A note on local $W^{1,p}$-regularity estimates for weak solutions of parabolic equations with singular divergence-free drifts

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by  Tuoc Phan

A NOTE ON LOCAL W1,p-REGULARITY ESTIMATES FOR WEAK SOLUTIONS OF
PARABOLIC EQUATIONS WITH SINGULAR DIVERGENCE-FREE DRIFTS
TUOCPHAN
7 Abstract. WeinvestigateweightedSobolevregularityofweaksolutionsofnon-homogeneousparabolicequa-
1 tionswithsingulardivergence-freedrifts. Assumingthatthedriftssatisfysomemildregularityconditions,we
0 establishlocalweightedLp-estimatesforthegradientsofweaksolutions.Ourresultsimprovetheclassicalone
2 totheborderlinecasebyreplacingthe L -assumptiononsolutionsbysolutionsintheJohn-NirenbergBMO
∞

n space. Theresultsarealsogeneralizedtoparabolicequationsindivergenceformwithsmalloscillationelliptic
a symmetriccoefficientsandthereforeimprovemanyknownresults.
J

2
1. Introductionandmainresults

]
P WeinvestigatelocalweightedLp-estimatesforthegradientsofweaksolutionsofparabolicequationswith
A
lowregularity ofthedivergence-free drifts. Atypical exampleistheparabolic equation
.
h
(1.1) u ∆u b u = 0, Rn (0, ),
t t
a − − ·∇ × ∞
m wherethedriftb : Rn (0, ) Rn isofdivergence-free, i.e. div(b(,t)) = 0indistributional sensefora.e.
× ∞ → ·
[ t. Due to its relevance in many applications such as in fluid dynamics, and biology, the equation (1.1) has

been investigated by many mathematicians (for example [15, 16, 28, 33]). Local boundedness, Harnack’s
1
v inequality, andHo¨lder’sregularity areestablished in[15,24,28,31,33]withpossible singulardrifts. Many
7 otherclassicalresultswithregulardriftscanbefoundin[14,17,18,19]. Ho¨lder’sregularityforthefractional
1
Laplacetypeequations oftheform(1.1)areextensively studiedrecently (see[7,13,29]).
4
Unlike the mentioned work, this note investigates the Sobolev regularity of weak solutions of (1.1) in
0
0 weighted spaces. Our goal is to establish local weighted estimates of Caldero´n-Zygmund type for weak
1. solutions of (1.1) with some mild requirements on the regularity of the drifts b. We study the following
0 parabolic equationthatismoregeneral than(1.1):
7
1 (1.2) u div[a(x,t) u] b u = div(F),
t
: − ∇ − ·∇
v
where a = (aij)n is a given symmetric n n matrix of bounded measurable functions, and F,b are given
Xi i,j=1 ×
vectorfieldswithdiv(b) = 0indistribution sense. Theexactrequired regularity conditions ofa,b,F willbe
r specified.
a
To state our results, we introduce some notation. For each r > 0, and z = (x ,t ) Rn R, we denote
0 0 0
Q(z )theparabolic cylinder inRn+1 ∈ ×
0
Q (z )= B (x ) Γ (t ), where Γ (t )= (t r2,t +r2), and B (x ) = x Rn : x x < r .
r ) r 0 r 0 r 0 0 0 r 0 0
× − { ∈ k − k }
Whenz = (0,0),wealsowrite
0
Q = Q (0,0), for 0 < r < .
r r
∞
Asweareinterested inthelocalregularity, wereduceourstudytotheequation
(1.3) u div[a(x,t) u] b u = div(F), in Q ,
t 2
− ∇ − ·∇
forgiven
a: Q : Rn n, b,F : Q Rn,
2 × 2
→ →
and
(1.4) div(b(,t)) = 0, indistribution sensein B , fora.e. t Γ .
2 2
· ∈
1
2 TUOCPHAN
Forthecoefficient matrixa,weassumethat
a = (aki)n : Q Rn n issymmetric,measurable
k,i=1 2 → ×
(1.5)  andthereexistsΛsuchthat
Λ 1ξ2 a(x,t)ξ,ξ Λξ2, fora.e. (x,t) Q , and ξ Rn.
− 2
Wealsorequirethatth|e|m≤athrixahasais≤mall| o|scillation. Therefore, w∈eneedthefol∀low∈ingdefinition.
Definition1.1. Leta : Q Rn n beameasurablematrixvaluedfunction. ForgivenR > 0,wedefine
2 ×
→
1
[a] = sup sup a(x,t) a¯ (t)2dxdt,
BMO(Q1) 0<ρ≤1(y,s)∈Q1 |Qρ(y,s)| ZQρ(y,s)| − Bρ(y) |
wherea¯ (t) = a(x,t)dxistheaverage ofainthesetU B .
