Optimal partial regularity of second-order parabolic systems under natural growth condition

R E S E A R C H

Open Access

Optimal partial regularity of second-order

parabolic systems under natural growth

condition

Shuhong Chen

_{and Zhong Tan}

*_{Correspondence: [email protected]} 2_{School of Mathematical Science,}

Xiamen University, Xiamen, Fujian 361005, China

Full list of author information is available at the end of the article

Abstract

We consider the regularity for weak solutions of second-order nonlinear parabolic systems under a natural growth condition whenm> 2, and obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particular, we get the optimal regularity by the method ofA-caloric approximation introduced by Duzaar and Mingione.

Keywords: nonlinear parabolic systems; natural growth condition;A-caloric approximation; optimal partial regularity

1 Introduction

Electrorheological fluids are special viscous liquids, that are characterized by their abil-ity to undergo significant changes in their mechanical properties when an electric field is applied. This property can be exploited in technological applications,e.g., actuators, clutches, shock absorbers, and rehabilitation equipment to name a few [].

A model was developed for these liquids within the framework of rational mechanics [, ]; it takes into account the complex interactions between the electro-magnetic fields and the moving liquid. If the fluid is assumed to be incompressible, it turns out that the relevant equations of the model are the system

div(E+P) = , (.)

curlE= , (.)

ρ

∂v

∂t –divS+ρ[∇v]v+∇φ=ρf+ [∇E]P, (.)

divv= , (.)

whereEis the electric field,Pis the polarization,ρis the density,vis the velocity,Sis the

extra stress,φis the pressure, andf is the mechanical force. In fact, in a model capable of explaining many of the observed phenomena, the extra stress has the form

S=α

 +|D|

p–

 _{– E}⊗_E+α

+α|E|

 +|D|

p–  _D

+α

 +|D|

p–

 _(DE⊗_E+E⊗_{DE), (.)}

where αij are material constants, and where the material function p depends on the strength of the electric field|E|_{and satisfies}

 <p∞≤p|E|≤p<∞. (.)

Since the material functionp, which essentially determinesS, depends on the magnitude of the electric field|E|_{, we have to deal with an elliptic or parabolic system of partial}

differential equations with the so-called non-standard growth conditions,i.e., the elliptic operatorSsatisfies

S(D,E)·D≥c +|E| +|D|

p_∞–

 |_D|_{, (.)}

_{S(D,E)≤c}

 +|D|

p_–

 |_E|_{. (.)}

Equality (.) of electrorheological fluids with the conditions (.) and (.) encouraged us to considered the partial regularity of a more simple and standard model as the follow-ing:

ui_t– n

α=

DαAαi(z,u,Du) =Bi(z,u,Du), i= , , . . . ,N, (.)

where⊂Rnis a bounded domain andT > ,z= (x,t) withx∈,  <t≤T, denote a point inQT=×(–T, ). Letu(z) = (u(z),u(z), . . . ,uN(z)) be a vector-valued function defined inQT. Denote byDuthe gradient ofu,i.e.,Du={Dαui}i=,...,N;α=,...,n.m>  is a real

number.

In order to define the weak solution of (.), one needs to impose some regularity con-ditions and constructer concon-ditions toAα

i andBi. For a vector fieldAαi :QT×RN×RnN, we shall denote the coefficients byAα

i(z,u,p) =Aαi(x,t,u,p) ifz= (x,t),u∈RN andp∈RnN. We assume that the functions (z,u,p) →Aα_i(z,u,p); (z,u,p) →∂Aαi

∂pj_β(z,u,p) are continuous inQT×RN×RnN and that the following growth and ellipticity conditions are satisfied:

(H) There exists a constantLsuch that

Aα

i(z,u,p)≤L

 +|p|m _{for allz}∈_Q

T,u∈Rnandp∈RnN.

(H) Aα

i(z,u,p)are differentiable functions inpand there exists a constantLsuch that

∂Aα

i ∂piβ

(z,u,p)≤L +|p|

m–

 _{for allz}∈_QT,u∈_Rn_andp∈RnN_.

(H) Aα_i is uniformly strongly elliptic, that is, for someλ> , we have

∂Aα

i ∂pi

β

(z,u,p)p˜iα

· ˜pjβ≥λ|˜p|

 +|p|m

–

 _{for allz}∈_QT,u∈_Rn_andp,p˜∈RnN_,

whereλ>  and ≤L<∞. Now we shall specify the regularity assumptions on

Aα

Aα i(z,u,p)

+|p| is Hölder continuous with respect to the parabolic metric

|x–x|₊|_t–t|

with Hölder exponentβ∈(, )but not necessarily uniformly Hölder continuous; namely we shall assume that:

(H) There exists a constantLsuch that

Aα_i(z,u,p) –Aα_i(z,u,p)≤Lθ|u|+|u|,|x–x|+|t–t|+|u–u| +|p|

m



for anyz= (x,t)andz= (x,t)inQT.uanduinRn_{and for allp∈R}nN_{, where} θ(y,s) =min{,K˜(y)sβ_},K˜ _{: [,∞) →(,∞)is a given non-decreasing function. Note}

that θ is concave in the argument. This is the standard way to prescribe

(non-uniform) Hölder continuity of the functionAα

i(z,u,p). We find it a bit difficult to handle, therefore, in many points of the paper, we shall use:

(H) Forβ∈(, )andK: [,∞)→[L,∞)monotone nondecreasing such that

Aα

i(z,u,p) –Aiα(z,u,p)≤K

|u||x–x|+|t–t|+|u–u|β +|p|m_,

valid for anyz= (x,t)andz= (x,t)inQT,uanduinRnandp∈RnN. (H) There exist constantsaandbsuch that

_{Bi(z,u,p)≤a|p|}m_{+b, sup} QT |

u|=V, aV<λ.

