th abov quation must hav only on ral ngativ root. By straightforward computations and using 4.6, w find that λ k = T i + T ζ + λ 4 ν i + ν ζ4 + O ζ 5, λ j k = σ j + O ζ, for j =, 3, 4, whr σ j j =, 3, 4 ar th roots of gλ =, satisfying 4 R σ j <, σ j = ν i + ν, σ σ 3 σ 4 = 4π ν + ν i, j= and λ 4 is to b dfind latr on. 4.8

23 Two fluid Eulr-Maxwll systm 3 Aftr chcking th cofficint of ζ 4 in 4.6, w can gt som information of λ 4 which is ncssary for th cofficint of ζ in λ λ 3 λ 4. In th cas λ = λ in 4.6, th cofficint of ζ 4 is ν i ν + 4π which implis that and λ 4 λ = λ λ 3 λ 4 = = + 4π ν i ν + 4π λ Ti + ν + T ν i +4π ν + ν i λ λ 4 Ti T + =, [ ] T i ν + T m ν iν i+ν i T i+t + TiT ν i+ν T i+t 4π ν + 4π ν i [ ν i ν + 4π + T i ν + T ν i miνi+mν m T i+t + T i T iν i+ν T it ζ 4 4π T i+t ζ T i+t ν i+ν ζ + λ 4 ζ4 + O ζ 5 4π ν i + ν Ti T = ζ 4π T i + T ν i + ν = 4π ν + ν i λ4 T i + T ζ + O ζ 3 λ + T it ν i + ν + 4π T i + T λ 4 T i + T Nxt, w stimat P ζ xactly. In th following, w dnot On can comput [Aζ] = [Aζ] a ij 4 4, [Aζ] 3 a 3 ij 4 4. λ T i ζ 4π 4π ν i ζ 4π T i ν i ζ + 4π ν i ζ ζ 4π ν ζ ζ T ζ 4π ν ζ 4π ν i ζ ζ T i ζ 4π + νi T ν ζ + 4π ν ζ ζ 4π 4π T ζ 4π + ν T i+t ] ζ + O ζ 3.,,

24 4 R. Duan, Q. Liu, C. Zhu and Ti ν i ζ + 4π ν i 4π ν i Ti ζ 3 + 4π ζ νi ζ 4π ζ [Aζ] 3 4π ν T ν ζ + 4π ν 4π ζ T ζ 3 + 4π ζ ν ζ =, a 3 3 a 3 3 a 3 33 a 3 34 a 3 4 a 3 4 a 3 43 a 3 44 whr a 3 3 = Ti ζ 3 + T i 4π a 3 3 = Ti 4π + T 4π T 4π i νi ζ + 4π νi + 4π 4π 4π 4π νi + 4π 4π ζ ζ, ζ a 3 33 = T i ν i ζ + 4π ν i ν 3 i, a 3 34 = 4π a 3 4 = Ti 4π a 3 4 = T a 3 43 = 4π ν i 4π ν, ζ + T 4π ζ 3 + T 4π T ν ν 4π ν i, a 3 44 = T ν ζ + 4π ν ν 3. 4π 4π ν + 4π 4π 4π ζ + 4π ζ ζ ζ ζ, ν + 4π 4π Not that w must dal with trms involving carfully, sinc thy contain singularity as k. By using 4.8, w stimat th numrator and dnominator of P ik, rspctivly, in th following way that and + P dn =: g l ζ l = 4π ν i + ν + g ζ + O ζ 3, l= P num =[Aζ] 3 λ + λ 3 + λ 4 [Aζ] + λ λ 3 + λ λ 4 + λ 3 λ 4 [Aζ] λ λ 3 λ 4 I =[Aζ] 3 + ν i + ν + λ [Aζ] + ν i ν = : f ij 4 4. Ti + T ζ + 4π ζ ζ, ζ ζ, + 4π λ λ + λ 3 + λ 4 [Aζ] λ λ 3 λ 4 I

