Everywhere regularity of solutions to a class of strongly coupled degenerate parabolic systems

576–586PARTIAL REGULARITY OF SOLUTIONS TO A CLASS OF STRONGLY COUPLED DEGENERATE PARABOLIC SYSTEMS Dung Le Department of Applied Mathematics University of Texas at San Antonio 6900 North

Supplement Volume 2005 pp 576–586

PARTIAL REGULARITY OF SOLUTIONS TO A CLASS OF STRONGLY COUPLED DEGENERATE PARABOLIC SYSTEMS

Dung Le Department of Applied Mathematics University of Texas at San Antonio

6900 North Loop 1604 West San Antonio, TX 78249, USA Abstract Using the method of heat approximation, we will establish partial reg-ularity results for bounded weak solutions to certain strongly coupled degenerate parabolic systems.

1 Introduction The aim of this paper is to study the partial regularity for weak solutions of nonlinear parabolic systems of the form

ut= div(a(x, t, u)Du) + f (x, t, u, Du), (1.1)

in a domain Q = Ω × (0, T ) ⊂ Rn+1, with Ω being an open subset of Rn, n ≥ 1 The vector valued functions u, f take values in Rm, m ≥ 1 Du denotes the spatial derivative of u Here, a(x, t, u) = (Aαβij ) is a matrix in Hom(Rnm, Rnm)

A weak solution u to (1.1) is a function u ∈ W21,0(Q, Rm) such that

ZZ

Q [−uφt+ a(x, t, u)DuDφ] dz =

ZZ

Q

f (x, t, u, Du)φ dz

for all φ ∈ C1

c(Q, Rm) Here, we write dz = dxdt

It has been known that, in the case of systems of equations (i.e m > 1), one cannot expect that bounded weak solutions of (1.1) will be H¨older continuous ev-erywhere (see [7]) Partial regularity for (1.1), when a is regularly elliptic, was considered by Giaquinta and Struwe in [5]

In this paper, we study the partial regularity for (1.1) when certain degeneracy

is present In particular, we consider the case when a ceases to be regular elliptic at certain values of u Strongly coupled systems of porous medium type are included here

For the sake of simplicity, we will only consider the homogeneous case f ≡ 0, and assume that a(x, t, u) depends only on u The nonhomogeneous case can be treated similarly modulo minor modifications In fact, we will assume the following structural conditions on (1.1)

(A.1): There exists a C1 map g : Rm → Rm, with Φ(u) = Dug(u), such that for some positive constants λ, Λ > 0 there hold

a(u)Du · Du ≥ λ|Dg(u)|2, |a(u)Du| ≤ Λ|Φ(u)||Dg(u)|

2000 Mathematics Subject Classification Primary: 35K65; Secondary: 35B65.

Key words and phrases Parabolic systems, Degenerate systems, Partial H¨ older regularity The author is partially supported by NSF Grant #DMS0305219, Applied Mathematics Program.

(A.2): (Degeneracy condition) Φ(0) = 0 There exist positive constants C1, C2

such that

C1(|Φ(u)| + |Φ(v)|)|u − v| ≤ |g(u) − g(v)| ≤ C2(|Φ(u)| + |Φ(v)|)|u − v| (A.3): (Comparability condition) For any β ∈ (0, 1), there exist constants

C1(β), C2(β) such that if u, v ∈ Rmand β|u| ≤ |v| ≤ |u|, then C1(β)|Φ(u)| ≤

|Φ(v)| ≤ C2(β)|Φ(u)|

(A.4): (Continuity condition) Φ(u) is invertible for u 6= 0 The map a(u)Φ(u)−1

is continuous on Rm\{0} Moreover, there exists a monotone nondecreasing concave function ω : [0, ∞) → [0, ∞) such that ω(0) = 0, ω is continuous at

0, and

|a(v)Φ(v)−1− a(u)Φ(u)−1| ≤ (|Φ(u)| + |Φ(v)|)ω(|u − v|2), (1.2)

|Φ(u) − Φ(v)| ≤ (|Φ(u)| + |Φ(v)|)ω(|u − v|2) (1.3) for all u, v ∈ Rm

In (A.4), we use ω to quantify our continuity hypothesis on a(u)Φ−1(u) We would like to remark that the existence of the function ω also comes from the con-tinuity of a(u)Φ−1(u) and Φ(u) ( see [4, page 169]) To avoid certain technicalities

in the presentation of our proof, we assume here (1.3) We will see at the end of this paper that it is not necessary