Finally, we remark a trial consequence of the continuity of∂Aαi

∂pj_β; this implies the existence of a functionω: [,∞)×[,∞) →[,∞) withω(t, ) =  for alltsuch thatt →ω(t,s) is nondecreasing for fixeds,s →ωm(t,s) is concave and nondecreasing for fixedt, and such that

(H) ∂A

α

i

∂pj_β(x,t,u,p) –

∂Aα_i

∂pj_β(x,t,u,p)

≤L +|p|+|p|

m– 

×ω|u|+|p|,|x–x|+|t–t|+|u–u|+|p–p|

for anyz= (x,t) andz= (x,t) inQT, anyu,uinRnandp,p∈RnNwhenever|u|+|p|+ |u–u|+|p–p| ≤M.

From (H) and (H) we immediately deduce the following:

Aα_i(z,u,p) –Aα_i(z,u,q)≤L +|p|+|q|

m–

 |_p–q|_{, (.)}

Aα_i(z,u,p) –Aα_i(z,u,q)(p–q)≥λ +|p|+|q|

m–

 |_p–q| _(.)

for allz∈QT,u∈RN _andp,q∈RnN_.

Definition . By a weak solution of (.) under the assumptions (H)-(H), we mean a vector-valued functionu∈Lm_{(–T, ;W},m_(,RN_))∩L∞_(QT;RN_{) such that}

ˆ

QT

Aα_i(z,u,Du)Dαϕi–uiϕi_t

dz=

ˆ

QT

Bi(z,u,Du)·ϕidz (.)

In [] Duzaar and Mingione considered the partial regularity of homogeneous systems of (.) withm≡ under the natural growth condition. In this paper, we extend their results to the case ofm> . We have to overcome the difficulty ofm> . Motivated by the works of Duzaar [, ], Chen and Tan [–] and Tan [], we use the technique of ‘A-caloric approximation’ to establish the optimal partial regularity of nonlinear parabolic systems (.). In fact, the use of the ‘A-caloric approximation lemma’ allows optimal reg-ularity, without the use of Reverse-Hölder inequalities and (parabolic) Gehring’s lemma. The method is based on an approximation result that we called the ‘A-caloric approxima-tion lemma’. This is the parabolic analogue of the classical harmonic approximaapproxima-tion lemma of De Giorgi [, ] and allows to approximate functions with solutions to parabolic sys-tems with constant coefficients in a similar way as the classical harmonic approximation lemma does with harmonic functions. And we can obtain the following theorem.

Theorem . Let u∈Lm(–T, ;W,m(,RN))∩L∞(QT;RN)be a weak solution to system

(.)under the assumptions(H)-(H)and the natural growth condition(H)and denote by Qthe set of regularity points of u in QT:

Q= z∈QT:Du∈Cβ,β/

O,RnN,O⊂QTis a neighborhood of z

.

Then Qis an open subset with full measure,and therefore

Du∈Cβ,β/Q,RnN, |QT\Q|= .

At the end of the section, we summarize some notions which we will be used in this pa-per. Forx∈Rn_{,t∈R, we denoteB(x,R) ={x∈R}n_{:|x–x|<R},Q((x,t),R) =B(x,R)×} (t–R_{,t). If v is an integrable function in Q(z,ρ) =Q}

ρ(z) =Bρ(x)×(t–ρ,t), z= (x,t), we will denote its average by (v)z,ρ=

ffl

Qρ(z)v dz=

 αnρn+

´

Qρ(z)v dz, where

αndenotes the volume of the unit ball inRn. We remark that in the following, when not crucial, the ‘center’ of the cylinder will be often unspecified,e.g.,Qρ(z) =Qρ; the same

convention will be adopted for balls inRn_{therefore denotingB(x,ρ) =B}

ρ(x). Finally, in

the rest of the paper, the symbolCwill denote a positive, finite constant that may vary from line to line; the relevant dependencies will be specified.

2 TheA-caloric approximation technique and preliminaries

In this section we introduce theA-caloric approximation lemma [] and some preliminar-ies. Recall a strongly elliptic bilinear formAα

i onRnNwith an ellipticity constantλ> , and upper bound>  means thatλ|˜p|_≤Aα

i(p˜,p˜),Aαi(p,p˜)≤|p||˜p|,∀p,p˜∈RnN, we define

A-caloric approximation function.

Definition . We shall say that a functionh∈L_{(–, ;W},_(B

ρ,RN)) isA-caloric onQρ

if it satisfies

ˆ

Qρ

hiϕ_ti–Aα

i

Dh,Dαϕi

dz=  for allϕ∈C_∞Qρ,RN

.

