Ann. I. H. Poincar AN 21 (2004) 2560www.elsevier.com/locate/anihpc

Regularity results for parabolic systems related to a classof non-Newtonian fluids

Rsultats de rgularit pour des systmes paraboliques lis une classe de fluides non Newtoniens

E. Acerbi a,, G. Mingione a, G.A. Seregin b

a Dipartimento di Matematica, Universit, via M. DAzeglio 85/a, 43100, Parma, Italyb Steklov Mathematical Institute, St. Petersburg Branch, 27, Fontanka, 191011, St. Petersburg, Russia

Received 28 October 2002; accepted 26 November 2002

Abstract

We consider a class of parabolic systems of the type:

ut diva(x, t,Du)= 0

where the vector field a(x, t,F ) exhibits non-standard growth conditions. These systems arise when studying certain classesof non-Newtonian fluids such as electrorheological fluids or fluids with viscosity depending on the temperature. For properlydefined weak solutions to such systems, we prove various regularity properties: higher integrability, higher differentiability,partial regularity of the spatial gradient, estimates for the (parabolic) Hausdorff dimension of the singular set. 2003 Elsevier SAS. All rights reserved.Rsum

Nous tudions une classe de systmes paraboliques du type :

ut diva(x, t,Du)= 0o le champ vectoriel a(x, t,F ) possde des conditions de croissance non standard. Ces systmes se prsentent dans ltudede certaines classes de fluides non Newtoniens comme les fluides electro-rhologiques ou les fluides dont la viscosit dpendde la temprature. Nous prouvons diffrentes proprits de rgularit pour des solutions faibles convenables de tels systmes :intgrabilit et diffrentiabilit amliores, rgularit partielle du gradient spatial et estimations pour la dimension de Hausdorff(parabolique) de lensemble des points singuliers. 2003 Elsevier SAS. All rights reserved.

* Corresponding author.

E-mail addresses: emilio.acerbi@unipr.it (E. Acerbi), giuseppe.mingione@unipr.it (G. Mingione), seregin@pdmi.ras.ru (G.A. Seregin).

0294-1449/$ see front matter 2003 Elsevier SAS. All rights reserved.doi:10.1016/j.anihpc.2002.11.002

26 E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 25601. Introduction

In recent years considerable attention was paid to the mathematical modelling of non-Newtonian fluids. Theseare fluids described by a set of equations including a stress tensor depending in a non-linear way by the gradient ofthe velocity. One of the first mathematical investigations of such models was carried out by Ladyzhenskaya in 1966(see [1719]); she considered the following system of equations that are known today as modified NavierStokesequations:{

ut diva(D(u))+D =div(u u)+ f,divu= 0, in (0, T ), (1.1)

where D(u) denotes the symmetric part of the gradient Du and the pressure. The main point in the previoussystem, as just mentioned, is that the monotone vector field a :R9 R9 depends in a non-linear way by D(u):

a(D(u)

) (1+ D(u)2) p22 D(u)+ terms with a similar growth (1.2)where p > 1. The basic analysis of such systems goes back to Ladyzhenskaya and J.L. Lions (see also [23]). Inparticular, Ladyzhenskaya was also able to prove an existence and uniqueness theorem provided p > 2 belongsto a certain range. Note that to find a set of equations for which uniqueness held was actually her main initialmotivation; indeed, when studying systems as (1.1), she was not thinking of non-Newtoninan fluids at all, insteadshe was looking for some alternative model system in Fluid dynamics, being of the opinion that the classicalNavierStokes equations are not quite correct even for Newtonian fluids if the gradient of the velocity is large, andthat they should be somehow modified; see also [33] for more comments. Recently, non-Newtonian fluids havebeen intensively analyzed by many authors and a considerable literature, related to their behavior, rapidly grew-up;for an account of the mathematical aspects of the theory see the book [28] (see also [27] and [5] for an updated listof references and the paper [11] for parabolic systems with non-linear growth). It is clear that for general systemsas (1.1), i.e. without additional structure assumptions on a, partial C0,-regularity of the spatial gradient Du is thebest possible result available, beside an estimate for the Hausdorff dimension of the singular set (the closed subsetoutside which Du is Hlder continuous). Indeed, even in the elliptic case, solutions to general elliptic systemsof the form diva(x,Du) = 0 are not everywhere regular, while estimates for the Hausdorff dimension of thesingular set are available (see for instance [26] for a detailed account of the problem in the elliptic case). It isanyway important to note that in the case of the usual NavierStokes equations a wider and deeper theory has beendeveloped (see [6,20,34,35]).

A new interesting kind of fluids of prominent technological interest has recently emerged: the so calledelectrorheological fluids. These are special fluids characterized by their ability to change in a dramatic way theirmechanical properties when in presence of an external electromagnetic field; for instance they are able to increasetheir viscosity by a factor 1000 in a few milliseconds. In the context of continuum mechanics these fluids have beenmodelled as non-Newtonian fluids. The basic studies can be found in the papers [29,30] while the fundamentalmathematical analysis for the model can be found in the monograph [31] (see also [32,9]). According to the modelproposed by Rajagopal and Ruicka, the system governing an electrorheological fluid looks to the nave eye exactlyas the one in (1.1) apart for the coupling with the external electromagnetic field E:

curl E= 0, div E= 0,ut diva(x, t,E,D(u))+D =div(u u)+ fdivu= 0.

in (0, T ), (1.3)

Here we want to stress the main feature of the previous model: the dependence of a upon E is described via avariable growth exponent, indeed in this case the relation in (1.2) is now substituted by:( ) ( 2) p(E)2a x, t,E,D(u) (x, t,E) 1+ D(u) 2 D(u)+ terms with similar growth (1.4)

E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560 27that is the exponent p is varying with the electromagnetic field E (this describes the changes of viscosity).Once noted that the system in (1.3) is uncoupled, one can first obtain E(x, t) from Maxwells equations so thatp p(x, t). This is going to tell us that actually the vector field a exhibits non-standard growth conditions:

c1|F |1 c2 a(x, t,F ): F c3|F |2 + c3, 1 < 1 < 2 1, (1.5)

in (0, T ). Indeed we prove partial C0, regularity of spatial gradient of weak solutions (defined in a suitablesense and named Energy Solutions; see Definition 2.1) to system (1.5), see Theorem 2.1. We remark that as faras partial C0, regularity of the spatial gradient is concerned, this is the first result for parabolic systems, undernon-standard growth conditions. We remark that even in the scalar case, the literature on the issue is not vast andwe mention here the nice paper by Lieberman [22], concerning everywhere regularity of the spatial gradient in thescalar case N = 1. We also observe that for the sake of brevity and in order to highlight the main ideas, we confinedourselves to the analysis of homogeneous systems but, in principle, the techniques developed here allow to treatmore general systems with a non-zero right-hand side, provided this satisfies suitable growth assumptions.