25 Two fluid Eulr-Maxwll systm 5 Notic that P dn = 4π ν i + ν g [ g ] ζ + O ζ 3. Lt us comput f ij i, j 4 as follows. For f, on has f ζ = T i ν i ζ + 4π Ti + ν i +ν i +ν +λ ζ 4π λ λ 3 λ 4 =: f l m ζ l, i l= whr f and thrfor, = 4π f =, ν i 4π 4π 4π ν i + ν + ν i + ν = 4π ν i, f = T i ν i + T i ν i + ν 4π m i T i + T ν i + ν T it ν i + ν + 4π T i + T λ 4 T i + T = 4π f ζ = 4π ν i + T i + T ν i + ν + T i ν T ν i 4π In a similar way, w can gt T i + T ν i + ν + T i ν T ν i λ ν i ν i + ν, ν i ν i + ν ζ + O ζ 3. and f ζ = 4π ν + 4π T i + T ζ + O ζ 4, ν i + ν f 3 ζ = 4π ζ + O ζ 3, f 4 ζ = 4π ζ, For f ζ, on has f ζ = 4π ν i + 4π T i + T ζ + O ζ 4. ν i + ν f ζ = T ν ζ + 4π T + ν + ν i + ν + λ ζ 4π λ λ 3 λ 4 = f l m ζl, l=

26 6 R. Duan, Q. Liu, C. Zhu whr f = 4π f =, ν 4π 4π 4π ν i + ν + ν i + ν = 4π ν, f = T ν + T ν i + ν 4π m T i + T ν i + ν T it ν i + ν + 4π T i + T λ 4 T i + T λ, Thrfor, = 4π f ζ = 4π ν + Similarly, on has T i + T ν i + ν T i ν T ν i 4π Morovr, it holds that f 3 ζ = Ti T i + T ν i + ν T i ν T ν i ν ν i + ν. f 3 ζ = 4π ζ, f 4 ζ = 4π ζ + O ζ 3. ζ 3 + T i 4π + ν i + ν + λ Ti Ti + ν i ν + T T i νi ζ + ν i ζ + 4π ζ ν i ζ 4π ζ + + 4π In th xprssion of f 3 ζ abov, sinc th cofficint of 4π 4π νi + 4π 4π ν ν i + ν 4π 4π + ν i + ν 4π ν i ζ + O ζ 3. νi + 4π 4π ζ ζ λ λ + λ 3 + λ 4 T i ζ 4π ζ ζ is vanishing, i.. ν i ν + 4π and th cofficint of ζ is givn by T i 4π T i νi + ν i + ν T i T i + T i 4π ν i ν i ν i + ν 4π ν i ν + + 4π Ti Ti + T 4π T i + T i 4π + ν i + ν ν i + ν = 4π T i + T ν i, ν i + ν it follows that f 3 ζ = 4π T i + T ν i ζ + O ζ 3. ν i + ν + 4π =, ζ ζ.

27 Two fluid Eulr-Maxwll systm 7 Similarly w can calculat f 3 ζ, f 33 ζ, f 34 ζ and f 4j ζ j =,, 3, 4 as follows: f 3 ζ = 4π T i + T f 33 ζ = O ζ, f 34 ζ = O ζ, ν ζ + O ζ 3, ν i + ν f 4 ζ = 4π T i + T ν i ζ + O ζ 3, ν i + ν f 4 ζ = 4π T i + T ν ζ + O ζ 3, ν i + ν f 43 ζ = O ζ, f 44 ζ = O ζ. Lt P j ik, P j ik, P 3 j ik, P 4 j ik b th four row vctors of P jik, j =,, 3, 4. According to th abov computations, w hav P ik = P dn P ik = P dn 4π 4π ν i + 4π 4π ν + P 3 ik = P dn ν i ν i+ν ζ + O ζ 3 T i+t ν i+ν ζ + O ζ 3 ζ + O ζ 3 ζ T i+t ν i+ν + Timν Tmiνi 4π ν + 4π 4π T i+t ν i+ν 4π 4π ν i + 4π T i+t ν i+ν ζ + O ζ 3 Timν Tmiνi ν ν i+ν ζ + O ζ 3 ζ 4π T i+t 4π T i+t 4π 4π ζ + O ζ 3 ν i ν i+ν ζ + O ζ 3 ν ν i+ν ζ + O ζ 3 O ζ O ζ T, T T,, and P 4 ik = P dn 4π T i+t 4π T i+t ν i ν i+ν ζ + O ζ 3 ν ν i+ν ζ + O ζ 3 O ζ O ζ T. Basd on th dfinitions of P, P, P 3,P 4, w hav th xprssions of ˆρ α ζ, û α, ζ