Systems that satisfy the above structural conditions include the porous medium type systems: a behaves like certain powers of the norm |u| In this case, one may consider g(u) = |u|α/2u for some α > 0

As we mentioned before, partial regularity results for the regular case, when g

is the identity map, were established in [5] There are three essential ingredients in their proof The first element is an inequality of Caccioppoli, or reverse-Poincar´e, type The second element of the proof is Campanato’s decay estimate for the averaged mean square deviation of solutions to systems of equations with constant coefficients Finally, the technique of freezing the coefficients was used One then had to control the deviation of the original solution of (1.1) from that of system of constant coefficients In this step, a variant version of the famous Gehring reverse H¨older inequality played a crucial role

Given the above assumptions, Cacciopoli’s inequality is not available More seriously, the crucial Gehring reverse inequality, and thus higher integrability of

Du, does not seem to hold anymore However, we are able to prove the following partial regularity result

Theorem 1 Let u be a bounded weak solution to (1.1) Set

Reg(u) = {(x, t) ∈ Ω × (0, T ) : u is H¨older continuous in a neighborhood of (x, t)} and Sing(u) = Ω × (0, T )\Reg(u) Then Sing(u) ⊆ Σ1SΣ

2, where

Σ1= {(x, t) ∈ Ω × (0, T ) : lim inf

R→0 |(u)Q R (x,t)| = 0},

Σ2= {(x, t) ∈ Ω × (0, T ) : lim inf

R→0

ZZ

Q |u − (u)Q R (x,t)|2 dz > 0}

Here, for each R > 0, QR(x, t) = BR(x) × (t − R2, t) and (u)QR(x,t)=

ZZ

QR(x,t)

u dz Moreover, Hn(Σ2) = 0, where Hn is the n-dimensional Hausdorff measure

We would like to describe briefly our approach here In order to deal with this degenerate situation, one needs to find another way to avoid the unavailable Gehring lemma Recently, the method of A-harmonic approximation has been successfully used by Duzaar et al [2, 3] to treat regular elliptic systems and p-Laplacian systems One of the advantages of this method is that it is more elementary and avoids the technical difficulties associated with the use of the Gehring lemma, the missing stone in our case Inspired by this method, we introduce in Section 2 its parabolic variance: the heat approximation Basically, the point is to show that a function which is approximately a solution to a heat equation in a parabolic cylinder will be

L2-close to some heat solution in a smaller cylinder We also present a parabolic version of Giaquinta’s lemma [2, Lemma A.1]

Finally, we prove Theorem 1 in Section 3 We apply a scaling argument to reflect the degeneracy of (1.1) and make use of the above heat approximation lemma in each scaled cylinders We start with the key assumption that the solution u is not averagely (in a scaled cylinder) too close to the singular point u = 0 We then derive a decay estimate for the averaged mean square deviation of g(u) This estimate allows us to show that u is still averagely far away from the singular point in a smaller cylinder, and the argument then can be repeated Moreover, we then obtain the decay estimates for the averaged mean square deviation of u in a sequence of scaled and nested cylinders It is then standard to conclude that u is locally H¨older continuous from these estimates

2 A-heat approximation In this section, we present the parabolic version of the A-harmonic approximation lemma formulated in [2] The proof is a straightforward modification of that for the elliptic case modulo some careful choice of the pertinent norms We present some details here for the sake of completeness

Fix (x0, t0) ∈ Rn+1 For each R > 0, we consider the cylinder QR = BR(x0) × (t0− R2, t0) Let V (QR) be the space of functions g ∈ W21,0(QR, Rm) with norm

kgkV (QR)= sup

t

R−n−2

Z

BR

g2(x, t) dx +

ZZ

QR |Dg|2dz (2.4) For A ∈ Bil(Hom(Rnm, Rnm)) and φ ∈ C1(QR) = C1(QR, Rm), we define