Lemma .(A-caloric approximation lemma) There exists a positive functionδ(n,N,λ,,

ε)≤  with the following property: Whenever A is a bilinear form on RnN, which is strongly ellipticity constant λ> and upper bound ,ε is a positive number,and u∈ L_{(–, ;W},_(B,RN_))with

ˆ

Q

|u|+|Du|dz≤, (.)

is approximatively A-caloric in the sense that

ˆ

Q

uϕt–A(Du,Dϕ)

dz≤δsup Q |

Dϕ| for allϕ∈C_∞Q,RN, (.)

then there exists an A-caloric function h such that

ˆ

Q

|h|+|Dh|dz≤, and

ˆ

Q

|u–h|dz≤ε. (.)

Actually, we could have directly applied Theorem  of [] with the choice X =

W,_(B,RN_),B=L_(B,RN_),R=W–l,_(B,RN_{),F= (vk)k}

∈N,p=  to conclude that (vk)k∈N is relatively compact inL_(Q

T,RN) =L(–, ;L(B,RN)).

Lemma . There exists a positive functionδ(n,N,λ,,ε)≤with the following property:

Whenever A is a bilinear form on RnN_{which is strongly ellipticity constantλ> and upper}

bound,εis a positive number,and u∈L_(t

–ρ,t;W,(Bρ(x),RN))with

ρ– Qρ(z)

|u|dz+ Qρ(z)

|Du|dz≤, (.)

is approximatively A-caloric in the sense that

_Qρ(z

)

uϕt–A(Du,Dϕ)

dz≤δ sup Qρ(z)

|Dϕ| for allϕ∈C_∞Qρ(z),RN

, (.)

then there exists h∈L_(t–ρ_,t;W,_(B

ρ(x),RN))A-caloric on Qρ(z)such that

ρ– Qρ(z)

|h|dz+ Qρ(z)

|Dh|dz≤, and ρ– Qρ(z)

|u–h|dz≤ε. (.)

Foru∈L_(Q

ρ(z),RN) we denote bylz,ρthe unique affine function (in space)l(z) =l(x)

minimizing l →ffl_Qρ(z

)|u–l|

_{dz, amongst all affine functionsa(z) =a(x) which are}

in-dependent oft. To get an explicit formula forlz,ρ, we note that such a unique minimum

point exists and takes the formlz,ρ(x) =ξz,ρ+νz,ρ(x–x), whereνz,ρ∈RnN. A

straight-forward computation yields thatffl_Qρ(z

)u·a(x)dz=

ffl

Qρ(z)lz,ρ(x)·a(x)dz, for any affine

function a(x) =ξ +ν(x–x) with ξ ∈RN andν∈RnN. This implies in particular that ξz,ρ=

ffl

Qρ(z)u dz= (u)z,ρandνz,ρ=

n+ ρ

ffl

Qρ(z)u⊗(x–x)dz.

Lemma . Let u∈L_(Q

ρ(z),RN),  <θ< ,and lz,ρrespectively lz,θρthe unique affine

functions minimizing l →ffl_Qρ(z

)|u–l|

_{dz respectively l →}ffl

Qθρ(z)|u–l|

_{dz.Then there}

holds

|νz,θρ–νz,ρ|≤

n(n+ ) (θρ)

Qθρ(z)

u– (u)z,ρ–νz,ρ(x–x) 

dz.

Moreover, ifDu∈L_(Q

ρ(z),RnN), we have

νz,ρ– (Du)z,ρ



≤n(n+ )

ρ

Qρ(z)

u– (u)z,ρ– (Du)z,ρ(x–x)

 dz.

3 Caccioppoli second inequality

In this section we prove Caccioppoli’s second inequality.

Theorem .(Caccioppoli second inequality) Let u∈Lm_{(–T, ;W},m_(,RN_))∩L∞_(Q T;

RN_{)be a weak solution to(.)under the assumptions(H)-(H)and the natural growth}

condition(H).Then,for any M> ,any affine function l(z) =l(x)independent of t and satisfying|l(z)|+|Dl| ≤M,and any Qρ(z)⊂⊂QTwith <ρ<R≤,we have

Qρ(z)

 +|Dl|

m–

 |_Du–Dl|_+|Du–Dl|m_dz

≤Ccac +|Dl|

m– 

QR(z)



(R–ρ)|u–l| _dz+

QR(z)



(R–ρ)m|u–l| m_dz

+K|l| +|Dl|

m

–β_Rβ_+b_+a|_Dl|m_R

.

Proof We take the test functionϕ=η_ξ_{(u–l), whereη(x)∈C}

(BR(x)) is a cut-off

func-tion in space such that ≤η≤,η≡ inBρ(x),|Dη| ≤ _(R–ρ). Whileξ∈C

_{(R) is a cut-off}

function in time such that, with  <σ<ρbeing arbitrary,

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ξ≡, on (t–ρ,t–σ),

ξ≡, on (–∞,t–R_{)∪(t,∞),}

≤ξ≤, onR,

ξt≤, on (t–ρ,∞),

|ξt| ≤_|R–_ρ|, on (t–R,t–ρ).