28 8 R. Duan, Q. Liu, C. Zhu and Ê ζ for k as follows: 4 ˆρ i ζ = xp λ j iktpj ikû k T j= = xp λ ikt P Û k T + O k xp λ ikt Û k 4 + xp λ j iktpj ikû k T, j= 4 ˆρ ζ = xp λ j iktpj ikû k T j= = xp λ ikt P Û k T + O k xp λ ikt Û k 4 + xp λ j iktpj ikû k T, j= 4 û i, ζ = xp λ j iktpj 3 ikû k T j= = xp λ ikt P Û k T + O k xp λ ikt Û k 4 + xp λ j iktpj 3 ikû k T, j= û, ζ = 4 xp λ j iktpj 4 ikû k T j= Hr Finally, noticing = xp λ ikt P Û k T + O k xp λ ikt Û k 4 + xp λ j iktpj 4 ikû k T. j= P ik P ik = P dn T iν T ν i T iν T ν i ν i ν i+ν ζ + O ζ 4 ν ν i+ν ζ + O ζ 3 O ζ 3 O ζ 3 T,

29 Two fluid Eulr-Maxwll systm 9 w also hav Ê = 4π ik k q α ρ α = 4π ik k ˆρ i ˆρ α = 4π ik k xpλ ikt P ik P ik Û k T 4π ik k 4 xp λ j ikt Pj ik Pj ik Û k T j= = xpλ ikt P 3 Û T + O k xpλ ikt Û k 4π ik k 4 xp λ j ikt Pj ik Pj ik Û k T. j= Error stimats Lmma 4.. Thr is r > such that for k r and t, th rror trm U U can b boundd as ˆρ α t, k ˆ ρt, k C k xp λ k t Û k + C xp λt Û k, 4. û α, t, k ˆū t, k C k xp λ k t Û k +C xp λt Û k + Ê k, 4. Ê t, k ˆĒ t, k C k xp λ k t Û k 4. +C xp λt Û k + Ê k, 4.3 whr C and λ ar positiv constants. Proof. It follows from th xprssions of ˆρ α ζ and ˆ ρζ that ˆρ i ζ ˆ ρζ = xp λ ikt P Û k T xp T i + T k t P Û k T ν i + ν 4 + O k xp λ ikt Û k + xp λ j iktpj ikû k T := ˆR ik + ˆR ik + ˆR 3 ik. W hav from 4.8 that λ ik + j= T i + T ν i + ν k = O k 4,

30 3 R. Duan, Q. Liu, C. Zhu and xp λ ikt xp T i + T k t ν i + ν = xp T i + T k t xp λ ikt + T i + T k t ν i + ν ν i + ν C xp T i + T k t k 4 t xp C k 4 t ν i + ν C k xp λ k t, as k. Thrfor, w obtain that ˆR ik C k xp λ k t Û k as k. Not that R λ ik λ k and xpλ ikt xp λ k t as k. Consquntly, w find that ˆR ik C k xp λ k t as k. Now it suffics to stimat ˆR 3 ik. Rcall R σ j < for j =, 3, 4. This togthr with 4.8 giv xpλ j ikt xp λt as k. Also notic Pj ik = O. Thus w hav ˆR 3 ik Cxp λt Û k as k. In a similar way, w can gt ˆρ k ˆ ρk C k xp λ k t Û k + C xp λt Û k. This thn provs th dsird stimat 4.. To considr th rst stimats, on has to prov that P 3 j U T k, P 4 j U T k, ik k 4 P j ik Pj ik U T k, j= ar all boundd. Notic that thos trms includ For j =, 3, 4, by using 4.8, w s that and P num P dn j = l jλ j ζ λ l ζ = O, j =[Aζ] 3 + ν i + ν + λ j [Aζ] Ti + ν i ν + T ζ + 4π λ λ 3 λ 4 I ζ ζ which is singular as k. + + λ j ν i + ν + λ j Aζ :=g j il

