C 0 Approximation on the Spatially Homogeneous Boltzmann Equation for Maxwellian Molecules*

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1 Alied Mhemics,,, doi:.46/m..666 Published Online December (h:// C Aroximion on he Silly Homogeneous Bolzmnn Equion or Mxwellin Molecules Absrc Minling Zheng School o Science, Huzhou Techer College, Huzhou, Chin E-mil: mlzheng@yhoo.com.cn eceived Augus 5, ; revised Ocober 5, ; cceed Ocober 9, In his er we sudy he viscosiy nlysis o he silly homogeneous Bolzmnn equion or Mxwellin molecules. e irs show h he globl exisence in ime o he mild soluion o he viscosiy equion Q(, ) v. e hen sudy he symoic behviour o he mild soluion s he coeiciens, nd n esime on is derived. Keywords: Viscosiy Bolzmnn Equion, Mild Soluion, Viscosiy Aroximion, Collision Kernel. Inroducion In his er we shll invesige he symoic roeries o he soluion o he viscosiy Bolzmnn equion or Mxwellin molecules Q, in, v () s he viscosiy coeiciens. Here, Q(, ) is he Bolzmnn collision oeror or Mxwellin molecules deined by is qudric orm Q, g g g bcossinddv uncion b is nonnegive nd coninuous, nd bcos sin,. Here he shorhnd, v',, v v', v re he re used; os-collisionl velociies corresonding o he re-collisionl velociies vv, resecively, which submi o he elsic collision lw v' vvv, () v v' v v, (,) denoes he sclr roduc. S he -D uni shere nd ( v' v)/ v' v. is he ngle beween v v nd,,. On hysiclly, Q sisies he symmerizion nd rnslion invrince. This work ws suored by Huzhou Nurl Science Foundion (8YZ6) nd Innovion Tem Foundion o Dermen o Educion o Zhejing Province (T94). For Mxwellin oenil Q cn be sli ino Q _ Q :,,, Q Q Q, cos Q b d dv S,, cos Q b ddv S nd The roblem o viscosiy roximion o he silly homogeneous Bolzmnn equion, nmely wheher he soluion o () converges o he soluion o he equion, in, Q () s, is very ineresed or mhemicl heory o Bolzmnn equion s well s rcicl licions. e know h he energy o he soluion o () is incresing wih he ime due o he diusion eec. e cnno exec h he soluion o () roches o he Mxwellin equilibrium in lrge ime. This observion hs recenly been shown by i-msumur []. In erly work o he uhors n exlici esime o in k ws derived which indices lso he deendence o ime []. I mus be sressed his resul excludes he cse o Mxwellin molecules. Acully, he roduce o momens or cuo oenil is no vlid or Mxwellin molecules. In his er we shll sudy he viscosiy roximion or Mxwellin molecules. Our gol is o Coyrigh Scies.

2 M.. ZHENG 55 sudy he exisence nd uniqueness o he globl soluion o he viscosiy equion () in ime, nd o esime exlicily in C -norm. The new ool is he Gglirdo-Nirenberg inequliy. e us menion some works bou he silly homogeneous Bolzmnn equion wih cuo oenil, see [-] or exmle. For he Mxwellin molecules Morgensern irs deduced he exisence nd uniqueness o he soluion in sce []. e lso remrk h he roximion wih diusion erm in velociy vrible ws resen in he work o DiPern-ions []. Now we comlemen he equion () nd () wih he sme iniil condiion: v v ( ),. (4) In he sequel we lwys ssume h ( v) ( ). (5) I mus be emhsized h he nonnegive hyohesis o () v is no necessry in resen er. m In he ollowing we denoe he C norm by m, nd mx nd mx su D m jm j, j j v Here is he muli-index. This er is orgnized s ollows. e inroduce mild soluion o he Cuchy roblem () nd (4) in Secion. e rove locl exisence o he mild soluion by he conrced ming rincile. In Secion, we roose he globl exisence o he mild soluion. Our min ool is he inerolion inequliies. Finlly, we sudy, he esime o in Secion 4 nd deduce he ollowing symoic exression Ae. The ocl Exisence k. In his secion we shll sudy he locl exisence o he soluion o he Cuchy roblem (), (4). Deiniion. Given. e cll is he mild soluion o he Cuchy roblem () nd (4), i C, ;, nd sisies v, Q, s, dsd, v G, v d G s, v (6) v G G, v ex, / 4 4 The ollowing is he locl exisence heorem., Theorem. Given. e ( ), nd sisy (5). Then here exiss T such h he Cuchy roblem () nd (4) hs unique mild soluion, T. In order o rove Theorem, le us recll well-known resul which is oen clled convoluion roery. Proosiion ([,4]). For ny, i ( ), g ( ) hen here exis consn C deenden on b only, such h, Q g C g The roo o Theorem Consider he ollowing sce,, ; C T T is deermined. Deined he ming : by v,, v (7) G v v G,, s v Q v ds v denoes he convoluion in vrible v. By nd he deiniion o he mild soluion (6), we hve, Q ds Mking use o he Pro. nd Gronwll s lemm, we obin he esime o. In erms o (7), we denoe, v by I nd I. Obviously, I,, By Pro. nd Young s inequliy, noing h G s, one obin I G Q, ds C s Here he nonnegive consn C deends on b only. In wh ollowing, we denoe C or vrious nonnegive consns indeenden o unless secil semens. On he oher hnd,, s I G Q ds C s ds C. (8) Thereore, Coyrigh Scies.