Related regularity results in the stationary case can be found in [2,3] (see also [8,1] for the variational case).Moreover, fluids with viscosity dependence described using non-standard growth conditions have been treated, inthe stationary case, in various settings; see for instance [5,13,14,12,4]; this paper also offers an approach which ispotentially useful to extend such results to the non-stationary case.

Finally we spend a few words about the techniques. In Section 3 we study the problem in cylinders where p(z)has small oscillations (cylinders of the type Q0). Here we prove an a priori estimate (Theorem 3.1), ensuringthe higher integrability of the spatial gradient in Q0; at this stage an interpolation-iteration procedure tailoredfor parabolic problems plays a central role (Lemmas 3.3 and 3.5). This is first done for a priori regular solutions(Section 3) and then adapted to the original one via approximation (Section 4); here we observe that such anapproximation argument works since the peculiar p(z)-growth structure of the problem is compatible with theusual convolution (see Appendix A and Lemma 4.1). In Section 5 we consider a blow up procedure exploitedonly in cylinders of the type Q0. The advantage is that we can use the higher integrability to choose a suitableexcess functional involving the maximum of p(z) in Q0, allowing to overcome the fact that the problem exhibitsnon-standard growth conditions (the use of such a maximal excess allows to treat, in a certain sense, the problemas one with standard polynomial growth). Finally we cover the full cylinder (0, T ) with small cylinders ofthe type Q0, and, using the results in Sections 5, we blow up the solution in each of these, thereby proving partial

regularity in each Q0; the conclusion follows (see the comments in Section 6).

28 E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 25602. Statements and notations

2.1. Basic notations

In the following will denote a bounded domain in Rn, and QT will be the cylinder (0, T ). Differentiationwith respect to the space variables will be denoted with a comma in lower indices, for example, f /xj = f,j . Forfunctions, vectors fields and matrix fields, respectively, we use the differential operatorsDf := (f,j ), Dv := (vi,j ),divA = (Aij,j ), all of which are understood in the sense of distributions; differentiation with respect to the timevariable will be denoted by t or by t : tf ft . The symbols Lp(;Rn) and W 1,p(;Rn) stand for the usualLebesgue and Sobolev spaces, and will be often abbreviated into Lp() and W 1,p(), respectively (or even Lpand W 1,p). The norm in Lp() will be abbreviated in || ||p, . The spaces C, are those of functions whichare -Hlder continuous in the space variables and -Hlder continuous in the time variable. We shall keep thestandard notation concerning balls and parabolic cylinders:

B(x,R) := {y Rn: |x y|0

Ps (F ).

Finally, in the following, the constant c will simply denote an unspecified, positive quantity, possibly changingfrom line to line, while only the critical connections will be remarked; more peculiar instances will be denoted byc, c and so on. In the rest of the paper we shall use Einsteins convention on repeated indices.

2.2. Systems and energy solutions

We are given an exponent function p : z QT p(z) (1,+) which is Lipschitz continuous with respect tothe space variables and /2-Hlder continuous with respect to time (in the following (x, t), (x0, t0) QT ), that is: ( )p(x, t) p(x0, t0) L |x x0| + |t t0|/2 , (0,1]; (2.1)

E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560 29moreover we shall assume that:2n

n+ 2 < 1 p(z) 2

30 E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560changes. Of course, the larger the number 2 1, the larger the class of fluids that the model is able to cover.Therefore we allow all possible large values of 2 1. This is also seemingly in contrast to what is usually donein the framework of non-standard growth conditions of (p, q) type (see [25], for instance) for elliptic systems,where a bound on the quantity q p in terms of that ratio q/p is usually assumed. Indeed it turns out (see [10])that:

q

p 1+

n

is in general a necessary condition for regularity, where , as in (2.6), is the Hlder continuity exponent withrespect to the x-variable. In particular q/p 1 when n+. The point here is that we fully use the p(z)-growth structure of the problem.

3. An a priori estimate

In this section we shall prove an a priori estimate that will be used in the sequel. This estimate is concerned withsolutions to a perturbed system, with standard polynomial growth and ellipticity conditions.

First of all let us recall some notations we shall keep for the rest of the section. In the following Q0 :=B(x0,R) (t1, t2) will be a fixed cylinder, with B(x0,R) . Then we shall put:

p2 := supQ0

p(z) 2, p1 := infQ0

p(z) 1 >2n

n+ 2while q > max{p2,2} will be a number specified below. Moreover (0,1) and v C0([t1, t2];L2loc()) Lq(t1, t2;W 1,qloc (;RN)) will be the solution to the perturbed parabolic system

Q0

vtw+ a(z,Dv) :Dwdz= 0, w C0(Q0;RN

), (3.1)

where

a(z,F ) := a(z,F )+ (1+ |F |2) q22 F. (3.2)

We will assume that the oscillation of p is not too large: precisely, in this section we assume that for some > 0

p2 p1 /2 < < 2(n+ 2)n if p2 2,

p2 p1 /2 < < min{

4n3,

12

(1 2n

n+ 2)}

if p2 < 2,(3.3)

and we set

q0 := p1 + 4n. (3.4)

We also define the number q according to:

q :={p2 + if p2 2,2+ if p2 < 2, (3.5)

and therefore in any case, by (3.3),q > 2, q0 > q > p2. (3.6)The main result of this section is the following a priori estimate for v :

E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560 31Theorem 3.1 (A priori estimate). Assume that (3.3) holds for some > 0, and let Q(z0,2 ) Q0; then thefollowing estimate holds for the functions v :

Q(z0,/2)

|Dv|q0 dz c n+2 + c(

1

)s1( Q(z0, )

|Dv |p1 dz)s2

, (3.7)

where

s1 = 84 (q p1)(n+ 2)n+ 2n

, s2 = 4 (q p1)n4 (q p1)(n+ 2)n+ 2n

and the constant c depends upon (n, 1, 2,L,) but is independent of and of the solution v .

For the sake of simplicity, from now on we shall suppose that, with a clear abuse (and even ambiguity) ofnotations:

Q0 QT :=A (0, T ); A := B(x0,R).The estimate will be reached through a series of lemmas. In the first one we derive a suitable Caccioppoli type

estimate which differs from the usual one due to the non-standard growth conditions we are assuming.

Lemma 3.1. Let , be two cut-off functions such that C0 (A;RN), 0 1, |D| 1 and W 1,((0, T )), (0)= 0 and t 0. Let Qt0 :=A (0, t0) and:

P :=ni=1

Da(z,Dv(z)

)Dv,i (z) :Dv,i (z),

I0(,) :=Qt0

2P dz, J0(,) := sup0

32 E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560If {g} denotes a family of standard, positive radially symmetric mollifiers, replacing w with w w g in theprevious equation yields:

QT

(hu)tw+(h(a(z,Du)

))Dw dz= 0.

In the last formulation we let the test function w := (hv) where C0 (Q0) is a non-negative cut-offfunction; this choice yields, after a simple integration by parts:

12

QT

t(hv)2 dz+

QT

(h(a(z,Dv

))):D(hv) dz

=QT

(h(a(z,Dv

))):D (hv) dz.