31 Two fluid Eulr-Maxwll systm 3 W hav to b carful to trat th third row and th fourth row involving straightforward to comput g j 3 as g j 3 ζ = Ti ζ 3 + T i 4π T i νi ζ 4π + 4π νi + 4π 4π ζ ζ Ti + ν i + ν + λ j ν i ζ + 4π ζ ν i ζ Ti + ν i ν + T 4π ζ + + 4π Th cofficint of T i ζ 4π ζ ζ ζ ζ. ζ ζ. It is + λ j ν i + ν + λ j in th abov xprssion of g j 3 ζ is furthr simplifid as 4π ν i σ j σ j ν i + ν + σ j 4π = 4π σ j ν + σ j, whr w rcall that σ j j =, 3, 4 ar th thr roots of gλ =. Thrfor, g j 3 = 4π σ j ν + σ j ζ ζ + O ζ = 4π W now turn to stimat g j 3. It follows that g j 3 ζ = Ti 4π + T 4π 4π ζ 4π + ν i + ν + λ j 4π ζ ν i ζ Ti + ν i ν + T 4π ζ + Th cofficint of ζ ζ which hnc implis that g j 3 = 4π σ j ν + σ j ik k + O k. νi + 4π 4π ζ ζ + 4π 4π + λ j ν i + ν + λ j in th abov xprssion is furthr simplifid as 4π ν i σ j + σ j ν i + ν + σ j 4π = 4π σ j ν + σ j, σ j ν + σ j ζ ζ + O ζ = 4π σ j ν + σ j ik k + O k. Chcking th third row of Aik, [Aik] and [Aik] 3, w can obtain that ζ ζ. g j 33 = O, gj 34 = O,

32 3 R. Duan, Q. Liu, C. Zhu as k. It is dirct to vrify that Pj 3 ikû k T =g j 3 ˆρ i + g j 3 ˆρ + g j 33ûi, + g j 34û, = 4π σ j ν + σ j ik k ˆρ i ˆρ + O Û k = σ j ν + σ j Ê k + O Û k, whr w hav usd th compatibl condition Ê = 4π ik k ˆρ i ˆρ. Thn th xprssions of û α, ζ and ˆū ζ imply that û i, ζ ˆū ζ = xp λ ikt P Û k T xp T i + T k t P Û k T ν i + ν In a similar way, w can gt + O k xp λ ikt Û k + 4 xp λ j iktpj 3 ikû k C k xp λ k t Û k + Cxp λt Û k + Ê k. û, k ˆūk C k xp λ k t Û k +C xp λt Û k + Ê k. This provs 4.. It now rmains to stimat ik k j= 4 P j ik Pj ik Û k T, 4.5 j= apparing in 4.9. Sinc th first row minus th scond row of I, Aik, [Aik] and [Aik] 3 ar rspctivly givn by and,,,,,, ζ, ζ, T i ζ 4π 4π, T ζ + 4π + 4π, ν i ζ, ν ζ, T i ν i ζ + 4π ν i + 4π ν, T ν ζ 4π ν i 4π ν, T i 4π ζ π ζ νi ζ, T 4π ζ 3 + 4π ζ + ν ζ,