3 56 M.. ZHENG I C,, This ollows he ming is closed. e,, by he bilineriy o Q,,, Q Q Q, Q Q, Q, C C Q, Q, C T T, (9) So, one deduces h he ming is loclly ischiz coninuous. By choosing T suibly, such h T, he Cuchy roblem () nd (4) exiss unique mild soluion.. The Globl Exisence o he ild Soluion In order o rove he globl exisence o he mild soluion bove, i suices o show h,, Firs, le us recll he N dimensionl Gglirdo-Nirenberg s inequliy: le qr,, j, m j re inegers nd j m. Suose h, m, ( i m j N / r is nonnegive ineger). Then here exiss consn C deenden on q, r, j, N m,, N such h or ny u D ( ), j Du C Du u r m j m m r N q q, emm. Given. e ( ), nd sisy (5). Then T,, T, he soluion o Cuchy roblem () nd (4) sisies Proo From (), we hve C T, d sgn dv d dv Q dv II sgn Q, dv sgn, () Obviously I. By Pro. nd Holder s inequliy,, Q dv Q C C () By hese esimes bove nd Gronwll s lemm, we derive C T, This inishes he roo o he lemm., emm 4. Given. e, nd sisy (5). Then T,, T, he soluion o Cuchy roblem () nd (4) sisies,, C T. Proo In erms o (6), or ny Thereore, s v, G G Q, ds., s G G Q ds () By Young s inequliy q, r s G G Q ds (), qr,,. q r re- Nex we esime secively. Noing h z nd Q, G s q,. ze e z G s s v G ( s) z s z e e s C s (4) Thereore q q s q s s G G G C s (5) r Coyrigh Scies.

4 By Gglirdo-Nirenberg s inequliy,,,, Q Q Q r Q, M.. ZHENG (6), r By he rnslion invrince o Q, i is esily o show h,,, Q Q Q So mking us o Pro. gin Q, C. (7) By (), i gives, s G G Q ds h is C ds (8) C T (9), Plugging (9) ino (7), gives, Q C. () By () nd (6), r, Q C C () Combining (5), () nd (), we deduce C s ds According o he emm, one hs () C s ds C s ds. () By Gronwll ye inequliy we obin he desired resul. Nex using he bsic heory o rbolic equion nd he riori esime bove, we hve he ollowing heorem., Theorem 5. Given. e, nd sisy (5). Then or ny T he Cuchy roblem () nd (4) exiss unique mild soluion such h 4., ;, C T C T, Esime nd, C Aroximion 57, In his secion we shll mke esime on nd deduce he exlici esime on he viscosiy roximion. Theorem 6. Given,. Then or ny, sisying (5) he mild soluion o he Cuchy roblem () nd (4) belongs o, C, ;. nd Proo By he equion o (), one hs d d Q,. (4) sgn dv sgn Q, dv I I (5) Nex we esime I nd I resecively. sgn I dv dv (6) I Q dv sgn,, Q dv, Q dv By Young s inequliy, or ny, I dv (7) Q, dv (8) 4 Emloying Young s inequliy gin, he second erm o he bove ormulion cn be esimed by Q, dv dv 4 (9) Tking, nd lugging (8), (9) nd (6) ino (5), i gives 4 Coyrigh Scies.