By (3.9) we can let 0 to get:1

2

QT

t|hv |2 +QT

ha(z,Dv) :Dhv dz=

QT

ha(z,Dv) :D hv dz.

Now we perform the following choice: (x, t) := (t)(t)2(x) where and are as in the statement and iscontinuous function defined as follows: with 0 < t0 < T and 0

E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560 33In order to estimate T2(h) from below, we observe that:

T2(h)Dhv :Dhv =

10

Dza(x, t,Dv + Dhv) dDhv :Dhv

L11

0

(1+ |Dv + hDv |2

) p(z)22 d |Dhv |2

c1(1+ Dv(x, t)2 + Dv(x + hei, t)2) p(z)22 |Dhv |2, (3.13)

where c c(n,L,1, 2) > 0. Finally we introduce a further notation:B(h) := [1+ Dv(x, t)2 + Dv(x + hei, t)2].

Using (3.11)(3.13) in (3.10) we obtain:A

(t)2(x)hv(x, t)2 dx +

Qt0

2T2(h)Dhv :Dhv dz

+Qt0

2[B(h)

p(z)22 + B(h) q22 ]|Dhv|2 dz

cQt0

2[B(h)

p(z)22 + B(h) q22 ]|Dhv||hv ||D|dz

+ cQt0

B(h)p21+/4

2[2|Dhv | + 2|D||hv|

]|h|dz+ c Qt0

t2|hv |2 dz. (3.14)

We estimate the terms coming from T1(h) (that is the fifth integral of the previous inequality) in the following way(using that |D| 1):

Qt0

B(h)p21+/4

2[2|Dhv| + 2|D||hv|

]|h|dz

Qt0

2B(h)p(z)2

2 |Dhv|2 dz+CQt0

2B(h)p(z)+

2 |h|2 dz+ cQt0

|D|2B(h) q12 |hv||h|dz.

Observe that in the previous estimate we have made crucial use of the fact that p2 p1 /2 (see (3.3)). Weremark that in all the previous estimates the constant c only depends on n,1, 2,L,. Now we plug-in the lastestimate in (3.14), using Young inequality to manage for the forth integral of (3.14), choosing small enough inthe previous relation and finally dividing up by |h|2 we obtain, using the definition of the number q and the factthat q > 2 (see (3.5)):

sup0

34 E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560Lemma 3.2. Let 1 C0 (A;RN) and 1 W 1,((0, T )) be two non-negative cut-off functions such that1(0)= 0 and t1 0; let 0 < t0 < T . Then:

Qt0

121 |Dv |q0 dz

c[

supt 0 by (3.6). Using Sobolevs embedding theorem, one has the inequality:

At

21H2 dx c

( t

D(1H) 2nn+2 dx) n+2n

c( At

|1DH | 2nn+2 dx) n+2

n + c( At

|D1H | 2nn+2 dx) n+2

n =: cA4 + cA5. (3.16)

By Hlder inequality we get, using also and recalling that from (3.6) and (3.15) it also follows (20+2p1)n/2=2, we get

A4 =[ At

(21(1+ |Dv |2) p122 D2v2) nn+2 (|Dv |20(1+ |Dv |2) 2p12 ) nn+2 dx] n+2n

( At

21(1+ |Dv |2) p(z)22 D2v2 dx)(

spt1

(1+ Dv(x, t)2) dx) 2n . (3.17)Moreover, again by Hlder inequality and since (1+ 0)(2n)/(n+ 2)= (p1n+ 4)/(n+ 2), we gain

A5 =(

2q0

)2( At

(|D1|2|Dv |p1) nn+2 |Dv | 4n+2 dx) n+2n

( At

|D1|2|Dv |p1 dx)(

spt1

Dv(x, t)2 dx) 2n . (3.18)Integrating on (0, t0) via (3.16), (3.17) and (3.18) we conclude the proof:

Qt0

211|Dv |q0 dz ct0

0

1

( spt1

(1+ Dv(x, t)2)dx) 2n (

At

21P + |D1|2|Dv |p1 dx)dt

c[

sup (

1+ Dv(x, t)2)dx] 2n [I0(1, 1)+ 1|D1|2|Dv |p1 dz].

0

E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560 35The next lemma gives a first form of the estimate in Theorem 3.1:

Lemma 3.3. Assume that > 0 is such that the cylinder Q(z0, )QT . Then:Q(z0,)

|Dv |q0 dz c[

1(R )2

Q(z0,R)

(1+ |Dv |q)dz] 2n+1 (3.19)

whenever 0

36 E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560The following is restatement of Lemma 6.1 from [16]:

Lemma 3.4. Let h : [/2, ] R be a non-negative bounded function, and let A,B,1 2 > 0. Assume that forany and R such that /2 < R :

h() 12h(R)+ A

(R )1 +B

(R )2 .Then there exists c c(1, 2), such that:

h(/2) c(

A

1+ B 2

).

Lemma 3.5. Assume there are numbers q1 1, > 1, 0, 1, (0,1) such that:

< 1. (3.21)

Assume that f Lq1(Q(z0, )) and:Q(z0,)

|f |q1 dz c1(R )

( Q(z0,R)

(1+ |f |)q1 dz) (3.22)

for all and R such that /2 < R < . Then there is a constant c c(c1, q1, ,,,n, ) such that:Q(z0,/2)

|f |q1 dz c (n+2) + c

( Q(z0, )

|f | q1(1) dz) ()

. (3.23)

Proof. Using Hlder inequality we have:Q(z0,R)

|f |q1 dz( Q(z0,R)

|f |q1 dz)

( Q(z0,R)

|f |q1 (1) dz)

.

Therefore, using (3.22) and the previous inequality, it follows that:Q(z0,)

|f |q1 dz c(R )

[ (n+2) +

( Q(z0,R)

|f |q1 dz) ]

c(n+2)

(R ) +c

(R )( Q(z0,R)

|f |q1 dz)

( Q(z0,R)

|f |q1 (1) dz) ()

. (3.24)

By (3.22) we can apply Young inequality to get:Q(z0,)

|f |q1 dz 12

Q(z0,R)

|f |q1 dz+ c(n+2)

(R )

+ c(

1R

) (

Q(z0,)

|f | q1(1) dz) ()

. (3.25)Finally the assertion follows applying Lemma 3.4 with the choice:

E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560 37A := c (n+2) , B := c( Q(z0, )

|f | q1(1) dz) ()

,

h(s) :=

Q(z0,s)

|f |q1 dz, 1 := , 2 := .

Remark 2. The constant and the quantities appearing in the previous lemma exhibit the following critical behavior:

lim /1c=+; (3.26)

this appears after using Young inequality in (3.24) in order to get (3.25), more precisely the constant c in (3.25)becomes unbounded as / 1. Therefore, when repeatedly applying this lemma, we shall take care that /stays uniformly bounded away from one, with respect to the parameters involved in the proof.