U U ⊂ 2
>
For the regularity of the vector field b, we need the following function space, which was introduced in
[20,25]
Definition1.2. For x Rn and r > 0, alocally square integrable function f defined inaneighborhood of
0
∈
B (x )issaidtobein 1,2(B (x ))ifthereisk [0, )suchthat
r 0 r 0
V ∈ ∞
(1.6) f(x)2ϕ(x)2dx k ϕ(x)2dx, ϕ C (B (x )).
Z | | ≤ Z |∇ | ∀ ∈ 0∞ r 0
Br(x0) Br(x0)
Wedenote
b 2 = inf k [0, )suchthat(1.6)holds .
k k 1,2(Br(x0)) { ∈ ∞ }
V
Inthispaper, thenumbers s,s (1, ),α > 0andλarefixedandsatisfying
′
∈ ∞
1 1
(1.7) + = 1, (n+2) λ s , α = λ(s 1).
′
s s − ≤ ≤ −
′
Wealsodenote Lp(Q,ω)theweightedLebesguespacewithweightω:
1/p
Lp(Q,ω) = f : Q R : f := f(x,t)pω(x,t)dxdt < , 1 < p < .
Lp(Q,ω)
→ k k Z | | ! ∞ ∞
n Q o
Atthismoment,wereferthereaders toSection2forthedefinition ofweaksolutions of (1.3),thedefinition
of of Muckenhoupt A weights, and the definition of fractional Hardy-Littlewood maximal functions .
q α
M
Ourmain result isthe following theorem onlocal weighted W1,p-regularity estimates for weaksolutions of
(1.3).
Theorem 1.3. LetΛ,M bepositive numbers, p (2, ), andω A . Let s,s ,λ,αbeasin(1.7). Then,
0 p/2 ′
∈ ∞ ∈
there exists a sufficiently small number δ = δ(Λ,M ,s,λ,[ω] ,p,n) > 0 such that the following holds:
0 Ap/2
Supposethatasatisfies(1.5), F L2(Q ),andb L (Γ , 1,2(B ))suchthat (1.4)holds,and
2 ∞ 2 2
∈ ∈ V
[[a]] < δ, b M .
BMO(Q1) k kL∞(Γ2,V1,2(B2)) ≤ 0
Thenforeveryweaksolutionuof (1.3),thefollowingestimateholds
(1.8) ZQ1|∇u|pω(z)dz ≤C(cid:20)kFkLpp(Q2,ω)+[u]sp′,λ,Q1(cid:13)Mα,Q2(|b|s)2/s(cid:13)Lp/p2/2(Q2,ω)+ω(Q1)k∇ukLp2(Q2)(cid:21),
(cid:13) (cid:13)
(cid:13) (cid:13)
aslongasitsrighthandsideisfinite. Here,[u] istheparabolic semi-Campanato’s normofuon Q ,
s′,λ,Q1 1
1/s
′
[u]s′,λ,Q1 = sup ρ−λ? |u(x,t)−u¯Qρ(z)|s′dxdt ,
0<ρ<1,z∈Q1 Qρ(z) 
andC > 0isaconstant depending onlyonΛ,M ,s,λ,p,nand[ω] .
0 Ap/2
LOCALGRADIENTESTIMATES,EQUATIONSWITHDIVERGENCE-FREEDRIFTS 3
We now point out a few remarks regarding Theorem 1.3. Firstly, observe that the standard Caldero´n-
Zygmundtheorycanbeapplied directlyto(1.3)toobtain
u C u b + ,
k∇ kLp(Q1) ≤ k kL∞(Q2)k kLp(Q2) ···
h i
aslongasu L (Q ). Theorem1.3improvesthisCaldero´n-Zygmundestimatetheoryfortheequation(1.3)
∞ 2
∈
totheborderlinecase,replacingtheassumptionu L (Q )byu BMO(Q ). Indeed,ifwetakeλ = 0(and
∞ 2 1
∈ ∈
thenα = 0),thentheestimate(1.8)reducesto
(1.9) u C u b + .
k∇ kLp(Q1,ω) ≤ k kBMO(Q1)k kLp(Q2,ω) ···
h i
Secondly, the weighted W1,p-regularity estimates are useful in some applications. For example, in [2, 3],
theweighted W1,p-regularity estimates arekeyingredients forproving theexistence anduniqueness ofvery
weaksolutionsofsomeclassesofellipticequations. Moreover,withsomespecificchoiceofω,theweighted
estimate (1.8) is known to produce the regularity estimates for u in Morrey spaces, see for example [1, 4,
∇
10,21]. Lastly,whenα > 0,because I ,theRieszpotential oforderα,weobservethatthefractional
α α
M ≤
Hardy-Littlewoodmaximalfunctionoforderαofb,i.e. (bs)2/s,ismoreregularthanb. Thisfactenables
α
M | |
theestimate (1.8)tobeuseful insomeapplications. Tosee this, wejust simply consider thestationary case
(i.e. u is time independent), n 3 and s = 2. Assume, for example, that b Ln, (B ) 1,2(B ), where
∞ 2 2
≥ ∈ ⊂ V
Ln, istheweak Ln-space, and assume also that F isregular enough. Then, itisproved in[28,33]that uis
∞
Ho¨lder. Therefore, [u] < withsomeλ > 0. Fromthis, and(1.7),weseethatα > 0,andwethencan
2,λ,B1 ∞
findsomesmallconstant ε > 0suchthat
0
(b) C b < , forall p < n+ε .