Thus, we obtain

ˆ

QR(z)

Aα_i(z,u,Du)D(u–l)iξηdz

= –

ˆ

QR(z)

Aα_i(z,u,Du)ξη∇η⊗(u–l)idz

+

ˆ

QR(z)

ui∂tϕidz+

ˆ

QR(z)

We further have

–

ˆ

QR(z)

Aα_i(z,u,Dl)Dα(u–l)iξηdz

= 

ˆ

QR(z) Aα

i(z,u,Dl)ξη∇η⊗(u–l)idz–

ˆ

QR(z) Aα

i(z,u,Dl)Dαϕidz

and

 =

ˆ

QR(z)

Aα_iz,l(z),DlDαϕidz.

Adding these equations and usinglt≡, we deduce

ˆ

QR(z)

Aα

i(z,u,Du) –Aαi(z,u,Dl)

D(u–l)ξηdz

= –

ˆ

QR(z)

Aα_i(z,u,Du) –Aα_i(z,u,Dl)ξη∇η⊗(u–l)dz

–

ˆ

QR(z)

Aα_i(z,u,Dl) –Aα_i(z,l,Dl)Dαϕidz

–

ˆ

QR(z)

Aα

i(z,l,Dl) –Aαi

z,l(z),Dl

Dαϕidz

+

ˆ

QR(z)

(u–l)i∂tϕidz+

ˆ

QR(z)

Bi(z,u,Du)ϕidz

≤I+II+III+IV+V. (.)

By (.) and Young’s inequality, we have

I≤ε ˆ

QR(z)

 +|Dl|

m–

 |_Du–Dl|_ξ_η_dz+ε–(m–)_Cm

ˆ

QR(z)

ξm|∇η|m|u–l|mdz

+C



ε ˆ

QR(z)

 +|Dl|

m–

 _ξ_|∇η|_|u–l|_dz

+ε ˆ

QR(z)

|Du–Dl|mξmm–_ηmm–_{dz. (.)}

By the condition (H) and Young’s inequality, we can get

II≤ε ˆ

QR(z)

ξη|Du–Dl|dz+

 ε+ 



|R–ρ|

ˆ

QR(z)

ξη|u–l|dz

+

 ε+ 



–β_K|_l|_{ +}|_Dl| m

–β_α nRn++

β

–β_{. (.)}

Similarly, we can estimateIIIas follows:

III≤ε ˆ

QR(z)

ξη|Du–Dl|dz+

ˆ

QR(z)

ξ|∇η||u–l|dz

+

 ε+ 

K|l| +|Dl|m+β_α

Using the fact thatξ≡ on (–∞,t–R_{)∪(t,∞), taking into account thatξ ξ}

t≤ for

t>t–ρ_and|ξ

t| ≤_|R–_ρ|, we infer

IV =

ˆ

QR(z)

(u–l)i∂tϕidz=

ˆ

QR(z)

|u–l|η∂t

ξdz+ 

ˆ

QR(z)

ξη∂t|u–l|dz

= 



ˆ

QR(z)

|u–l|η∂t

ξdz=

ˆ

QR(z)

|u–l|ηξ ξtdz

≤ 

|R–ρ|

ˆ

QR(z)

|u–l|dz, (.)

and forμpositive to be fixed later, we have

V =

ˆ

QR(z)

a|Du|mξη|u–l|dz+

ˆ

QR(z)

|u–l| R–ρξ η

ξ ηb(R–ρ)dz

≤

ˆ

QR(z) a

( +μ)|Du–Dl|m+

 +  μ

|Dl|m

ξη|u–l|dz

+ 

ε ˆ

QR(z)

_| u–l| R–ρξ η

 dz+ε



ˆ

QR(z)

ξηbRdz

≤aV( +μ)

ˆ

QR(z)

ξη|Du–Dl|mdz+ ε ˆ

QR(z)

|u–l|

R–ρξ η

 dz +ε  a  +  μ 

|Dl|m+b

αnRn+. (.)

By (.) we have

ˆ

QR(z)

Aα

i(z,u,Du) –Aαi(z,u,Dl)

Dα(u–l)iξηdz

≥λ ˆ

QR(z)

 +|Du|+|Dl|

m–

 |_Du–Dl|_ξ_η_dz

≥λ ˆ

QR(z)

 +|Dl|

m–

 |_Du–Dl|_ξ_η_+|Du–Dl|m_ξ_η_{dz. (.)}

Combining (.)-(.) in (.) and noting thatR

β

–β ≤_Rβ _(R≤_{), that }



–β _{> , that} [K(|l|)( +|Dl|)m+β_]≤_[K(|_l|_{)( +}|_Dl|₎m_]



–β _(forK≥_{), choosingεsufficiently small and} taking into account that aV≤λ, thatξ≡ fort∈[t–ρ,t–σ], thatη≡ onBρ(x),

we infer that

ˆ t–σ

t–ρ

ˆ

Bρ(x)

 +|Dl|

m–

 |_Du–Dl|_+|Du–Dl|m_{dx dt}

≤C +|Dl|

m– 

ˆ

QR(z)



|R–ρ||u–l| _dz+

ˆ

QR(z)



|R–ρ|m|u–l| m_dz

+C

K|l| +|Dl|m–β_α

nRn++β+C a  +  μ 

|Dl|m+b

αnRn+.

4 The proof of the main theorem

The next lemma is a prerequisite for applying theA-caloric approximation technique.