Proof of Theorem 3.1. We want to apply Lemma 3.5 in the context of Lemma 3.3, therefore in Lemma 3.5 weperform the following choice of the various quantities involved:

f := |Dv |, := (n+ 2)/n, := 2(n+ 2)/n, q1 := q (3.27)and finally

:= p1 + 4/nq

.

Now, as suggested by the statement, we are going to distinguish between the cases p2 < 2 and p2 2 checkingin both cases that the choice of the previous quantities allows to apply Lemma 3.5.

Case p2 < 2. In this case, since p1 > 2nn+2 we first obtain that stays uniformly bounded away from 1 withrespect to all the parameters involved: (L,p1,p2, ). Indeed:

p1 + 4/n2+ 4/n3 >

(2n

n+ 2 +4n

)(2+ 4

n3

)1> 1.

Let us define as the solution to the equation:

q(1 ) = p1, (3.28)

i.e.:

= (q p1)q p1 =

n(q p1)4

. (3.29)

Then we check, in order to apply Lemma 3.5, that (0,1). Indeed, it turns out that also stays bounded awayfrom 1 (from above, this time) uniformly with respect to all the parameters involved; using in turn (3.29) and (3.3)it follows:

0< = n(q p1)4

p1 + 4/n2+

n4(2 1 + )p1 + 4/n2

= n4

[2 2n

n+ 2 +(

(1 2n

n+ 2))]

p1 + 4/n2

=: n(

2n)

2+ 4/n

2nn+2 wehave that the crucial quantity appearing in (3.21) stays uniformly bounded away from 1: / < b < 1 with bdepending on (n, 1, 2,L) but not on and which, in turn, can be fixed a priori depending only on n and thedistance 1 2nn+2 (in the case p2 < 2). Finally observe how also the exponents s1 and s2 critically depend on thedistance 1 /; indeed it turns out that:

s1 = 1 / , s2 = (1 /)1 / .

Case p2 2. As before we observe that stays bounded away from 1:

= p1 + 4/np2 + = 1+

4/n (p2 p1)p2 + > 1+

4/n 4/(n(n+ 2))2 + 2/(n(n+ 2)) > 1,

where we used both inequalities in (3.3)1. Then we check that , again defined by (3.28), still belongs to (0,1) andis uniformly bounded away from 1. Let us observe that:

q p1 = (p2 + ) p1 2 < 4(n+ 2)n 0 the existence of a solution w with the properties

stated above is a well established fact [21]. From now on will abbreviate a sequence of positive numbers {n}n3such that n 0; from time to time we shall pass to a subsequence that will be still denoted by . We shall assumethat 0< 0 := (1/1000)min{dist(B0, ), t1,t1}.

Lemma 4.1. If w is a solution to problem (4.3)(4.6), then:

supt1

E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560 4112w ( , t)2L2(B0) +

B0(t1,t )

a( )(z,Du +Dw ) : (Du +Dw )dz

c

B0(t1,t )

[(a(z,Du)

):Dw + a( )(z,Du +Dw ) :Du

]dz. (4.9)

On the other hand, writing

a( )(z,F ) : F =(a( )(z,F ) a( )(z,0)

) : F + a( )(z,0) : Fusing (2.3)(2.5) we easily derive also the following estimate:

12w ( , t)2L2(B0) +

Q0

|Dv |p(z) + ( )|Dv |q dz

cQ0

1+ (a(z,Du))

|Dw | + |Dv |dz+ c

Q0

((1+ |Dv |2

) p(z)12 + ( )(1+ |Dv |2) q12 )|Du |dz. (4.10)

We recall that, since p(z) is Hlder continuous we have that (see Appendix A):Q0

Du (z)p(z) dx c QT

1+ Du(z)p(z) dx

42 E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560Proof. Indeed from Lemma 4.2 it follows that

w 0 in L2(Q0;Rn

). (4.13)

Now we observe that by (2.5)(a(z,F2) a(z,F1)

) : (F2 F1) c(1+ |F2|2 + |F1|2) p(z)22 |F2 F1|,for any F1,F2 RnN , where the constant c depends only on L, 1 and n. From this inequality it follows that ifp1 2 then:

Q0

|Dw |p1 dz cQ0

(1+ |Dw |2 + |Du |2

) p122 |Dw |2 dz cI ( ). (4.14)

If p1 < 2, instead, we must argue in a different manner, using Hlder inequality:Q0

|Dw |p1 dz=Q0

(|Dw |2(1+ |Dw |2 + |Du |2) p122 ) p12 (1+ |Dw |2 + |Du |2) 2p12 p12 dz( Q0

(1+ |Dw |2 + |Du |2

) p122 |Dw |2 dz

) p12( Q0

(1+ |Dw |2 + |Du |2

) p12 dz

) 2p12

c[I ( )

] p12 0 (4.15)

as 0. So in any case we have that Dw 0 in Lp1(Q0;Rn). Finally, keeping into account that u ustrongly in L1loc(QT ;RN) and Du Du strongly in Lp1loc(QT ;RnN), the statement follows using (4.13). Proof of Lemma 4.2. From (4.6) it is possible to derive the following identity:

12w ( , t)2L2(B0) +

tt1

B0

(a(z,Dv ) a(z,Du )) :Dw dz+ ( )t

t1

B0

(1+ |Dv |2

) q22 |Dv |2 dz

( )t

t1

B0

(1+ |Dv |2

) q22 Dv :Du dz

+t

t1

B0

((a(z,Du)

) a(z,Du)) :Dw dz

+t

t1

B0

(a(z,Du) a(z,Du )

) :Dw dzfor all t (t1, t2) and, therefore:

I ( ) c(I1( )+ I2( )+ I3( )

), (4.16)where:

E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560 43I1( ) := ( )Q0

(1+ |Dv |2

) q22 |Dv Du |dz,

I2( ) :=Q0

((a(z,Du)) a(z,Du))Dw |dz,

I3( ) :=Q0

a(z,Du) a(z,Du )|Dw |dz.By Hlder inequality we have:

I1( )(( )

Q0

1+ |Dv |q dz) q1

q(( )

Q0

1+ |Du |q dz)

and therefore I1( ) 0, by (4.2). Now, with (0,1) we use Youngs inequality in (4.16) to deduce the estimate:I2( )+ I3( )

Q0

|Dw |p(z) + |Dv |p(z) dz+ c(I4 + I5), (4.17)

where this time we have set:

I4( ) :=Q0

(a(z,Du)) a(z,Du) p(z)p(z)1 dz,

I5( ) :=Q0

a(z,Du) a(z,Du ) p(z)p(z)1 dz.Observe that, by the results in Appendix A, it follows that I4( )+ I5( ) 0 when 0. Finally the statementfollows looking at (4.17) and letting first 0 and then letting 0, keeping into account the boundednessresult in (4.8) and the choice (4.2) of ( ). Proof of Theorem 4.1. We take a parabolic cylinder Q(z0,2 ) Q0 (z0 (x0, t0)) as in Theorem 3.1; thenwe apply the approximation procedure described above to Q0. At this stage we can use the a priori estimate ofTheorem 3.1 for each function v . Indeed by (4.5) it follows that v solves the problem