Mα,B2 | | Lp(B2) ≤ k kLn,∞(B2) ∞ 0
(cid:13) (cid:13)
(cid:13) (cid:13)
Therefore,(1.8)givesth(cid:13)eestimate(cid:13)of u withsome p [2,n+ε ). Thisestimatewith p > nisuseful
k∇ kLp(B1) ∈ 0
in[12]toprovetheregularity,anduniquenessofveryweakW1,q-solutionofthestationaryequationof (1.3),
with1 < q< 2. Detailsofthisdiscussion anditsapplication canbealsofoundin[26].
Wefinallywouldliketopointoutthatthespace 1,2(Rn)isalreadyappearedin[20,25,33]. Inparticular,
in[33],thespace L ( 1,2(Rn))isusedtostudytheVboundedness ofweaksolutionoftheequation(1.1). For
∞t V
n 3,thespace 1,2(Rn)isalreadyappeared in[20,25]. Moreover, itisknownthat(see[25])
≥ V
(1.10) Ln(Rn) Mp,p(Rn) 1,2(Rn) 2< p n,
⊂ ⊂ V ∀ ≤
andtherefore
L (Ln(Rn)) L (Mp,p(Rn)) L ( 1,2(Rn)),
∞t ⊂ ∞t ⊂ ∞t V
whereMp,p(Rn)denotes the homogeneous Morrey space. Specifically, for0 < p nand0 < λ < p,the
function f Lp (Rn)belongs tothespaceMp,λ(Rn)if ≤
∈ loc
1
p
f = sup rλ n f(x)p < .
k kMp,λ(Rn) Br(x0)⊂Rn( − ZBr(x0)| | ) ∞
Weuseperturbationapproachintroducedin[6]toproveTheorem1.3. Ourapproachisalsoinfluencedby
[5,11,21,23,32]. Toimplementtheapproach,weintroducethefunction B(x,t)= [u] b(x,t) s,which
s′,λ,Q1| |
isinvariantunderthestandarddilation,andtranslation. Thisfunctionalsocapturesthecancellationduetothe
(cid:0) (cid:1)
divergence-free ofthevectorfieldb,whichisthemainreasonsothattheestimate(1.9)holdstheborderline
case. TheresultsonthedoublingpropertyandreverseHo¨lder’sinequalityfortheMuckenhouptweightsdue
toR.R.Coifman,andC.Feffermanin[8]arealsousedfrequently toderivetheweightedestimates.
We conclude the section by introducing the organization of the paper. Section 2 gives definitions, nota-
tions, andsomepreliminaries results needed inthepaper. Somesimpleenergy estimates forweaksolutions
of (1.3) is given in Section 3. The main step in the perturbation technique, the approximation estimates, is
carriedoutinSection4. Section5isabouttheproofofTheorem1.3.
4 TUOCPHAN
2. Definitionsofweaksolutions, andpreliminariesonweightedinequalities
2.1. Definitionsofweaksolutions. Foreachz = (x ,t ) Rn R,andforanyparabolic cylinder Q (z ),
0 0 0 R 0
∈ ×
wedenote∂ Q (z )theparabolic boundary ofQ (z ),i.e.
p R 0 R 0
∂ Q (z )= (B (x ) t R2 ) (∂B (x ) [t R2,t +R2]).
p R 0 R 0 0 R 0 0 0
×{ − } ∪ × −
Thefollowingstandarddefinitions ofweaksolutions arealsorecalled.
Definition 2.1. Let Q be a parabolic cube. For every f L2(Q ),F,b L2(Q )n, wesay that u is a weak
r r r
∈ ∈
solutionof
u div[a u] b u = div(F)+ f, in Q ,
t r
− ∇ − ·∇
ifu L2(Γ ,H1(B )),u L2(Γ ,H 1(B )),and
r r t r − r
∈ ∈
u,ϕ dt+ a u, ϕ b uϕ dxdt = [fϕ F, ϕ ]dxdt,
ZΓrh t iH−1(Br),H10(Br)) ZQrhh ∇ ∇ i− ·∇ i ZQr −h ∇ i
forallϕ φ C (Q ) :φ = 0on∂ Q .
∞ r p r
∈ { ∈ }
Thefollowingdefinitionofweaksolution isalsoneededinthepaper.