Lemma . Let u∈Lm(–T, ;W,m(,RN))∩L∞(QT;RN)be a weak solution to(.)under the assumptions(H)-(H).Then for any M> ,we have

_Qρ(z

)

(u–l)iϕi_t–∂A

α

i

∂pj_β

z,l(z),DlDui–DliDαϕi

dz

≤CEuω(M+ ,) _+++H(M)ρβ _sup

Qρ(z)

|Dϕ|,

for any Qρ(z)⊂⊂QTandϕ∈C∞(Qρ(z),RN)withρ≤and any affine function l(z) = l(x)independent of time,satisfying|l(z)|+|Dl| ≤M.Here CEu=CEu(M,L,m)and we write

=(z,ρ,Dl) = +|Dl|m–

Qρ(z)

|Du–Dl|dz+ Qρ(z)

|Du–Dl|mdz,

(z,R,l) = +|Dl|

m– 

QR(z)



|R–ρ||u–l| _dz+

QR(z)



|R–ρ|m|u–l| m_dz,

=(z,ρ),

H(s) =K˜(s)( +s)m 

–β_{, forK}˜_{(s) =max K(s),a,b.}

Proof Without loss of generality, we can assume thatsup_Qρ(z)|Dϕ| ≤. From (.) and the fact thatffl_Qρ(z

)A

α

i(z,l(z),Dl)Dαϕidz=  and

ffl

Qρ(z)lϕtdz= , we deduce

Qρ(z)

(u–l)iϕ_ti–∂A

α

i ∂pjβ

z,l(z),DlDui–DliDαϕi

dz

= Qρ(z)

Aα_iz,l(z),Du–∂A

α

i ∂pj_β

z,l(z),DlDui–DliDαϕidz

+ Qρ(z)

Aα_i(z,u,Du) –Aα_i(z,l,Du)Dαϕidz

+ Qρ(z)

Aα

i(z,l,Du) –Aαi

z,l(z),Du

Dαϕidz

– Qρ(z)

Bi(z,u,Du)ϕidz

=I+II+III+IV.

In turn, we split the first integral as follows:

I= 

|Qρ(z)|

ˆ

s

(· · ·)dz+ 

|Qρ(z)|

ˆ

s

(· · ·)dz=I+I,

We proceed estimating the two resulting pieces. As forI, using (H), the fact thats →

ωm(t,s) is concave and Jensen’s inequality (note thatm_m–>_), we get

I = 

|Qρ(z)|

ˆ

s

ˆ 



∂Aα

i

∂pjβ

z,l(z),Dl+τ(Du–Dl)

–∂A

α

i

∂pjβ

z,l(z),Dl

dτ(Du–Dl)Dαϕidz

≤L

Qρ(z)

 +|Dl|+|Du–Dl|m–_{ωM+ ,}|_Du–Dl||_Du–Dl|_dz

≤L +|Dl|

m– 

Qρ(z)

ωmM+ ,|Du–Dl|dz 

m

·

Qρ(z)

 +|Du–Dl|

m–

 |_Du–Dl|mm–_dz

m–

m

≤L +|Dl|

m–  _ω

M+ ,

Qρ(z)

|Du–Dl|dz

·

Qρ(z)

|Du–Dl|dz

m

(m–)

m–

m

+

Qρ(z)

|Du–Dl|mdz m–

m

≤L +|Dl|

m–  _ω

M+ ,

Qρ(z)

|Du–Dl|dz

·

Qρ(z)

|Du–Dl|dz+ Qρ(z)

|Du–Dl|mdz 



.

To estimateI, we preliminarily observe that, using Hölder inequality,

|s| ≤

ˆ

s

|Du–Dl|dz≤ ˆ

s

dz 

ˆ

s

|Du–Dl|dz 



≤|s| ˆ

Qρ

|Du–Dl|dz 



,

and therefore

√ |s|

|Qρ(z)| ≤

Qρ

|Du–Dl|dz 



.

Similarly, we also have

√ |s|

|Qρ(z)| ≤

Qρ

|Du–Dl|m_dz

 

Using (H), (H) and the previous inequality, we then conclude the estimate ofI as follows:

|I| ≤ L |Qρ(z)|

ˆ

s

 +|Du|m _{+ +}|_Dl|m–|_Du–Dl|_dz

≤_| L Qρ(z)|

ˆ

s

 +|Dl|m_{ +}|_Du–Dl|m_dz

+ L

|Qρ(z)| ˆ

s

 +|Dl|

m–  _dz

 ˆ

s

 +|Dl|

m–

 |_Du–Dl|_dz 



≤L(M+ )m

√ |s|

|Qρ(z)|

Qρ(z)

|Du–Dl|mdz 



+ L|s|

|Qρ(z)|

( +M)m

+ L

√ |s|

|Qρ(z)|

 +|Dl|

m– 

Qρ(z)

 +|Dl|

m–

 |_Du–Dl|_dz 



≤L( +M)m

× +|Dl|

m– 

Qρ(z)

|Du–Dl|dz+ Qρ(z)

|Du–Dl|mdz

.

Combining the estimates found forIandI, we have

|I| ≤L +Mm–_{ω(M+ ,)}√_{+ L( +M)}m_.