5 58 M.. ZHENG d dv d 4 Q, dv dv () By () nd Gronwll s lemm, nd he Schuder heory, we conclude he desired resul. Now, we consider wheher he mild soluion o he Cuchy roblem () nd (4) converges o he soluion o () nd (4) in C -norm s. The ollowing heorem is our min resul., Theorem 7. e C ( ) nd sisying (5),. For ny T nd, se is he mild soluion o he viscosiy equion Q, in, T () in nd is he soluion o Q, in, T () in Then, k,, Ae () A nd k re consns indeenden o. Furhermore, or ny i, (4) min log, T k A Proo By he heorem bove nd he resul o silly homogenous Bolzmnn equion, we know h,, v,. e w, hen w Q Thereore, Q w sgn w sgn w Q, Q,. Noing h,,,,,, Q Q Q Q (5) C (6) By he esime on in Secion, we know he esime (6) is uniorm in. Thereore,, Q Q C w (7) Nex, we esime. Indeed, by (6) or ny muli- indices,, we hve, D G D v d G s, D Q, s, v dsd I I (8) G, v dv, I. Mking use o eibniz ormul, or DQ, QD, D (9) noing h, I C ds. (4) Noing h So,, Q ds C dsct (4) By Gronwll s inequliy we cn deduce he bound o nd he bound is indeenden o. Togeher hese esime wih (5) we hve Thereore, w Q, Q, C C w (4) w Ae This inishes he roo o Theorem Acknowledgemens I m greul o he reviewer or he meningul suggesions. 6. eerences [] H.. i nd A. Msumur, Behviour o he Fokker- Plnck-Bolzmnn Equion ner Mxwellin, Archive or ionl Mechnics nd Anlysis, Vol. 89, No., 8, [] M. Zheng nd X. P. Yng, Viscosiy Anlysis on he k Coyrigh Scies.

6 M.. ZHENG 59 Silly Homogeneous Bozmnn Equion, Asymoic Anlysis, Vol. 5, 7,. -8. []. Arkeryd, On he Bolzmnn Equion, Archive or ionl Mechnics nd Anlysis, Vol. 45, 97,. -4. [4]. Arkeryd, Sbiliy in or he Silly Homogeneous Bolzmnn Equion, Archive or ionl Mechnics nd Anlysis, Vol., 988, [5] T. Elmroh, Globl Boundedness o Momens o Soluion o he Bolzmnn Equion or Forces o Ininie nge, Archive or ionl Mechnics nd Anlysis, Vol. 8, 98,. -. [6]. Desvillees, Some Alicions o he Mehod o Momens or he Homogeneous Bolzmnn Equion nd Kc Equion, Archive or ionl Mechnics nd Anlysis, Vol., 99, [7] S. Mischler nd B. ennberg, On he Silly Homogeneous Bolzmnn Equion, Annles de l Insiu Henri Poincr e - Anlyse non lin eire, Vol. 6, No. 4, 999, [8] B. ennberg, Enroy Dissiion nd Momen Producion or he Bolzmnn Equion, Journl o Sisicl Physics, Vol. 86, 997, [9] B. ennberg, Sbiliy nd Exonenil Convergence in or he Silly Homogeneous Bolzmnn Equion, Nonliner Anlysis, Theory, Mehods & Alicions, Vol., 99, [] C. Mouho nd C. Villni, egulriy Theory or he Silly Homogeneous Bolzmnn Equion wih Cuo, Archive or ionl Mechnics nd Anlysis, Vol. 7, No., 4, [] C. Mouho, e o Convergence o Equlibrium or he Silly Homogeneous Bolzmnn Equion wih Hrd Poenils, Communicions in Mhemicl Physics, Vol. 6, 6, [] D. Morgensern, Generl Exisence nd Uniqueness Proo or Sil Homogeneous Soluion o he Mxwell- Bolzm-nn Equion in he Cse o Mxwellin Molecules, Proceedings o he Nionl Acdemy o Sciences, Vol. 4, 954, []. J. DiPern nd P.. ions, On he Fokker-Plnck- Bolzmnn Equion, Communicions in Mhemicl Physics, Vol., No., 988,. -. [4] I. M. Gmb, V. Pnerno nd C. Villni, On he Bolzmnn Equion or Diusively Excied Grnulr Medi, Communicions in Mhemicl Physics, Vol. 46, No., 4, Coyrigh Scies.