33 Two fluid Eulr-Maxwll systm 33 on can comput 4.5 by 4.4 as ik k P j ik P j ik Û k T = P dn j = = P dn j ik k g j gj ˆρ i + g j gj ˆρ + g j 3 gj 3 û i, + g j 4 gj 4 û, Pj dn 4π + + ik σ j k ˆρ i ˆρ + O Û k σ j Ê k + O Û k, which is boundd whn k. Thn th xprssions of Ê ζ and ˆĒ ζ imply that Ê ζ ˆĒ ζ = xpλ ik P 3 Û T xp T i + T k t P 3 Û k T ν i + ν + O k xpλ ik Û k 4π ik 4 k xp λ j ikt Pj ik Pj ik Û k T j= C k xp λ k t Û k + C xp λt Û k + Ê k. This provs 4.3 and thn complts th proof of Lmma 4.. Nxt, w considr th proprtis of ˆρ α ζ, û α, ζ and Ê ζ as k. It follows from 3.7 that { Cxp λ k Ût, k t Ûk, k r, Cxp λ k t Ûk, k r. 4.6 Hr r is dfind in Lmma 4.. Combining 4.6 with 4.6, 4.7 and 4.8, w hav th following pointwis stimat for th rror trms ˆρ α k ˆ ρk, û α, k ˆūk and Ê k ˆĒk as k. Lmma 4.. Lt r > b givn in Lmma 4.. For k r and t, th rror U Ū can b boundd as ˆρ α t, k ˆ ρt, k C xp λ k t Û k + C xp λt [ˆρ i k, ˆρ k], û α, t, k ˆū t, k C xp λ k t Û k + C xp λt k [ˆρ i k, ˆρ k], Ê t, k ˆĒ t, k C xp λ k t Û k + C xp λt k [ˆρ i k, ˆρ k], whr C and λ ar positiv constants. Basd on Lmma 4. and 4. togthr with 6, th tim-dcay proprtis for th diffrnc trms ρ α ρ, u α, ū and E Ē ar statd as follows.

34 34 R. Duan, Q. Liu, C. Zhu Thorm 4.. Lt p, r q, l, and lt m b an intgr. Suppos that [ρ α, u α, ] is th solution to th Cauchy problm Thn U = [ρ α, u α, ] and E satisfy th following tim-dcay proprty: m ρ α t ρt L q C + t 3 p m+ q U L p + Cxp λt U L p + C + t l m+[l+3 r q ]+ U L r + Cxp λt m+[3 r q ]+ [ρ i, ρ ] L r, m u α, t ū t L q C + t 3 p m+ q U L p + Cxp λt U L p + C + t l m+[l+3 r q ]+ U L r + Cxp λt m++[3 r q ]+ [ρ i, ρ ] L r, and m E t Ē t L q C + t 3 p m+ q U L p + Cxp λt U L p + C + t l m+[l+3 r q ]+ U L r + Cxp λt m++[3 r q ]+ [ρ i, ρ ] L r, for any t, whr C = Cm, p, r, q, l and [l + 3 r q ] + is dfind in Spctral rprsntation for lctromagntic part Asymptotic xpansions and xprssions for B Taking th curl for th quations of t u i,, t u,, t E in 4.9 and using B = B, it follows that t u i, E + ν i u i, =, t u, + E + ν u, =, 4.7 t E + c B + 4π u i, u, =, t B + c E =. Taking th tim drivativ for th fourth quation of 4.7 and thn using th third quations to rplac t E givs tt B c B 4πc u i, u, =. 4.8 Furthr taking th tim drivativ for 4.8 and rplacing t u i, and t u, giv ttt B c B t + 4π + t B + 4πcν i u i, ν u, =. 4.9 Hr w hav rplacd E by c tb. Furthr taking th tim drivativ for 4.9 and rplacing t u i, and t u, givs tttt B c B tt + 4π + tt B 4π ν i + ν t B 4πcν i u i, ν u, =. 4.3