For the remaining pieces, using (H), we deduce

|II| ≤Kl(z) Qρ(z)

|u–l|β +|Dl|+|Du–Dl|

m

 _dz

≤m

Qρ(z)

 ρ|u–l|

_dz+

Qρ(z)

|Du–Dl|mdz

+K|l| +|Dl|

m

–β_ρ

β

–β

.

Here we have used that K≥ and the assumption that ρ≤. Using again (H) and Young’s inequality, we estimate

|III| ≤K|l|

Qρ(z)

ρβ +|Dl|β +|Dl|+|Du–Dl|m _dz

≤m

H(M)Rβ₊

Qρ(z)

|Du–Dl|mdz

,

and

|IV| ≤

Qρ(z)

bR dz+ m– Qρ(z)

aR|Dl|mdz+ m–a

Qρ(z)

Noting the definition ofHand combining the estimates just found forI,II,IIIandIV, we obtain

Qρ(z)

(u–l)iϕi_t–∂A

α

i

∂pjβ

z,l(z),Dl

Dui–DliDαϕi

dz

≤C(L,M,m)ω(M+ ,) _+++H(M)ρβ_.

A simple scaling argument yields the result for generalϕ.

The next lemma is a standard estimate for weak solutions to linear parabolic systems with constant coefficients [], Lemma ..

Lemma . Let h∈L(t–ρ,t;W,(Bρ(x),RN))be a weak solution in Qρ(z) =Bρ(x)×

(t–ρ_{,t)of the following linear parabolic system with constant coefficients:}

Qρ(z)

hiϕ_ti–Aα_iDh,Dαϕi

dz= , ∀ϕ∈C_∞Qρ(z),RN

,

where the coefficients Aα

i satisfy Aαi(p,p)≥λ|p|,Aαi(p,p˜)≤L|p||˜p|for any p,p˜∈RnN.Then

h is smooth in Qρ(z)and there exists a constant Cpa=Cpa(n,N,L/λ)≥such that

ψ(z,θρ)≤Cpaθψ(z,ρ), ∀ <θ< .

Here we write

ψ(z,σ) =  σ

Qρ(z)

h– (h)z,σ – (Dh)z,σ(x–x) 

dz.

In the following we consider a weak solutionuof the nonlinear parabolic system (.) on a fixed sub-cylinderQρ(z)⊂QTandρ≤.

Lemma . Given M> and <β<α< ,there existθ∈(,_)andδ∈(, ]depending only on n,N,λ,L,β,αand m such that if

ωM+ ,˜(z,ρ,lz,ρ)

+

˜

(z,ρ,lz,ρ)≤

δ ,

on Qρ(z)⊂QTfor some <ρ≤and such if

_lz

,ρ(z)+|Dlz,ρ| ≤M,

then

˜

(z,θρ,lz,θρ)≤θ

α˜

(z,ρ,lz,ρ) +Cρ

β H(M)

for

˜

(z,ρ,lz,ρ) =(z,ρ,lz,ρ) +H

_(M)ρβ

Proof GivenM> . And we shall always considerρ≤. We first want to apply Lemma . onQρ/(z) tou–l, wherel(z) =l(x) is an affine function independent oftsatisfying|l(z)|+ |Dl| ≤M. We observe thathas the following property:

(z,ρ/,l)≤n+m+ +|Dl|

m– 

Qρ(z)

u_ρ–ldz+ Qρ(z)

u_ρ–lmdz

= n+m+_(z

,ρ,l). (.)

From Caccioppoli’s second inequality, we infer

(z,ρ/,l)≤Ccacm(z,ρ,l) + H(M)ρβ=Ccac˜ ˜(z,ρ,l). (.)

From Lemma . we therefore get, for anyϕ∈C_∞(Qρ/(z),RN), that

Qρ/(x)

(u–l)iϕ_ti–∂A

α

i ∂pjβ

z,l(z),DlD(u–l)iDαϕi

dz

≤ ˜CEuω(M+ ,˜)

˜

+˜ +ρβH(M)· sup Qρ/(z)

|Dϕ|, (.)

whereCEu˜ =CEu˜ (L,M,m).

For givenε>  to be specified later, we letδ=δ(n,N,λ,L,ε)∈(, ] to be constant from Lemma .. Defineγ =CEu˜ (z,ρ) + δ–_H_(M)ρβ_andw=γ–_(u–l).

Then from (.) we deduce that, for allϕ∈C∞_(Qρ/(z),RN), the following holds:

Qρ/(x)

wiϕi_t–∂A

α

i

∂pjβ

z,l(z),DlDwDαϕi

dz

≤

ωM+ ,˜(z,ρ,lz,ρ)

+

˜

(z,ρ,lz,ρ) +

δ 

sup Qρ/(z)

|Dϕ|. (.)

Moreover, we estimate, using Caccioppoli’s second inequality, (.) and (.),

(ρ/)– Qρ/(x)

|w|dz+ Qρ/(x)

|Dw|dz≤

n+m+_+Ccac˜ ˜ C

Eu

≤, (.)

provided we have chosenC˜Eu large enough.