Definition2.2. LetQ beaparaboliccube. Forevery f L2(Q ),F,b L2(Q )n,andforg L2(Γ ,H1(B )),
r r r r r
∈ ∈ ∈
wesaythatuisaweaksolutionof
u div[a u] b u = div(F)+ f, in Q ,
t r
− ∇ − ·∇
( u = g, on ∂pQr,
ifuisaweaksolution of
u div[a u] b u = div(F)+ f, in Q ,
t r
− ∇ − ·∇
inthesenseof Definition2.1andu g φ L2(Γ ,H1(B ) :φ = 0on∂ Q .
r r p r
− ∈ { ∈ }
2.2. Munckenhoupt weights and Hardy-Littlewood maximal functions. For each 1 q < , a non-
negative, locally integrable function µ : Rn+1 [0, )issaidtobeintheclassofparabol≤ic A of∞Mucken-
q
→ ∞
houptweightsif
q 1
1 −
[µ]Aq := sup ? µ(x,t)dxdt! ? µ(x,t)1−qdxdt! < ∞, if q > 1,
r>0,z Rn+1 Qr(z) Qr(z)
∈
[µ] := sup µ(x,t)dxdt µ 1 < if q = 1.
A1 r>0,z Rn+1 ?Qr(z) !(cid:13) − (cid:13)L∞(Qr(z)) ∞
∈ (cid:13) (cid:13)
(cid:13) (cid:13)
ItiswellknownthattheclassofA -weightssatisfiesthereverseHo¨lder’sinequality andthedoubling prop-
p
erties, see for example [8, 9, 30]. In particular, a measure with an A -weight density is, in some sense,
p
comparablewiththeLebesguemeasure.
Lemma2.3([8]). For1 < q< ,thefollowing statementsholdtrue
∞
(i) If µ A , then for every parabolic cube Q Rn+1 and every measurable set E Q, µ(Q)
q
[µ]Aq∈|QE| pµ(E). ⊂ ⊂ ≤
(ii) If µ (cid:16)A| |(cid:17), then there isC = C([µ] ,n) and β = β([µ] ,n) > 0 such that µ(E) C E βµ(Q), for
∈ q Aq Aq ≤ |Q|
everyparabolic cube Q Rn+1 andeverymeasurable set E Q. (cid:16)| |(cid:17)
⊂ ⊂
Let us also recall the definition of the parabolic fractional Hardy-Littlewood maximal operators which will
beneededinthepaper
Definition 2.4. Let α R, the parabolic Hardy-Littlewood fractional maximal function of order α of a
locallyintegrable functi∈on f onRn isdefinedby
( f)(x,t) = supρα f(y,s) dyds.
Mα ? | |
ρ>0 Qρ(x,t)
LOCALGRADIENTESTIMATES,EQUATIONSWITHDIVERGENCE-FREEDRIFTS 5
If f isdefinedinaregionU Rn R,thenwedenote
⊂ ×
f = (χ f).
α,U α U
M M
Moreover,whenα = 0,wewrite
f = f, f = f.
0 U 0,U
M M M M
The following boundedness of the Hardy-Littlewood maximal operator is due to Muckenhout [22]. For
theproofofthislemma,onecanfinditin[9,30].
Lemma2.5. Assumethatµ A forsome1 < q< . Then,thefollowings hold.
q
∈ ∞
(i) Strong(q,q): ThereexistsaconstantC =C([µ] ,n,q)suchthat
Aq
C.
Lq(Rn+1,µ) Lq(Rn+1,µ)
kMk → ≤
(ii) Weak(1,1): ThereexistsaconstantC =C(n)suchthatforanyλ> 0,wehave
C
(x,t) Rn+1 : (f) > λ f(x,t)dxdt.
(cid:12)(cid:12)(cid:8) ∈ M (cid:9)(cid:12)(cid:12) ≤ λ ZRn+1| |
(cid:12) (cid:12)
2.3. Some useful measure theory lemmas. Wecollect some results needed in the paper. Our first lemma
isthestandardresultininmeasuretheory.
Lemma 2.6. Assume that g 0 is a measurable function in a bounded subset U Rn+1. Let θ > 0 and
̟ > 1begivenconstants. Ifµ≥isaweightinL1 (Rn+1),thenforany1 p< ⊂
loc ≤ ∞
g Lp(U,µ) S := ̟pjµ( x U : g(x) > θ̟j )< .
∈ ⇔ { ∈ } ∞
Xj 1
≥
Moreover,thereexistsaconstantC > 0suchthat
C 1S g p C(µ(U)+S),
− ≤ k kLp(U,µ) ≤
whereC depends onlyonθ,̟and p.
Thefollowing lemmaiscommonly used, anditisaconsequence oftheVitali’s covering lemma. Theproof
ofthislemmacanbefoundin[21,Lemma3.8].
Lemma 2.7. Let µ be an A weight for some q (1, ) be a fixed number. Assume that E K Q are
q 1
∈ ∞ ⊂ ⊂
measurablesetsforwhichthereexistsǫ,ρ (0,1/4)suchthat
0
∈
(i) µ(E) < ǫµ(Q (z))forallz Q ,and
1 1
∈
(ii) forallz Q andρ (0,ρ ],ifµ(E Q (z)) ǫµ(Q (z)),then Q (z) Q K.
