Maximal Regularity and Asymptotic Behavior of Solu

Title:MaximalRegularityandAsymptotiBehaviorofSolutionsfortheCahn-HilliardEquationwithDynamiBoundaryConditionsProposedrunninghead:Cahn-HilliardequationAuthors:JanPrussDepartmentofMathematisandComputerSieneMartin-Luther-University60120HalleGermanypruessmathematik.uni-halle.deReinhardRakeDepartmentofMathematisandStatistisUniversityofKonstanz78457KonstanzGermanyreinhard.rakeuni-konstanz.deSongmuZhengInstituteofMathematisFudanUniversity200433ShanghaiP.R.Chinaszhengfudan.a.nMailingaddress:Prof.Dr.ReinhardRakeDepartmentofMathematisandStatistisUniversityofKonstanz78457KonstanzGermanyVersionofMay27,2003MaximalRegularityandAsymptotiBehaviorofSolutionsfortheCahn-HilliardEquationwithDynamiBoundaryConditionsJanPrussReinhardRakeSongmuZhengDepartmentofMathematisDepartmentofMathematisandComputerSieneandStatistisMartin-Luther-UniversityUniversityofKonstanz60120Halle78457KonstanzGermanyGermanypruessmathematik.uni-halle.dereinhard.rakeuni-konstanz.deInstituteofMathematisFudanUniversityShanghai200433P.R.Chinaszhengfudan.a.nAbstratThispaperdealswiththeCahn-Hilliardequationt=;=+3;(t;x)2J;subjettotheboundaryonditions1st=sjjgs+h;=0;andtheinitialondition(0;x)=0(x)whereJ=(0;1),andRnisaboundeddomainwithsmoothboundary=G,n3,ands;s;gs0;hareonstants.ThisproblemhasalreadybeenonsideredinthereentpaperofRakeandZheng[13℄whereglobalexisteneanduniquenesswereobtained.InthispaperwerstobtaintheresultsonmaximalLp-regularityofsolutionandprovethatthesolutiondenesaC0-semigroupinertainSobolevspaes.Wethenstudytheasymptotibehaviorofthesolutionsofthisproblemandprovetheexisteneofglobalattrators.11IntrodutionLetRnbeaboundeddomainwithboundary=oflassC4,andonsiderthefollowingboundaryvalueproblemfortheCahn-Hilliardequation.t=;=+3;t0;x2;(1.1)=0;t0;x2(1.2)tsjju++gs=h;t0;x2(1.3)=0t=0;x2:(1.4)Here(x)denotestheouternormalofatx2,jjmeanstheLaplae-Beltramioperatoron,ands;s;gs0;hareonstants.Thisproblemarisesfromthestudyofspinodaldeompositionofbinarymixturesthatappears,forexamples,inoolingproessesofalloys,glassesorpolymermixtures(seeCahnandHilliard[2℄,Novik-CohenandSegal[10℄,Kenzleretal.[8℄,andthereferenesitedtherein.)Boundaryondition(1.3)isusuallyalledthedynamiboundaryonditionsineitalsoinvolvesthetimederivativeofdependentfuntion.Itisderivedwhentheeetiveinterationbetweenthewall(i.e.,theboundary)andtwomixtureomponentsareshort-ranged(seeKenzleretal.[8℄).ThisproblemwasreentlystudiedbyRakeandZheng[13℄andtheglobalexisteneanduniquenessofsolutionwasprovedthere.Furthermore,itwaspointedoutthatfort0thesolutionisC1.However,itwasnotlearwhetherthesolutiondenesaC0-semigroupintheSobolevspaeVintroduedinthatpaper.Inthepresentpaperwefurtherinvestigatethisproblem.Morepreisely,weprovemaximalLp-regularityofsolutionwhihimpliesthatthesolutiondenesaC0-semigroupinertainSobolevspaes.Furthermore,weprovetheexisteneofaglobalattratorforthisproblem.Thispaperisorganizedasfollows.InSetion2,werststudyalinearproblemassoiatedwithouroriginalproblem(1.1){(1.4),andweestablishmaximalLp-regularityresults.Thenweonsidertheorrespondingnonlinearproblem(1.1){(1.4),andproveinSetions3and4loalwell-posednessaswellasglobalwell-posednessinadierentphasemanifoldMthanV,whihturnsoutthatthesolutiondenesaC0-semigroupinM.InthenalsetionweprovetheexisteneofaglobalattratorinM2(forp=2)aswellasinV.2TheLinearProblemInthissetionwestudythefollowinglinearizedversionof(1.1).tv+2v=f;t0;x2;v=g;t0;x2(2.1)1stvsjjv+v+gsv=h;t0;x2v=v0t=0;x2:Herethefuntionsf,g,haswellastheinitialvaluev0aregiven;s;0,andgs0aregivenonstants(=1intheoriginalsystem).2LetJ=[0;T℄and1p1.Wearelookingforsolutionsinthelassv2H1p(J;Lp())\Lp(J;H4p());whihisthenaturallassfor(2.1)intheLp-setting.Thenbywell-knowntraetheorems(f.[9℄,[1℄,[4℄)thedataf,g,v0neessarilysatisfyf2Lp(J);g2W1=41=4pp(J;Lp())\Lp(J;W11=pp());v02W44=pp():Asusual,hereandinthesequelWspdenotethefrationalSobolevspaes.Furthermore,thetraesofvandvonsatisfyvj2W11=4pp(J;Lp())\Lp(J;W41=pp());andvj2W3=41=4pp(J;Lp())\Lp(J;W31=pp()):Thisleavessomehoieforthesettingofthedynamiboundaryondition.Thepossibilityoflowestorderisthehoieh2Lp(J;W21=pp()).LookingatthedynamialboundaryonditionasaheatequationonJ,thiswillresultinvj2H1p(J;W21=pp())\Lp(J;W41=pp()):Thisregularityimpliesthatthetraeofvjatt=0neessarilysatisesv0j2W43=pp().Theotherextremepossibilityonsistsintakingtheregularityofthenormalderivativeofvasthebasiregularity,i.e.wemayonsiderthelassh2W3=41=4pp(J;Lp())\Lp(J;W31=pp()):Thisleadstovj2W7=41=4pp(J;Lp())\H1p(J;W31=pp())\Lp(J;W51=pp()):Thenneessarilyv0j2W53=pp()andtheompatibilityonditionsv0j=gjt=0;forp5;andsjjv0jv0jgsv0j+hjt=02W35=pp();forp5=3;musthold.Moregenerally,anyhoieofthespaeforhofthetypeh2Wsp(J;Lp())\Lp(J;Wrp());0s3=41=4p;21=pr31=p;willwork.Observethatforsuhr;stheinequality2srisvalid.Theorrespondingtraespaeforv0jnowbeomesWr+22=pp(),andinases1=palsothetimederivativetvjhastraeatt=0,whihbelongstoWr(11=sp)p().Hereisthemainresultonmaxim