Assuming the smallness condition,

ωM+ ,˜(z,ρ,lz,ρ)

+

˜

(z,ρ,lz,ρ)≤

δ

, (.)

satisfied. Then (.) and (.) allow us to apply Lemma .,i.e., they yield the existence of

h∈L_(t–ρ_,t;W,_(B

ρ(x),RN)) solving the ∂Aα

i

∂pj_β-heat equation onQρ/(z) and satisfy-ing

(ρ/)– Qρ/(x)

|h|dz+ Qρ/(x)

and

(ρ/)– Qρ/(x)

|w–h|dz≤ε. (.)

From Lemma . we recall thathsatisfies, for any  <θ < , thea prioriestimate (note thatCpa=Cpa(n,N,λ,L)≥)

(θρ/)–

Qθρ/(x)

h– (h)z,θρ/– (Dh)z,θρ/(x–x)

 dz

≤Cpaθ(ρ/)– Qρ/(x)

h– (h)z,ρ/– (Dh)z,ρ/(x–x)

 dz

≤Cpaθ

(ρ/)–

Qρ/(x)

|h|+(h)z,ρ/



+(Dh)z,ρ/



≤Cpaθ

(ρ/)– Qρ/(x)

|h|dz+ Qρ/(x)

|Dh|dz

≤Cpaθ.

Here we have used that|(h)z,ρ/|≤

ffl

Qρ/(x)|h|

_{dz, and|(Dh)z}

,ρ/|≤

ffl

Qρ/(x)|Dh|

_dz

and (.). Combining the previous estimate with (.), we deduce

(θρ/)–

Qθρ/(x)

w– (h)z,θρ/– (Dh)z,θρ/(x–x)

 dz

≤(θρ/)–

×

Qθρ/(x)

|w–h|dz+

Qθρ/(x)

h– (h)z,θρ/– (Dh)z,θρ/(x–x)

 dz

≤Cpaθ–n–ε+θ. (.)

Recalling back (u–l) viaw=u–l

γ , we arrive at

(θρ)–

Qθρ/(x)

u–l–γ(h)z,θρ/+ (Dh)z,θρ/(x–x)

 dz

≤Cpaθ–n–ε+θγ. (.)

Next we use the minimizing property oflz,θρ/

(θρ/)–

Qθρ/(z)

|u–lz,θρ/|dz≤Cpa

θ–n–ε+θγ. (.)

At the same time, from (.), we can see that: For ≤m≤n+  (n≥), we have  <

m<m∗, where

m∗=

⎧ ⎨ ⎩

m(n+)

n–m+ ifn+  >m,

m∗>m ifm=n+ 

Using Sobolev’s, Caccioppoli’s and Young’s inequalities together with (.), we have

(θρ/)–m

Qθρ/(z)

u–l–γ(h)z,θρ/+ (Dh)z,θρ/(x–x)

m

dz

≤(θρ/)–ms

(θρ/)–

Qθρ/(z)

u–l–γ(h)z,θρ/+ (Dh)z,θρ/(x–x)

 dz

(–s)m



·

Qθρ/(z)

u–l–γ(h)z,θρ/+ (Dh)z,θρ/(x–x)

m∗

dz ms

m∗

≤Cpaθ–n–ε+θγ

(–s)m



Qθρ/(z)

Du–Dl–γ(Dh)z,θρ/

m

dz s

. (.)

Using Lemma ., Caccioppoli’s inequality, (.), (.), (.) and Young’s inequality, we obtain

ˆ

Qθρ/(z)

Du–Dl–γ(Dh)z,θρ/mdz 

m

=|Qθρ/|



m

Qθρ/(z)

_{Du–Dl– (Dh)z}

,ρ

m

dz 

m

+

Qθρ/(z)

(Dh)z,ρ–γ(Dh)z,θρ/

m

dz 

m

≤ |Qθρ/|



m_{Ccac+H(M)(θρ)}β



m

+

n(n+ ) (θρ)

Qθρ/(z)

u–l–γ(Du)z,ρ(x–x) 

dz 



≤ |Qθρ/|



m_C

cac

+H(M)(θρ)βm ₊

√ n(n+ )

θρ

≤ |Qθρ/|



m_θ–_C˜



m

cac+ 

n(n+ )γm_{. (.)}

From (.) and (.), we conclude

(θρ/)–m

Qθρ/(z)

u–l–γ(h)z,θρ/+ (Dh)z,θρ/(x–x)

m

dz

≤C

( –s)m



θ–n–ε+θγ+sm

m∗θ

–(n+m_m)m∗_γm_m∗

≤Cθγ, (.)

providedγ(m∗–m)/m≤θ+(n+m)m∗/mand we fixedε=θn+. That it is to say,

(θρ/)–m

Qθρ/(z)

Combining (.) and (.) yields the desired estimate

(z,θρ/,lz,θρ/)≤Cθ

_{(z,ρ,l) + δ}–_H_(M)ρβ

(.)

for C =C+ Cpa. Given β <α < , we choose  <θ <  such that α_Cθ _≤θα

with θ =θ(n,m,N,λ,L,α,β). This also fixes the constants ε=ε(n,m,N,λ,L,α,β) and

δ=δ(n,m,N,λ,L,α,β)∈(, ]. Thus we have shown Lemma ..

In the following, we want to iterate Lemma .. That is,

Lemma . For M> and Qρ(z)⊂⊂QT,suppose that the conditions

(i) |lz,ρ|+(Dl)z,ρ≤M;

(ii) ρ≤ρ(M);

(iii) ˜(ρ)≤ ˜(M)

are satisfied.Then,for every j∈N,we have

˜

z,θjρ,lz,θρ

≤θαj˜(z,ρ,lz,ρ) +C(M)

θjρβH(M)

and

|lz,θjρ|+(Dl)z,θjρ≤M.

