a new approach to the regularity of solutions for parabolic equationsnew

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1、AEWARACHTTHEREGUAR
TYFSUT
SFRARAB
CEUAT
SACH
ESCHER,ARUSS,ADG
ER
S
ETTDedi atedtoerryGoldsteinontheoasionofhis60thbirthdayAbstra t.
nthisnotewedes ribeanewapproa htoestablishregular-itypropertiesforsolutionsofparaboli equations.
tisbasedon

2、maximalregularityandtheimpli itfun tiontheorem.1.
ntrodu tion
nthisnotewedes ribeanewapproa htoestablishregularitypropertiesforawidearrayofparaboli evolutionequations.
tisbasedonthetheoryofmaximalregularity.Thethrustofthisapproa hismanifold.
tallowstosolveagivenpart

3、ialdierentialequationwithoutlossofderiva-tives,thuspermittingtohandlefullynonlinearequations.
tallowstoresorttotheimpli itfun tiontheoremtostudyfurtherprop-ertiesofsolutions,su hassmoothdependen eongivendata.
tallowstostudytheregularityofsolutionsbymerelyapplyings

4、 alingargumentsin onjun tionwiththeimpli itfun tiontheorem.
nordertoexplainthemainideaofourapproa h,letus onsiderthemodelnproblemofafamilyofgraphsf(t)=graph(u(
;t));0tTgoverR,evolvinga ordingtothemean urvature owuuijuÆu=0;u(0)=u;(1.1)tijij021+jrujwhere1

5、i;jn,andwhereÆdenotestherone kerdelta.Equation(1.1)isijaquasilinearparaboli evolutionequationofse ondorder.Toe onomizenotationwesetuuijF(u):=Æuijij21+jrujandrestateequation(1.1)asu+F(u)=0;u(0)=u:(1.2)t02j+snetE:=bu (R),j=0;1,bethelittleHolderspa esdenedi

6、n(2.8).ThejmappingFisrealanalyti ,thatis,!F2C(E;E):(1.3)1012.ESCHER,.RUSS,ADG.S
ETTGiventhatFisdierentiable,one an onsiderthelinearizedproblem0v+F(u)v=f;v(0)=v;(1.4)t00whereF(u)istheFre hetderivativeofFatu2E.extweintrodu ethe1anisotropi spa es1E(
):=C(
;E)

7、;E(
):=C(
;E)C(
;E);00101where
=[0;T℄isaxedinterval.Clearly,thetra eoperator:E(
)!E,011v7!v(0)islinearand ontinuous.
t anbeshown,andthisistheessentialpartoftheanalysis,thatthelinearproblem(1.4)enjoysthepropertyofmaximalregularity.Bydenition,thismeansthat0(+F(u);)

8、2
som(E(
);E(
)E)(1.5)t01010foranyfun tionu2E.Thatis,thelinearmapping(+F(u);)isatopologi al1t0isomorphismbetweentheindi atedspa e