1 0 ρ ρ ρ 1
∈ ∈ ∩ ≥ ∩ ⊂
Thenwithε = ε(20)nq[µ]2 sothatthefollowingestimateholds
1 Aq
µ(E) ǫ µ(K).
1
≤
3. Caccioppoli’stypeestimates
Suppose thatasatisfies (1.5),andb L (Γ , 1,2(B ))n L2(Q )n withdiv(b) = 0. Inthis section, letu
∞ 2 2 2
∈ V ∩
beaweaksolutionof
u div[a(x,t) u] b(x,t) u = div(F), in Q .
t 2
− ∇ − ·∇
Also,letvbeaweaksolutionof
v div[a¯ (t) v] = 0, Q ,
t − B7/4 ∇ 7/4
( v = u, ∂pQ7/4.
The meanings for weak solutions of these equations are given in Definition 2.1 and Definition 2.2, respec-
tively. Wewillderivesomefundamental estimatesforuandv.
6 TUOCPHAN
Lemma3.1. Letw = u v,thenthereexistsaconstantC depending ononlyΛ,nsuchthat
−
sup w2(x,t)dx+ w2dxdt
t∈Γ7/4ZB7/4 ZQ7/4|∇ |
C b 2 +1 u2dxdt+ F2dxdt .
≤ "(cid:16)k kL∞(Γ2,V1,2(B2)) (cid:17)ZQ7/4|∇ | ZQ7/4| | #
Proof. Notethatwisaweaksolution of
w div[a¯ (t) w+(a a¯ (t)) u] b u = div(F), in Q ,
t − B7/4 ∇ − B7/4 ∇ − ·∇ 7/4
( w = 0, on ∂pQ7/4.
Multiplying theequationwithw,andusingtheintegration bypartsin x,weseethat
1 d
w2(x,t)dx+ a¯ (t) w, w dx
2dt Z Z h B7/4 ∇ ∇ i
B7/4 B7/4
= (a a¯ (t)) u, w dx+ [b u]wdx F wdx.
−Z h − B7/4 ∇ ∇ i Z ·∇ −Z ·∇
B7/4 B7/4 B7/4
Then,byintegrating thisequalityintimeandusingtheellipticity condition (1.5),weobtain
1
sup w2dx+Λ 1 w2dxdt
−
(3.1) 2 Γ7/4 ZB7/4 ZQ7/4|∇ |
(a a¯ (t)) u, w dxdt + b uwdxdt+ F w dxdt.
≤ Z |h − B7/4 ∇ ∇ i| Z | ·∇ | Z | ·∇ |
Q7/4 Q7/4 Q7/4h i
Wenow estimate terms by terms ofthe right hand side of (3.1). FromHo¨lder’s inequality, and the Young’s
inequality,andthefactthatw = 0on∂ Q ,thesecondtermintherighthandsideof (3.1)canbeestimated
p 7/4
as
1/2 1/2
b w udxdt b2w2dxdt u2dxdt
Z | || ||∇ | ≤ (Z | | ) (Z |∇ | )
Q7/4 Q7/4 Q7/4
1/2 1/2
(3.2) b w2dxdt u2dxdt
≤ k kL∞(Γ2,V1,2(B2))(ZQ7/4|∇ | ) (ZQ7/4|∇ | )
Λ 1
− w2dxdt+C(Λ) b 2 u2dxdt.
≤ 6 ZQ7/4|∇ | k kL∞(Γ2,V1,2(B2))ZQ7/4|∇ |
Ontheotherhand,bytheboundedness ofain(1.5),andtheHo¨lder’sinequality, weconclude that
Λ 1
(a a¯ ) u, w dxdt C(Λ) u2dxdt+ − w2dxdt, and
Z |h − B7/4 ∇ ∇ i| ≤ Z |∇ | 6 Z |∇ |
Q7/4 Q7/4 Q7/4
Λ 1
F wdxdt C(Λ) F2dxdt+ − w2dxdt.
Z | ·∇ | ≤ Z | | 6 Z |∇ |
Q7/4 Q7/4 Q7/4
Collectingalloftheestimates, weobtainfrom(3.1)that
1
sup w2(x,t)dx+Λ 1 w2dxdt
−
2 Γ7/4 ZB7/4 ZQ7/4|∇ |
Λ 1
− w2dxdt+C b 2 +1 u2dxdt+ F2dx .
≤ 2 ZQ7/4|∇ | hk kL∞(Γ2,V1,2(B2)) iZQ7/4|∇ | ZQ7/4| | !
Therefore,
sup w2(x,t)dx+ w2dxdt
t∈Γ7/4ZB7/4 ZQ7/4|∇ |
C(Λ) b 2 +1 u2dxdt+ F2dxdt .