Moreover,the limit

z= lim

j→∞(Du)z,θjρ/ exists,and the estimate

 +|Dl|

m– 

Qr(z)

|Du–z|

_dz+

Qr(z)

|Du–z|

m_dz

≤C

r

ρ/

α

(z,ρ,lz,ρ) +rβH(M)

is valid for a constant C=C(n,N,λ,α,L,β,M,m).

Proof For fixedzwe shall denotelz,ρ≡lρ. For givenM>  (andβ<α< ), we

deter-mineδ=δ(M),θ=θ(M) andC=C(M) according to Lemma .. Then we can find

˜

(M) >  sufficiently small such that

ωM+ , ˜(M)

+

˜

(M)≤

δ

 (.)

and

˜

(M)≤

M_θn+_{( –θ}α₎

Given this, we can also findρ(M)∈(, ] so small that, writing

C(M) = C(M) θβ_–θα,

we have

C(M)ρ(M)βH(M)≤min

δ

,˜(M),

Mθn+( –θβ₎

(n+ )

. (.)

Now, suppose that the conditions (i), (ii) and (iii) are satisfied onQρ(z)⊂QT. Then, for j= , , , . . . , we shall show

(I)j ˜(z,θjρ,lz,θρ)≤θαj˜(z,ρ,lz,ρ) +C(M)(θjρ)βH(M),

(II)j l_z,θj_ρ(z)+(Dl)_z

,θjρ≤M.

Note first that (I)jcombined with (ii), (iii) and (.) yields

(I)j ˜θjρ≤˜(M).

Moreover, we have ρ ≤ρ(M)≤ and |lz,ρ|+|(Dl)z,ρ| ≤M. There we can apply

Lemma . to conclude that (I) holds. Furthermore, using Lemma ., (iii) and (.),

we deduce

|lz,θρ|+(Dl)z,θρ≤M+

_Qθρ(z

)

(u–uρ)dz +

n(n+ )

(θρ)

Qθρ(z)

|u–lρ|dz 



≤M+

Qθρ(z)

|u–lρ|dz 



+

n(n+ )

(θρ)

Qθρ(z)

|u–lρ|dz 



≤M+ +

√ n(n+ )

√

θn+ ˜

 _(z

,ρ)

≤M,

i.e., (II)holds. We now assume that (I)ιand (II)ιforι= , , . . . ,j–  hold. We can apply

Lemma . to calculate

˜

θjρ≤θαj˜(ρ) +C(M)θjρβθ–β j–

ι=

θ(α–β)ι

≤θαj˜(ρ) + C(M) θβ_–θα

θjρβ

=θαj˜(ρ) +C(M)θjρβ,

showing (I)j. To show (II)jwe estimate

|l_z,θj_ρ|+(Dl)_z

,θjρ ≤M+

j

ι=

Qθ ιρ(z)

|u–lθι–_ρ|dz

 

+ j

ι=

n(n+ ) (θι_ρ)

Qθ ιρ(z)

|u–lθι–_ρ|dz

≤M+ +

√ n(n+ )

√

θn+

j

ι=

_z,θι–_ρ

≤M+

√ n(n+ )

√

θn+

j–

ι=

θαι˜_{(ρ) +C} (M)

θι_ρβ

≤M+√n+  θn+

_˜(ρ)

 –θα +

C(M)ρβ

 –θβ

≤M.

Here we have used in turn Lemma ., the definition of(θι–_{ρ) and (I)}

ιforι= , , . . . ,j–.

Since|Dlθj_ρ| ≤M. We are in a position to apply Theorem .. We obtain

θjρ/, (Du)θjρ/

≤θjρ, (Du)θjρ

≤Ccac(M)˜θjρ

≤Ccac(M)θαj˜(ρ) +C(M)θjρβ. (.)

We now consider  <r≤ρ/. We fixk∈N∪ {}withθk+ρ/ <r≤θkρ/. Then the previous estimate implies

r, (Du)r

≤θ–n– +|Dl|

m– 

Q_θk_ρ/(z)

Du– (Du)θk_ρ/

 dz

+ Q

θkρ/(z)

Du– (Du)θk_ρ/mdz

≤θ–n–Ccac(M)θαk˜(ρ) +C(M)θkρβ

≤θ–n–Ccac(M)

θ–α

r

ρ/

α ˜

(ρ) +C(M)βθ–βrβ

≤θ–n––αCcac(M)

r

ρ/

α ˜

(ρ) +C(M) + 

rβ

≤Cdec(M)

r

ρ/

α ˜

(ρ) +rβ

.

Next, we show that ((Du)θjρ/)j∈N is a Cauchy sequence inRnN. ForK>jwe deduce

(Du)θj_ρ/– (Du)_θk_ρ/≤ k

ι=j+

(Du)θι_ρ/– (Du)_θι–_ρ/

≤√θ–n–

k–

ι=j

θι_ρ/(z

)

_{Du– (Du)θ}ι_ρ/dz

 

=√θ–n–

k–

ι=j