35 Two fluid Eulr-Maxwll systm 35 Taking th summation of 4.3, 4.9 ν i + ν and 4.8 ν i ν yilds tttt B + ν i + ν ttt B c B tt + 4π + tt B + ν i ν tt B c ν i + ν B t + 4π ν + ν i t B ν i ν c B =. In trms of th Fourir transforn x of th abov quation, on has tttt ˆB + νi + ν ttt ˆB + c k + 4π + 4π + ν i ν tt ˆB + ν i + ν c k + 4π ν + 4π ν i t ˆB + νi ν k c ˆB =. 4.3 Initial data ar givn as ˆB t= = ˆB, t ˆB t= = cik Ê,, tt ˆB t= = c k ˆB + 4πc ik û i, ik û,, ttt ˆB t= = c k + 4π + cik m Ê, 4πcν i ik û i, + 4πcν ik û,. Th charactristic quation of 4.3 rads λ 4 + ν i + ν λ 3 + c k + 4π + + c ν i + ν k + 4π + ν i ν λ ν + ν i 4.3 λ + ν i ν c k =. For th roots of th abov charactristic quation and thir basic proprtis, on has c ν i ν λ k = 4π ν i + ν k + O k 4, 4.33 λ j k = σ j + O k, for j =, 3, 4, as k. Hr w not that σ j j =, 3, 4 with R σ j < ar th solutions to gλ = with gλ still dfind in 4.7. On can st th solution of 4.3 to b 4 ˆB = c j ik xp{λ j ikt}, 4.34 j= whr c i i 4 ar to b dtrmind by 4.3 latr. In fact, 4.34 implis c c ˆB t= M c c 3 := λ λ λ 3 λ 4 c λ λ λ 3 λ 4 c 3 = t ˆB t= tt ˆB t=, 4.35 c 4 λ 3 λ 3 λ 3 3 λ 3 4 c 4 ttt ˆB t=

36 36 R. Duan, Q. Liu, C. Zhu whr th right-hand trs givn in trms of 4.3 by ˆB t= t ˆB t= tt ˆB t= = cik 4πcik 4πcik c k ttt ˆB t= 4πcν i ik 4πcν ik c k + 4π + cik 4.36 It is straightforward to chck that dt M = j<i 4 λ i λ j, as long as λ j k ar distinct to ach othr, and M M M 3 M 4 M = M M M 3 M 4 dt M M 3 M 3 M 33 M 43, M 4 M 4 M 34 M 44 û i, û, Ê, ˆB whr M ij is th corrsponding algbraic complmnt of M. Notic that 4.35 togthr with 4.36 giv c û i, c c 3 = M cik 4πcik 4πcik c k û, Ê,, c 4 4πcν i ik 4πcν ik c k + 4π + cik ˆB which aftr plugging M, implis c = = j<i 4 λ i λ j [4πcM 3 4πcM 4 ν i ik û i, + 4πcM 3 + 4πcM 4 ν ik û, + M + M 4 c k + 4π + M c k ] M 3 ˆB M ˆB λ i λ j + O k Û,. j<i 4 + cik Ê,. W dduc that M λ i λ j j<i 4 has th following asymptotic xpansion as k : j<i 4 M + λ i λ j = c l k l, l=

37 Two fluid Eulr-Maxwll systm 37 whr λ λ 3 λ 4 M = λ λ 3 λ 4 λ 3 λ 3 3 λ 3 = λ λ 3 λ 4 4 λ i λ j. j<i 4 By straightforward computations, c = holds tru and this implis that c ik = ˆB + O k Û, [,, ] T Error stimats In this sction, w first giv th rror stimats for B B, and thn apply th nrgy mthod in th Fourir spac to th diffrnc problm for 4.9 and 4. to gt th rror stimats for u α, ū α, and E Ē. It should b pointd out that it is also possibl to carry out th sam strnuous procdur as in th prvious sction to obtain th rror stimats on u α, ū α, and E α, Ēα,. Th rason why w choos th Fourir nrgy mthod is just for th simplicity of rprsntation, sinc th stimats on u α, ū α, and E α, Ēα, can b dirctly obtaind basing on th stimat on B B. Lmma 4.3. Thr is r > such that for k r and t, ˆBt, k ˆ Bt, k C k xp λ k t + xp λt Û,, 4.38 whr C and λ ar positiv constants. Proof. It follows from 4.34 and 4.3 that ˆBt, k ˆ Bt, 4 c ν i ν k = c j ik xp{λ j ikt} xp 4π ν i + ν k t j= =c ik ˆB xp{λ ikt} { + ˆB c xp{λ } ν i ν ikt} xp 4π ν i + ν k t 4 + xp{λ j ikt}c j ik j= := ˆR ik + ˆR ik + ˆR 3 ik. ˆB k Using 4.33 and 4.37, on has ˆR ik C k xp λ k t Û,, ˆR ik C k xp λ k t ˆB, ˆR 3 ik Cxp λt Û,. This provs 4.38 and thn complts th proof of Lmma 4.3.