≤ hk kL∞(Γ2,V1,2(B2)) iZQ7/4|∇ | ZQ7/4| | !
LOCALGRADIENTESTIMATES,EQUATIONSWITHDIVERGENCE-FREEDRIFTS 7
Theproofiscomplete. (cid:3)
Thefollowingversionoflocalenergyestimateforw = u visalsoneeded.
−
Lemma3.2. Thereexists aconstant C = C depending only onΛ,nsuch that forw = u v, and for every
0
−
smooth,non-negative cut-offfunction ϕ C (Q )with0 < r 7/4,thereholds
∈ 0∞ r ≤
sup w2ϕ2dx+ w2ϕ2dxdt
t∈Γr ZBr ZQr|∇ |
C b 2 +1 w2 ϕ2+ ∂ϕ2+ ϕ2]dxdt+ F2ϕ2dxdt
≤ 0(hk kL∞(Γ2,V1,2(B2)) iZQr h | t | |∇ | ZQr| |
1/2
+ vϕ b ϕ w2ϕ2dxdt + vϕ 2 a a¯ (t)2dxdt .
k∇ kL∞(Q7/4)k kL∞(Γ2,V(B2))k∇ kL2(Q7/4)(ZQr| | ) k|∇ | kL∞(Qr)ZQr| − B7/4 | 

Proof. Wewrite Q = Q ,B= B ,andΓ = Γ . Notethatwisaweaksolution of
r r r
w div[a w+(a a¯ ) v] b w b v = div(F), in Q .
t− ∇ − B7/4 ∇ − ·∇ − ·∇ 7/4
Byusingwϕ2 asatestfunction oftheequation ofw,weobtain
1 d
w2(x,t)ϕ2(x,t)dx+ a w, w ϕ2dx
2dt Z Z h ∇ ∇ i
B B
= a w, (ϕ2) wdx (a a¯ (t)) v,ϕ2 w+2wϕ ϕ dx
−Z h ∇ ∇ i −Z h − B7/4 ∇ ∇ ∇ i
(3.3) B B
+ [b w]wϕ2dx+ [b v]wϕ2dx
Z ·∇ Z ·∇
B B
F, (wϕ2) + w2ϕϕdx.
t
−Z h ∇ i Z
B B
Noteagainthatthesecondterminthelefthandsideof (3.3)canbeestimatedusing(1.5)as
a w, w ϕ2dxdt Λ 1 w2ϕ2dxdt.
−
Z h ∇ ∇ i ≥ Z |∇ |
Q Q
Also,fromtheintegration bypartsin x,anddiv(b) = 0,wealsohave
1
[b w]wϕ2dx = [b (w2)]ϕ2dx = [b ϕ]ϕw2dx.
Z ·∇ 2Z ·∇ −Z ·∇
B B B
Hence,(3.3)implies
1 d
w2(x,t)ϕ2(x,t)dx+Λ 1 w2ϕ2dx
−
2dt Z Z |∇ |
B B
a w, (ϕ2) wdx+ (a a¯ (t)) v,ϕ2 w+2wϕ ϕ dx
≤ Z |h ∇ ∇ i | Z |h − B7/4 ∇ ∇ ∇ i|
B B
+ [b ϕ]w2ϕ2dx+ [b v]wϕ2dx
Z | ·∇ | Z | ·∇ |
B B
+ F, (wϕ2) +2w2ϕϕ dx.
t
Z |h ∇ i| | |
Bh i
8 TUOCPHAN
Byintegrating thisinequality intime,andusingthe L -boundofafrom(1.5),weinferthat
∞
1
sup w2(x,t)ϕ2(x,t)dx+Λ 1 w2ϕ2dxdt
−
2 t Γr ZB ZQ|∇ |
∈
2 w ϕ ϕwdxdt+ a a¯ v ϕ2 w +2wϕ ϕ dxdt
≤ Z |∇ ||∇ || | Z | − B7/4||∇ | |∇ | | | ||∇ |
(3.4) Q Q h i
+ b ϕw2ϕ2dxdt+ b v wϕ2dxdt
Z | ||∇ | Z | ||∇ || |
Q Q
+ F, (wϕ2) +2w2ϕϕ dxdt.
t
Z |h ∇ i| | |
Qh i
Wenowpayparticularattentiontothetermsintherighthandsideof (3.4)involving b,asothertermscanbe
estimatedexactlyasinLemma3.1. ByusingtheHo¨lder’sinequality andYoung’sinequality, weseethat
1/2 1
2
w2ϕb ϕdxdt b2w2ϕ2 w2 ϕ2dxdt
Z | ||∇ | ≤ (Z | | ) (Z |∇ | )
Q Q Q
1/2 1
2
b (wϕ)2dxdt w2 ϕ2dxdt
≤ k kL∞(Γ7/4 V1,2(B7/4))(ZQ|∇ | ) (ZQ |∇ | )
ǫ w2ϕ2dxdt+C(ǫ) b 2 w2 ϕ2dxdt,
≤ ZQ|∇ | k kL∞(Γ7/4 V1,2(B7/4))ZQ |∇ |
foranyarbitrary ǫ > 0. Similarly,wealsoobtain
1/2 1/2
b v wϕ2dxdt vϕ b2ϕ2dxdt w2ϕ2dxdt
Z | ||∇ || | ≤ k∇ kL∞(Q7/4)(Z | | ) (Z | | )
Q Q Q
1/2
vϕ b ϕ w2ϕ2dxdt .