Q0

v tw+ a(z,Dv ) :Dwdz= 0 w C0(Q0;RN

)which is of the type in (3.1). Therefore the a priori estimate in (3.7) is valid for v with a constant c that does notdepend on . Using the result of Lemma 4.3 and the lower semicontinuity of integrals it follows:

Q(z0,/2)

|Du|q0 dz lim inf

Q(z0,/2)

|Dv |q0 dz

c n+2 + limc

(1

)s1( Q(z0, )

|Dv |p1 dz)s2

= c n+2 + c(

1)s1(

|Du|p1 dz)s2Q(z0, )

44 E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560and the proof of the higher integrability is finished; now we turn to the higher differentiability part. The estimateof Theorem 3.2 holds uniformly with respect to . From this fact it immediately follows that the sequence {v }is bounded in

Lp(t0 2/16, t0;W 2,p

(B(x0, /4);RN

)), p :=min{p1,2}

since Dv Du strongly in Lp1(Q(z0, /4);RN), up to (not relabelled) subsequences we have that D2v D2u weakly in Lp(Q0;Rn2N). Finally, by well known lower semicontinuity theorems (in fact, Dv convergesstrongly, D2v converges weakly and the integrand is convex with respect to the D2v argument) and the estimateof Theorem 3.2 it follows:

Q(z0,/4)

(1+ |Du|2) p(z)22 |D2u|2 dz lim inf

Q(z0,/4)

(1+ |Dv |2

) p(z)22 |D2v |2 dz

c(n, 1, 2,L, s1, )( Q(z0, )

1+ |Du|p1 dz)s2

.

From this (4.1)1 and (4.1)3 follow via a standard covering argument.

5. A decay estimate

In this section we consider an Energy Solution u to the system (2.7) in Q0, under the assumptions described inSection 2, where Q0 B(x0,R) (t1, t2) QT and Q0 satisfies all the requirements described in Section 3, inparticular those in Theorem 3.1. Therefore we shall suppose that the numbers p1,p2, q, q0 and are exactly asthe ones described in (3.3)(3.5) and (3.6). In this situation we are able to apply Theorem 4.1 (which is actually aconsequence of Theorem 3.1) from Section 4 to gain both higher integrability and higher differentiability for thespatial gradient of u. Finally we perform a last reduction, since the result of Theorem 4.1 is only local with respectto Q0: up to passing to subcylinders Q Q0 and proving Propositions 5.1 and 5.2 in Q, we shall suppose that(4.1) hold globally in Q0. Summarizing, we get, essentially by Theorem 4.1, that:

Q0

|Du|q0 dz 2.

E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560 45Let us recall that a regular point z QT for the function u is a point such that the gradient Du is Hldercontinuous for any exponent 1 < (with respect to the standard parabolic metric) in a neighborhood of it. A pointwhich is not regular will be called a non-regular point. The following proposition allows to characterize the regularpoints via the excess functional U(z0,R). In the following the exponent is the one introduced in (2.1).

Proposition 5.1. Assume that conditions of Section 3 are satisfied with respect to a cylinder Q0 := B(x0,R) (t1, t2). Let z0 Q0 such that there exists a sequence of radii {Rk}kN, Rk 0, such that

limkU(z0,Rk)= 0, sup

kN

(Du)z0,Rk

46 E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560and the new functions vk , defined on Q1 according to:

vk(l) := u(z)Ak(x xk) (u)zk,Rk

kRk.

With such a notation, the following relations easily follow:

U(zk, Rk)= kV k( ), V k( ) := V k1 ( )+ V k2 ( ), (5.8)where:

V k1 ( ) :=1

(Q()

vk (Dvk) y (vk) 2 dl)1/2 +( Q()

Dvk (Dvk) 2 dl)1/2,V k2 ( ) :=

q21k

(Q()

Dvk (Dvk) q dl)1/2.Consequently (vk)1 = 0 and (Dvk)1 = 0. Moreover, from the definition of k :

1= V k(1)+ R1k

k(Q1

|vk |2 dl)1/2

+(Q1

|Dvk |2 dl)1/2

+ q21k

(Q1

|Dvk |q dl)1/2

(5.9)

while (5.7) turns to:V k( ) > 2c1(M). (5.10)

Regarding the functions vk , it follows that each one is an Energy Solution to the system:Q1

(vksw+ k :Dyw)dl = 0 w C0(Q1;RN

), (5.11)

where we set

k := 1k[a(xk +Rky, tk +R2k s,Ak + kDvk) a(xk, tk,Ak)

] (5.12)According to (5.9), since q > 2, up to not relabelled subsequences:

vk v in L2(Q1;RN

)Dvk Dv in L2

(Q1;RnN

)

q2q

k vk 0 in Lq

(Q1;RN

)

q2q

k Dvk 0 in Lq

(Q1;RnN

)zk (xk, tk) (x, t)=: z in Rn RAk A in RnNkDv

k 0 a.e. in Q1q2k |Dvk |q 0 a.e. in Q1.

(5.13)

Without loss generality we can assume that z Q1; (let us just comment that by (5.13)2 it follows that, up tosubsequences, (q2)/qk |Dvk | 0 a.e. and therefore (5.13)8 follows ).

Using standard lower semicontinuity results for integral functionals, it follows:(v)1 = 0, (Dv)1 = 0, |A|M, V (1) 1, (5.14)

E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560 47where for any 0 < 1

V () := 1

(Q

v (Dv) y (v) 2 dl)1/2 +( Q

Dv (Dv) 2 dl)1/2.Now we are going to let k pass to the limit in the systems (5.11); in the next step we will prove that the limitfunction v is a solution to the following linear parabolic system with constant coefficients:

Q1

vsw+ aF

(z,A)Dv :Dwdl = 0 w C0(Q1;RN

). (5.15)

By (2.4) and (2.5) it follows that the following ellipticity conditions are satisfied:

c1(M,L)||2 aF

(z,A) c(M,L)||2,

for any RnN , where c(M,L) c(M) is a bounded constant essentially depending on M only. Therefore,taking into account the standard regularity theory for linear parabolic systems with constant coefficients (see [7]),we conclude that the function v is smooth in Q1 and that there is a constant c1 c1(M), which from this momentwe choose to appear in (5.7), such that:

V () c1(M)V (1) c1(M), 0 < < 1. (5.16)Now if we prove that

V h() V () (5.17)this, via (5.10), implies that

V () 2c1(M),which contradicts (5.16), and the proof will be finished.

We now prove (5.15), while (5.17) will be proved in step 3.Step 2: proof of (5.15). Let us first derive some preliminary estimates. We split the vector field in (5.12) as

follows:

k := k1 + k2 ,where:

h1 := 1k[a(xk +Rky, tk +R2k s,Ak + kDvk) a(xk, tk,Ak + kDvk)

],

k2 := 1k[a(zk,Ak + kDvk) a(zk,Ak)

].