38 38 R. Duan, Q. Liu, C. Zhu Nxt, in ordr to gt th rror stimats for u α, ū α, and E Ē, w writ ũ α = u α, ū α,, Ẽ = E Ē, B = B B. Combining 4.9 with 4., thn [ũ α, Ẽ] satisfis m α t ũ α q α Ẽ + m α ν α ũ α = m α t ū α,, t Ẽ c B + 4π q α ũ α = t Ē Lmma 4.4. Thr is r > such that C k xp λ k t + xp λt Û,, for k r, û α, t, k ˆū α, t, k C xp λ k t Û k C xp λ k t k ˆB k, for k r, and C k xp λ k t + xp λt Û,, for k r, Ê t, k ˆĒ t, k C xp λ k t Û k C xp λ k t k ˆB k, for k r, whr C and λ ar positiv constants. Proof. It is straightforward to obtain th rror stimats for k r du to 3.7, 4. and 4.3. In th cas k r, th dsird rsult can follow from th Fourir nrgy stimat on th systm Indd, aftr taking th Fourir transforn x, 4.39 givs m α tˆũ α q α ˆẼ + m α ν αˆũ α = m α tˆū α,, t ˆẼ cik ˆ B + 4π q αˆũ α = t ˆĒ. 4.4 By taking th complx dot product of th first quation of 4.4 with ˆũ α, taking th complx dot product of th scond quation of 4.4 with ˆẼ, and taking th ral part, on has d dt = m α ˆũ α + d 4π dt ˆẼ + m α ν α ˆũ α m α R tˆū α, ˆũ α 4π R t ˆĒ ˆẼ + 4π Rcik ˆ B ˆẼ,

39 Two fluid Eulr-Maxwll systm 39 which by using th Cauchy-Schwarz inquality with < ɛ <, implis d m α ˆũ α + dt 4π ˆẼ + m α ν α ˆũ α ɛ ˆũ α + ɛ ˆẼ + C ɛ tˆū α, + C ɛ ik ˆ B + C ɛ t ˆĒ By taking th complx dot product of th first quation of 4.4 with q α ˆẼ, rplacing t ˆẼ by th scond quation of 4.4 and taking th ral part, on has t Rm αˆũ α q α ˆẼ + q α ˆẼ = q α Rm αˆũ α cik ˆ B + 4π q α R m αˆũ α q αˆũ α + q α R m αˆũ α t ˆĒ + Rm α ν αˆũ α q α ˆẼ + Rm α tˆū α, q α ˆẼ, which by using th Cauchy-Schwarz inquality with < ɛ <, implis t Rm αˆũ α q α ˆẼ + ˆẼ q α C ɛ ˆũ α + ɛ ˆẼ + C ɛ tˆū α, + C ɛ ik ˆ B + C ɛ t ˆĒ, 4.44 for < ɛ <. W now dfin Et = m α ˆũ α + 4π ˆẼ + κ Rm αˆũ α q α ˆẼ, for a constant < κ to b dtrmind. Notic that as long as < κ is small nough, thn Et ˆũ α + ˆẼ 4.45 holds tru. On th othr hand, th sum of 4.43 and 4.44 κ givs t Et + λ ˆũ α + ˆẼ C tˆū α, + C ik ˆ B + C t ˆĒ C k 6 xp{ λ k t} ˆ B + C k C k xp{ λ k t} + xp{ λt} Û, C k 4 xp{ λ k t} Û, + C k xp{ λt} Û,, 4.46