≤ k∇ kL∞(Q7/4)k kL∞(Γ7/4,V(B7/4))k∇ kL2(Q7/4)(ZQ| | )
Othertermscanbeestimatedsimilarly. Then,collectingalltheestimatesandchooseǫ sufficientlysmall,we
obtainthedesired result. (cid:3)
4. Approximation estimates
We apply the ”freezing coefficient” technique to establish the regularity estimates for weak solutions of
(1.3). Todothis,weapproximate theweaksolution uoftheequation
(4.1) u div[a u] b u = div(F) in Q ,
t 2
− ∇ − ·∇
bytheweaksolutionvoftheequation
v div[a¯ (t) v] = 0, in Q ,
(4.2) t − B7/4 ∇ 7/4
( v = u, on ∂pQ7/4
Again, the meanings for weaksolutions ofequations (4.1) -(4.2)are given inDefinition 2.1 and Definition
2.2, respectively. Weessentially follow the method in our recent work [11, 23], which in turn is influenced
by [5, 6, 27, 32]. We first begin with the standard result on the regularity of weak solution of the constant
coefficientequation (4.2).
Lemma4.1. Thereexists aconstantC depending onlyontheellipticity constant Λandnsuch thatifvisa
weaksolution of
v div[a¯ (t) v] = 0 in Q ,
t − B7/4 ∇ 7/4
then
1/2
v C v(x,t)2dxdt .
k∇ kL∞(Q3) ≤ ? |∇ | !
2 Q7/4
Our next lemma confirms that we can approximate in L2(Q ) the solution u of (4.1) by the solution v of
7/4
(4.2)ifthecoefficients andthedataaresufficiently closetoeachothers.
LOCALGRADIENTESTIMATES,EQUATIONSWITHDIVERGENCE-FREEDRIFTS 9
Lemma 4.2. Let M ,Λ > 0 and s > 1, be fixed. Then, for every ǫ > 0, there exists δ > 0 depending on
0
only ǫ,Λ,n,M ,s such that the following statement holds true: For every a,b,F such that if (1.5) holds,
0
b M ,,and
k kL∞(Γ2,V1,2(B2)) ≤ 0
1/2 1/2
a a¯ (t)2dxdt + F2dxdt
(? | − B7/4 | ) (? | | )
(4.3) Q7/4 Q2
1/s 1/s
′
+ bsdxdt uˆ s′dxdt δ
(? | | ) (? | | ) ≤
Q2 Q2
withuˆ = u u¯ ,theneveryweaksolution uof (4.1)with
− Q2
u2dxdt 1,
? |∇ | ≤
Q2
theweaksolution vof (4.2)satisfies
u v2dxdt ǫ, and v2dxdt C(Λ,M ,n).
? | − | ≤ ? |∇ | ≤ 0
Q7/4 Q7/4
Proof. Notethatoncetheexistence isproved,itfollowsfromLemma3.1andtheassumption (4.3)that
w2dxdt C[M +1].
0
Z |∇ | ≤
Q7/4
Fromthis,andusing(4.3),weinferthat
v2dxdt w2dxdt+ u2dxdt C(Λ,M ,n).
? |∇ | ≤ ? |∇ | ? |∇ | ≤ 0
Q7/4 Q7/4 Q7/4
Therefore, we only need to prove the existence of δ. We use the contradiction argument as this method
works well for nonlinear equations, and non-smooth domains. Assume that there exist M ,Λ > 0,s,s ,λ,
0 ′
andǫ > 0beasintheassumption suchthatforeveryk N,thereare F ,a ,b ,suchthat
0 k k k
∈
1/2 1/2
a a¯ (t)2dxdt + F 2dxdt
(? | k − k,B7/4 | ) (? | k| )
(4.4) Q7/4 Q2
1/s 1/s
′ 1
+ b sdxdt uˆ s′dxdt , for uˆ = u u¯
(? | k| ) (? | k| ) ≤ k k k − k,Q2
Q2 Q2
andaweaksolution u of
k
(4.5) ∂u div[a u ] b u = div(F ), Q ,
t k k k k k k 2
− ∇ − ·∇
satisfying
(4.6) u 2dxdt 1,
? |∇ k| ≤
Q2
butfortheweaksolutionv of
k
∂v div[a¯ (t) v] = 0, in Q ,
(4.7) t k − k,B7/4 ∇ 7/4
( vk = uk, on ∂pQ7/4,
wehave
(4.8) u v 2dxdt ǫ .