We obtain some estimates for these objects that will be useful below. Using (2.6), (3.5), (5.5) and (5.9) we have:

|k1 | c1k Rk(1+ |Ak + kDvk |

) p212 log

(2+ |Ak + kDvk |

) c(M,)R1k

(1+ |kDvk |q1

) c(M,)R1k

(1+

2(q1)q

k (q2) q1

q

k |Dvk |q1) (5.18)

so that, by (5.9),|k1 |

qq1 dl 0. (5.19)Q1

48 E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560For k2 we have (using a standard algebraic lemma to treat the second integral):

|k2 | =

10

a

F(zk,Ak + kDvk)Dvk d

c

10

(1+ |Ak + kDvk |2

) p(zk )22 d |Dvk |

c(M)(1+ |Dvk | + q2k |Dvk |q1

). (5.20)

From the last inequality and (5.18) we also find:|k| c(M)[1+ |Dvk | + q2k |Dvk |q1]

and, with (5.9) and using the fact that q > 2, we get|k| qq1 c(M). (5.21)

In order to get (5.15), it is clear that it suffices proving that:

Ik :=Q1

k :DwdlQ1

a

F(z,A)Dv :Dwdl =: I,

for any w C0 (Q1;RN). To this aim we write:

Ik =Q1

k1 :Dwdl +Q1

(k2

a

F(zk,Ak)Dvk

):Dwdl

+Q1

a

F(zk,Ak)Dvk :Dwdl =: I 1k + I 2k + I 3k .

By (5.19) and (5.13) it follows that I 1k 0 and I 3k I. It remains to prove that I 2k 0. This can be rapidlyseen as follows: fix 0 < < 1 and determine, by Egorov theorem and (5.13)7, AQ1 such that |Q1 \A|< andkDv

k 0 uniformly in A; then break I 2k in two pieces according to:

I 2k =

Q1\A(. . .) dl+

A

(. . .) dl =: I 4k + I 5k .

Using the uniform continuity of Da on bounded subsets we conclude that I 5k 0 (since Dvk is bounded in L1and Dw is smooth); in order to treat I 4k it suffices to observe that, by (5.20) and (5.21):

|I 4k | c(M)DwL

Q1\A

(1+ |Dvk | + q2k |Dvk |q1

)dl

c(M)[|Q1 \A| 12 +

q2q

k |Q1 \A|1q

( Q1

q2k |Dvk |q dl

)1 1q]

and keeping into account (5.9) and that q > 2, we achieve that I 4k 0 letting first k+ and then 0.

Step 3: proof of (5.17).

E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560 49In the following we shall extract a compactness information from the existence of second spatial derivatives. Inorder to do so we shall differentiate the system.

Warning. We warn the reader that the following calculations will be, at the beginning, a bit sloppy. Indeed,in order to proceed in a rigorous way, we should use the same procedure (smoothing plus difference quotientarguments) introduced for the proof of Lemma 3.1. This would take too much time, yielding only technical changes,so we shall proceed formally, differentiating the systems rather than taking difference quotients and assuming theexistence of Dxa rather than (2.6), which is just Lipschitz continuity, therefore the existence a.e. of Dxa. Moreprecisely we assume that:Dxa(z,F ) L(1+ |F |2) p(z)12 log(2+ |F |). (5.22)This additional assumption can be easily removed with extra technical-routine efforts, either, as mentionedbefore, following the procedure of Lemma 3.1 or by a simple approximation method based on mollification ofa with respect to x . The only difference with respect to Lemma 3.1 is that since now we are dealing directlywith the original solution, which is, according to definition only a L2(0, T ;L2loc(;RN)) function (and notC0((0, T );L2loc(;RN)) as the approximating solutions of Lemma 3.1), then, where previously we used sup,this will be replaced by ess sup. We shall proceed this way for the sake of brevity.

Accordingly, we consider the following differentiated system:Q1

[vk,isw k,i :Dw

]dl = 0 w C0

(Q1;RN

). (5.23)

In the previous formula we take the test function w = 2vk,i where is as in the proof of Lemma (3.1), W 1,0 ((1,0)) such that t(t) 0 and (t)= 0 in a small right neighborhood of 1, while W 1,q0 (B1)has the property that |D| 1; they are usual cut-off functions to be chosen later. As a result of manipulationssimilar to the ones at the beginning of Lemma 3.1 we obtain:

ess sup1

50 E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560In (5.24) we are going to use the growth and ellipticity conditions stated in the formulas (2.4), (2.5), (5.22), and weestimate, as in (3.12):

log(1+ |Ak + kDvk |2

) c()

(1+ |Ak + kDvk |2

)/4. (5.25)

Moreover we shall denote p(xk +Rky, tk +R2k s) p(). Using Cauchy-Schwartz and Young inequalities, (5.25)and the fact that Rk/k stays bounded from above, we obtain, with (0,1):

ess sup1

E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560 51Then, using also Sobolev embedding theorem we gain the inequalities:B

H 2 dy c( )[(

B

|DH | 2nn+2 dy) n+2

n +(

B

H dy

)2](5.29)

while using Hlder inequality we get:( B

H dy

)2 c

( B

|Dvk |p1 dy)(

B

|Dvk | 4n dy). (5.30)

Since 4/n 2, estimate (5.26) leads to the following bound, which is uniform with respect to s (( 1+2 )2,0):( B

H(y, s) dy

)2 c(M)c( )

B

Dvk(y, s)p1 dy, (5.31)where the constant in the previous inequality is again independent of k N. Summarizing (5.29)(5.31) and usingin a standard way Hlder inequality, we obtain:

B

H 2 dy c(M)c( ){

(q2)q0q

k

[B

(1+ |Ak + kDvk |2

) p122 |D2vk |2 dy

]

[ B

(1+ |Ak + kDvk |2

) 2p12

n2 |Dvk |0n dy

] 2n +

B

|Dvk |p1 dy}

c(M)c( ){B

P k dy

[B

(1+ |Ak + kDvk |2

) 2p12

n2 |Dvk |0n |Dvk | dy

] 2n

+B

|Dvk |p1 dy}, (5.32)

where we have set := n(q0 q)/q > 0. Now we check that: < 2, < 0n. (5.33)

Indeed, as for the first of (5.33), we notice that if p1 < 2 then, since q > 2:

n

(q0q 1

)< n

(q02 1

)(3.4)= n

2

(p1 + 4

n 2

)< 2.

On the other hand if p1 2 then:

n

(q0q 1

)< n

(q0p1

1)= np1

4n= 4p1 2.