40 4 R. Duan, Q. Liu, C. Zhu for k r, whr w hav usd th xprssions of ū α,, Ēα, in 4., th xprssion of ˆ B in 4.3 and Lmma 4.3. Multiplying 4.46 by xpλt and intgrating th rsulting inquality ovr, t yild that Et xp λt ˆũ α + ˆẼ + Cxp λt xpλs k 4 xp λ k s Û, + C k xp λs Û, ds Cxp λt Û, + C k 4 xp λ k t Û, Thrfor, 4.4 and 4.4 follows from 4.47 by noticing This thn complts th proof of Lmma 4.4. From Lmma 4.4 togthr with 6, on has Thorm 4.. Lt p, r q, l, and lt m b an intgr. Suppos that U = [u α,, E, B] is th solution to th Cauchy problm Thn on has th following tim-dcay proprty: and m u α, t ū α, t L q C + t 3 p m+ q U L p + Cxp λt U L p + C + t l m+[l+3 r q ]+ U L r + Cxp λt m++[3 r q ]+ B L r, m E t Ē t L q C + t 3 p m+ q U L p + Cxp λt U L p + C + t l m+[l+3 r q ]+ U L r + Cxp λt m++[3 r q ]+ B L r, m Bt Bt L q C + t 3 p m+ q U L p + Cxp λt U L p + C + t l m+[l+3 r q ]+ U L r + Cxp λt m+[3 r q ]+ B L r, for any t, whr C = Cm, p, r, q, l and [l + 3 r q ] + is dfind in 3.8. W now dfin th xpctd tim-asymptotic profil of [ρ α, u α, E, B] to b [ ρ, ū α, Ē, B], whr ρ and B ar diffusion wavs, and [ū α, Ē] is givn by ū α = ū + ū α,, Ē = Ē + Ē. Combining Thorm 4. with Thorm 4., on has Corollary 4.. Lt p, r q, l, and lt m b an intgr. Suppos that Ut = tl U is th solution to th Cauchy problm with initial data U = [ρ α, u α, E, B ] satisfying 3.7. Thn U = [ρ α, u α, E, B] satisfis th following tim-dcay proprty: m ρ α t ρt L q C + t 3 p m+ q U L p + Cxp λt U L p + C + t l m+[l+3 r q ]+ U L r + Cxp λt m+[3 r q ]+ [ρ i, ρ ] L r,

41 Two fluid Eulr-Maxwll systm 4 m u α t ū α t L q C + t 3 p m+ q U L p + Cxp λt U L p +C+t l m+[l+3 r q ]+ U L r+cxp λt m++[3 r q ]+ [ρ i, ρ, B ] L r, m Et Ēt L q C + 3 t p m+ q U L p + Cxp λt U L p +C+t l m+[l+3 r q ]+ U L r+cxp λt m++[3 r q ]+ [ρ i, ρ, B ] L r, and m Bt Bt L q C + t 3 p m+ q U L p + Cxp λt U L p + C + t l m+[l+3 r q ]+ U L r + Cxp λt m+[3 r q ]+ B L r, for any t, whr C = Cm, p, r, q, l and [l + 3 r q ] + is dfind in 3.8. Corollary 4.. Undr th sam assumptions of Corollary 4., it holds that m ρ α t L q C + t 3 p m+ q U L p + Cxp λt U L p + C + t l m+[l+3 r q ]+ U L r + Cxp λt m+[3 r q ]+ [ρ i, ρ ] L r + C + t 3 p q m [ρi, ρ ] L p, m u α t L q C + t 3 p m+ q U L p + Cxp λt U L p + C + t l m+[l+3 r q ]+ U L r + Cxp λt m++[3 r q ]+ [ρ i, ρ, B ] L r + C + t 3 p m+ q [ρ i, ρ, B ] L p, m Et L q C + t 3 p m+ q U L p + Cxp λt U L p + C + t l m+[l+3 r q ]+ U L r + Cxp λt m++[3 r q ]+ [ρ i, ρ, B ] L r and + C + t 3 p m+ q [ρ i, ρ, B ] L p, m Bt L q C + t 3 p m+ q U L p + Cxp λt U L p + C + t l m+[l+3 r q ]+ U L r + Cxp λt m+[3 r q ]+ B L r + C + t 3 p q m B L p, for any t, whr C = Cm, p, r, q, l and [l + 3 r q ] + is dfind in 3.8.