? | k − k| ≥ 0
Q7/4
Sincea¯ (t)isaboundedsequenceinL (Γ ,Rn n),wecanalsoassumethatthereisa¯(t)inL (Γ ,Rn n))
k,B7/4 ∞ 7/4 × ∞ 7/4 ×
suchthata¯ ⇀ a¯ weakly-*inL (Γ ;Rn n). Thismeansthatforeachvectorξ Rn,andforallfunction
k,B7/4 ∞ 7/4 × ∈
φ L1(Γ ),wehave
7/4
∈
(4.9) a¯(t)ξ,ξ φ(t)dt = lim a¯ (t)ξ,ξ φ(t)dt.
ZΓ h i k ZΓ h k,B7/4 i
7/4 →∞ 7/4
10 TUOCPHAN
Also,foreachk N,letw = u v ,weseethatw isaweaksolutionof
k k k k
∈ −
∂w div[a¯ w +(a a¯ ) u ] b u = div[F ], Q ,
(4.10) t k− k,B7/4∇ k k − k,B7/4 ∇ k − k ·∇ k k 7/4
( wk = 0, ∂pQ7/4.
From(4.4),and(4.6),wecanapplyLemma3.1toyield
(4.11) sup w 2dx+ w 2dxdt C, k N.
k k
Γ7/4 | | ZQ7/4|∇ | ≤ ∀ ∈
Thisestimate,togetherwith(4.4),(4.6),andthePDEin(4.10),weconcludethat w isaboundedsequence
k k
{ }
in (Q ),where
7/4
E
(Q ) = g L2(Γ ,H1(B )) :g L2(Γ ,H 1(B ),g = 0on∂ Q .
7/4 7/4 7/4 t 7/4 − 7/4 p 7/4
E { ∈ ∈ }
Therefore,bythecompactembedding (Q )֒ C(Γ ,L2(B )),andbypassingthroughasubsequence,
7/4 7/4 7/4
E →
wecanassumethatthereisw (Q )suchthat
7/4
∈ E
w wstrongly inL2(Q ), w ⇀ wweaklyinL2(Q ),
(4.12) k → 7/4 ∇ k ∇ 7/4
( ∂twk ⇀ ∂twweakly-*inL2(Γ7/4;H−1(B7/4)), and wk wa.e. inQ7/4.
→
From(4.8)and(4.12),itfollowsthat
(4.13) w2dxdt ǫ .
? ≥ 0
Q7/4
Moreover, due to the boundary condition w = 0 on ∂ Q , and (4.12), we also conclude that, in the trace
k p 7/4
sense,
(4.14) w = 0, ∂ Q .
p 7/4
Weclaimthatwisaweaksolution of
w div[a¯(t) w] = 0, Q ,
(4.15) t − ∇ 7/4
( w = 0, ∂pQ7/4
Fromthis, andbythe uniqueness oftheweaksolution ofthisequation, weinfer thatw = 0andthis contra-
dictsto(4.13). Thus,itremainstoprovethatwisaweaksolution of (4.15). Toprovethis,wepassthelimit
ask of (4.10). By(4.14),weonly needtofindthelimitsask foreach termintheweakform of
→ ∞ → ∞
the equation (4.10). Let us fix a test function ϕ C (Q ) with ϕ = 0 on ∂ Q . Then, it is easy to see
∞ 7/4 p 7/4
∈
from(4.4),and(4.6)that
lim F ϕdxdt = 0, lim (a a¯ (t)) u , ϕ dxdt = 0.
k→∞ZQ7/4 k ·∇ k→∞ZQ7/4h k − k,B7/4 ∇ k ∇ i
Furthermore,from(4.12),wealsofindthat
lim ∂w ,ϕ dt = ∂w,ϕ dt.
k ZΓ h t k iH−1(B7/4),H01(B7/4) ZΓ h t iH−1(B7/4),H1(B7/4)
→∞ 7/4 7/4
Fortheterminvolving b ,sincediv(b )= 0,wecanusetheintegration bypartsin xtowrite
k k
[b u ]ϕdxdt = b ϕ uˆ dxdt, uˆ = u u¯ .
Z k ·∇ k −Z ·∇ k k k − k,Q2
Q7/4 Q7/4h i
Then,byHo¨lder’sinequality and(4.6),seethat
1/s 1/s
′
[b u ]ϕdxdt ϕ b sdxdt uˆ s′dxdt
(cid:12)(cid:12)(cid:12)ZQ7/4 k ·∇ k (cid:12)(cid:12)(cid:12) ≤ k∇ kL∞(Q7/4)(ZQ2| k| ) (ZQ2| k| )
(cid:12) (cid:12)
(cid:12)(cid:12) (cid:12)(cid:12) |Q7/4| ϕ 0, as k .
≤ k1/2 k∇ kL∞(Q7/4) → → ∞