As for the second inequality in (5.33), we observe that: < 0n q0

q 1 < q0

2 1 2 < q,

which is true.By (5.33), using Hlder inequality in (5.32), we get the following estimate, again uniform in s (((1 + )/2)2,0):

52 E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560B

H 2 dy c(M)c( ){B

P k dy

(B

|Dvk |2 dy)

n

[ B

((1+ |Ak + kDvk |2

) 2p12

n2 |kDvk |0n

) 22 dy

] 2n +

B

|Dvh|p1 dy}. (5.34)

We are finally ready to prove (5.27). We first observe that (2 p1)n/2+ 0n= 2; therefore integrating (5.34) on(((1+ )/2)2,0) we get:

Q

(

q2q

k |Dvk |)q0 dl =

Q

H 2 dl

(5.26) c(M)c( )

[Q

Pk dl +Q

1+ q2k |Dvk |q dl]

(5.13),(5.26) c(M)c( )

and (5.27) is completely proved.Now we observe that if we let fk := q2k |Dvk |q then (5.13)8 implies that fk 0 a.e. while (5.27) implies that

{fk} is bounded in Lq0/q(Q) with q0/q > 1; therefore it follows thatq2k

Q

|Dvk |q dl 0,

so

V k2 ( ) 0. (5.35)In order to fully prove (5.17) it remains to prove that

V k1 ( ) V (). (5.36)To this aim, first we assume that p1 2. Then it follows from (5.26) that:

Q

|D2vk |2 dl c(M)c( ). (5.37)

In the second case we assume p1 < 2 and, again by (5.26), we have the bound:Q

|D2vk |p1 dl (

Q

(1+ |Ak + kDvk |2

) p122 |D2vk |2 dl

) p12

(

Q

(1+ |Ak + kDvk |2

) p12 dl

) 2p12 c(M)c( ) (5.38)

with the constants independent of k. Using the multiplicative inequality we gain:( |Dvk |0 dl

) 10 c

( |D2vk|p1 dl +

|Dvk |p1 dl

) p1

ess sup(

|Dvk |2 dy) (1)

2,Q Q Qs(((1+ )/2)2,0)

B

E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560 53where

= n/(n+ 2), 0 = p1(n+ 2)/n > 2, (5.39)the last inequality following directly by the lower bound in (2.2). By (5.26) and (5.38) we obtain yet another bound:

Q

|Dvk |0 dl c(M)c( ). (5.40)

Now, in order to gain compactness, we are going to estimate the time derivatives. To do this, let w C0 (QT ;RN);we first estimate

Q

k :Dwdl

0((1+ )/2)2

Dw( , s)L(B)

k( , s)L1(B) ds

ckL

qq1 (Q)

( 0((1+ )/2)2

Dw(, s)qL(B)

) 1q

(5.21) c(M)

( 0((1+ )/2)2

Dw( , s)qL(B)

) 1q

. (5.41)

By the usual Sobolev inequality:

DwL(B) cDwW 2,l0 (B), l > (n+ 2)/2,we deduce from (5.41) and Eq. (5.11), that:

{svk}k is bounded in Lq

q1(((1+ )/2)2,0; (W 2,l0 (Q;RN))). (5.42)

Now, let us consider again separately the previous cases. If p1 2 then from (5.37) and an AubinLions typecompactness result (see for instance [36], Theorem 2.1, chapter 3) it follows that, up to not relabelled subsequences,

Dvk Dv in L2(Q;RnN ). (5.43)If p1 < 2, from (5.38) and again an AubinLions type result, we first obtain that, again up to subsequences,

Dvk Dv in Lp1(Q;RnN)and then by (5.40) we get (5.43), again interpolating L2 between Lp1 and L0 , since by (5.39) it follows that0 > 2.

In a similar (actually, much easier) way it follows that:vk v in L2(Q;RnN ). (5.44)

We are ready to finish: indeed (5.43), (5.44) imply (5.36), that togehter with (5.35) completely establish (5.17) andthe proof of Lemma 5.1 is now complete.

6. The main result proved

In this section we are going to prove Theorem 2.1. We explain the way we proceed. All the results developed

in Sections 35 have been obtained in a particular situation i.e.: in the basic cylinder Q0 a special bound (the one

54 E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560described before the statement of Theorem 3.1) for the oscillations of the function p(z), measured in terms of thesize p2 p1, is valid. In order to use these results for the proof of Theorem 2.1, the first thing we do is reducing tothe previous situation: up to passing to a subcylinder compactly contained in QT , we shall cover QT with a finitenumber of small subcylinders in which the oscillations of the exponent function p(z) are small enough to meet theconditions of Theorem 3.1. Then we proceed working in each of these subcylinders, blowing up the solution andthereby proving partial regularity of the solution in each subcylinder. The final result then follows by a coveringargument.

Proof of Theorem 2.1. We take an increasing sequence of cylinders Qh QT such that Qh QT . If we provethat the Du is partially regular in each of the Qh we are clearly done, since in each of them the set of non-regularpoints will be negligible (see also the argument below). Therefore we can reduce ourself to prove the statementof Theorem 2.1 in a single Qh; such a cylinder will denoted by Q. We fix (n, 1) > 0 according to therestrictions proposed in Section 3, right-hand side inequalities of (3.3), then we determine (p(z),) suchthat:

|z1 z2| p(z1) p(z2)< /2. (6.1)

Then we take a finite covering of Q, {Q(k)}, with Q(k) QT , such that, by (6.1):diamQ(k) < , oscQ(k) p(z) < /2 k. (6.2)

Now, each Q(k) is small enough to meet the conditions imposed in Sections 35, that is the ones in the left-handside inequalities of (3.3), and we can apply Proposition 5.2 to each Q(k); therefore there exists an open subsetQ0(k) Q(k) such that Du is Hlder continuous in Q0(k) with any exponent 1 < and |Q(k) \Q0(k)| = 0. Then weset

Q0 :=k

Q0(k)

and we observe that Du is Hlder continuous in Q0 (which is obviously open) with any exponent 1 < and|Q \ Q0| = 0, thereby proving Theorem 2.1.

7. Estimates for the singular set

For the sake of simplicity, in this section we shall assume (5.22).Remarks on the Integrability of Solutions. Here we are going to observe that given a parabolic cylinder Q0

as considered in Sections 35, and therefore with the bounds in (3.3) being in force, the function |D2u| + |tu|is in certain Lebesgue space L0loc(Q0). Let us recall that by the results in Section 4 (and again up to passing tosubcylinders as in Section 5) we have:

Q0

|Du|q0 dz 1. Then we let:

0 :=1 if p2 < 2,2 if p1 2,min{1,2} if p1 < 2 p2.

E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560 55In any case 0 2. Now we wish to show that (7.1) implies:|D2u| + |tu| L0(Q0). (7.2)

First we remark that:

|D2u| {L1loc(Q0) if p1 < 2,

L2loc(Q0) if p1 2.(7.3)

Indeed, if p1 2 then, obviously:

|D2u|2 (1+ |Du|2) p(z)22 |D2u|2.If p1 < 2 then also 1 < 2 and Young inequality yields:

|D2u|1 = [(1+ |Du|2) p124 |D2u|(1+ |Du|2) 2p14 ]1 c

[(1+ |Du|2) p122 |D2u|2 + (1+ |Du|2) q02 ]

since q0 = p1 + 4/n= (2 p1)1/(21) and (7.3) follows by (7.1). Now we show that:

|tu| {L2loc(QT ) if p2 2,L2loc(QT ) if p2 > 2.

(7.4)

In order to establish (7.4) we use the equation:ut = diva(z,Du) (7.5)

and the fact that, via Theorem 3.2 and (7.1), it can be differentiated (locally) with respect to the space variables.Using (2.4), (2.6) and (3.12), we obtain:

|tu| c[(

1+ |Du|2) p(z)22 |D2u| + (1+ |Du|2) p212 log(1+ |Du|2)] c

[(1+ |Du|2) p(z)22 |D2u| + (1+ |Du|2) q12 ] (7.6)

where q has been defined in (3.5). Now, consider the case p2 2. Then since p(z) 2 it also follows that:

|tu|2 c[(

1+ |Du|2) p(z)22 |D2u|2 + 1+ |Du|2(q1)].We must check that 2(q 1) q0 = p1 + 4/n; recall that q = 2+ if p2 2 ((3.6)) therefore, using the fact that2n/(n+ 2) < p1 it suffices to show that 1 + n/(n+ 2)+ 2/n which follows from the upper bound on in(3.3). So, by (7.1), it follows that |tu| L2loc in the case p2 2. Next, we consider the case p2 > 2. According to(7.6) we have, using Young inequality (observe that when p2 > 2 then 2 < 2):

|tu|2 c[(

1+ |Du|2) (p(z)2)24 ((1+ |Du|2) p(z)22 |D2u|2)22 + 1+ |Du|(q1)2] c

[(1+ |Du|2) p(z)22 |D2u|2 + (1+ |Du|) (p(z)2)222 + |Du|(q1)2]. (7.7)

In order to use again (7.1) and making all the terms appearing in the right-hand side of (7.7) integrable, we checkthat

(p(z) 2)222

(p2 2)222 q0, (q 1)2 q0.

The first inequality follows from the definition of 2. As for the second one:2q0(q 1)2 = (p2 + 1)p2 2+ q0 q0, (7.8)

56 E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560which is equivalent to p2 p1 + 2 4/n that follows immediately from (3.3); therefore all the quantitiesappearing on the right-hand side of (7.7) are integrable by (7.1). In turn, by (7.7) we have |tu| L2loc(QT ) therebycompleting the proof of (7.4). Finally (7.2) follows by (7.3) and (7.4).

We are ready to give a first result for the estimate of the singular set, which is local, in the sense that it gives anestimate on the singular set when we just look at Q0:

Theorem 7.1. Under the previous assumptions on the cylinder Q0 suppose that:

q 0(n+ 2)n+ 20 . (7.9)

Then, with the notation of Proposition 5.2 it follows that for any > 0Pn+20+ (Q0 \Q0)= 0.

Proof. From Proposition 5.1 it follows that Q0 \ (Q0T Q0)0 1 2 where

0 :={z Q0: lim sup

0

(Du) =+},1 :=

{z Q0: lim inf

0 U1(z, ) > 0},

2 :={z Q0: lim inf

0 U(z,) > 0},

therefore it suffices to prove that Pn+20+ (i)= 0 for i {0,1,2} for any > 0. For 2 2, we observe that(7.9) implies the following inequality:

U(z0,R)R(

Q(z0,R)

|D2u|0 + |tu|0 dz+) 1

0dz

+[R

(

Q(z0,R)

|D2u|0 + |tu|0 dz+) 1

0] q

2. (7.10)

Nest we recall that if S Q0 denotes the set of points z0 Q0 such that:lim sup0

R(0n2)

Q(z0,R)

|D2u|0 + |tu|0 dz > 0,

from a well known measure density result (first developed in the context of elliptic systems, see [26] for commentsand extensions) adapted to the parabolic geometry of rescaled cylinders it follows Pn+20(S)= 0; therefore, by(7.9), (7.10) and the same reduction lemma it follows that Pn+20(1) = Pn+20(2) = 0. In a similar way,via the same measure density arguments it is possible to prove that Pn+20+ (0)= 0.

As stated before, the previous theorem provides an estimate which is local, that is, the size of the singular setdepends on the peculiar choice of the cylinder Q0, via the number 0. In order to get a global estimate for thesingular set we find uniform bounds for 0. This is the aim of the next theorem.

Remark 4. Before going on with the proof we make some remarks on the possible choice of the number . It isclear from the statement of Theorem 3.1 that the constant c in (3.7) blows up when 0. Anyway the higher

integrability exponent q0, which is the one of the gradient, does not change directly: it remains linked to p1 via the

E. Acerbi et al. / Ann. I. H. Poincar AN 21 (2004) 2560 57same relation: q0 = p1 + 4/n. The only way q0 may depend on appears in the proof of Theorem 2.1; there, p1changes with Q(k), which, in that scheme, depends on .

In the following we shall assume, without loss of generality, that:2n

n+ 2 < 1 < 2. (7.11)

Theorem 7.2. Define > 1 as follows:

:=min{

2(1 + 4/n)2+ 4/n ,

2(2 + 4/n)22 2+ 4/n

}(7.11) 2, (7.12)

and suppose that

2 0Pn+2+ (QT \Q0T )= 0. (7.14)

Remark 5. Observe that when 1 = 2 = p(z) = 2 then = 2, as for the usual estimates concerning parabolicsystems with linear growth (see, for instance, [7]).

Proof of Theorem 7.2. The proof will be achieved using Theorem 7.1. We fix > 0 and we observe that it sufficesto consider so small that (7.13) continues to hold with /2 in the place of (observe that the function inthe right-hand side of (7.13) is increasing with respect to ). Then we can find > 0 satisfying the right-handinequalities in (3.3) and such that:

2 + < ( /2)(n+ 2)n+ 2 ( /2) and

2(2 + 4/n)22 2+ 4/n+ > /2. (7.15)

Now, with respect to such an , we find a covering family of cylinders {Q(k)} as in the proof of Theorem 2.1,conditions (6.1), (6.2); our aim is to apply the estimate of the singular set stated in Theorem 7.1 in each cylinderQ(k), with respect to the constant 0 0(Q(k)) (note that 0 depends on the cylinder Q(k)). Now let us provethat, in each Q(k):

/21 if p2(Q(k)) < 22 if p1(Q(k)) 2min{1,2} 0 if p1(Q(k)) < 2 p2(Q(k))

= 0(Q(k)). (7.16)

We prove that the first two inequalities in (7.16) are true for all values of p1 and p2, therefore the third one follows.The first inequality is trivial since 1 always, it remains to check the second:

2 = 2q0p2 2+ q0

(3.3) 2q0

p1 + 2+ q0 =2q0

2q0 2 4/n+ = 1+ 2+ 4/n

2q0 2 4/n+ 1+2+ 4/n

2p2 2+ 4/n+ 1+ 2+ 4/n

22 2+ 4/n+ =2(2 + 4/n)

22 2+ 4/n+ (7.15) /2.

It follows that:(7.15) ( /2)(n+ 2) (7.16) 0(n+ 2)q 2 +