LA(g, φ, QR) =

ZZ

QR [A(Dg, Dφ) − gφt] dz, and

∆(LA, g, QR) = sup{|LA(g, φ, QR)| : φ ∈ C1

c(QR), sup

Q R

|Dφ| ≤ 1

R, |φt| ≤ 1

R2}

We shall consider the set of A-heat functions

H(A, QR) = {H ∈ V (QR) : LA(H, φ, QR) = 0, ∀φ ∈ Cc1(QR)}

The following A-heat approximation lemma is the parabolic version of [2, Lemma 2.1]

Lemma 2.1 Consider fixed λ, Λ > 0 For any given ε > 0 there exists δ ∈ (0, 1] that depends on λ, Λ, ε and has the following property:

for any A ∈ Bil(Hom(Rnm, Rnm)) satisfying

A(u, u) ≥ λ|u|2, |A(u, v)| ≤ Λ|u||v| for all u, v ∈ Rnm, (2.5) for any g ∈ V (Qρ) satisfying

kgkV (Q ρ )≤ 1, and |∆(LA, g, Qρ)| ≤ δ, (2.6) then there exists v ∈ H(A, Qρ) such that

ρ−2 ZZ

Q ρ

|v − g|2 dz ≤ ε and

ZZ

Q ρ

Proof We assume first that ρ = 1 If the conclusion is false, we can find ε > 0, {Ak} each satisfying (2.5) and {gk} ⊂ V (Q1) such that

ZZ

Q 1

|vk− gk|2 dz ≥ ε for all vk∈ H(Ak, Q1) with

ZZ

Q 1

|Dvk|2 dz ≤ 1, (2.8) and

|LA k(gk, φ, Q1)| ≤ 1

ksupQ 1

(|Dφ| + |φt|), any φ ∈ C1

c(Q1) (2.10)

By (2.9), we can extract a weakly convergent sequence still denoted by {gk},

g ∈ V (Q1) and A such that:

gk → g weakly in V (Q1), gk → g in L2(Q1), Ak → A, and kgkV (Q 2 )≤ 1 For φ ∈ C1

c(Q1), we have

LA(g, φ, Q1) = LA(g − gk, φ, Q1) +

ZZ

Q 1

(A − Ak)(Dgk, Dφ) dz + LA k(gk, φ, Q1)

Letting k → ∞ and using (2.10), we see that g ∈ H(A, Q1) We then consider the solution vk in Q1 of the problem

LA k(vk, φ, Q1) = 0 for all φ ∈ C1

c(Q1), with vk = g on the parabolic boundary of Q1 Set φk = vk− g We have

λ ZZ

Q 1

|Dvk− Dg|2 dz ≤

ZZ

Q 1

Ak(Dφk, Dφk) dz

On the other hand,

Q 1

Ak(Dφk, Dφk) dz =

ZZ

Q 1

Ak(Dvk, Dφk) dz −

ZZ

Q 1

Ak(Dg, Dφk) dz

= −

Z

B 1

vkφk dx +

ZZ

Q 1

vk(φk)t dz −

ZZ

Q 1

Ak(Dg, Dφk) dz

= −

Z

B 1

vkφk dx +

ZZ

Q 1

vk(φk)t dz −

ZZ

Q 1

A(Dg, Dφk) dz +

ZZ

Q 1

(A − Ak)(Dg, Dφk) dz

=

ZZ

Q 1

(A − Ak)(Dg, Dφk) dz −

Z

B 1

φ2k dx +

ZZ

Q 1

φk(φk)tdz

=

ZZ

Q 1

(A − Ak)(Dg, Dφk) dz −12

Z

B 1

φ2k dx

≤ |(A − Ak)|

ZZ

Q 1

|Dg||Dφk| dz

We conclude that kDvk−DgkL 2 (Q 1 )→ 0 Since vk = g on the parabolic boundary

of Q1, a simple use of the Poincar´e inequality shows that kvk − gkL 2 (Q 1 ) → 0 Let Vk = vk/mk, where mk = max{kDvkkL 2 (Q 1 ), 1} Since lim kDvkkL 2 (Q 1 ) = kDgkL 2 (Q 1 )≤ 1, we have mk→ 1 Thus, kVk−gkL 2 (Q 1 )→ 0 and

ZZ

Q 1

|DVk|2 dz ≤

1 This and the fact that kgk− gkL(Q 1 ) → 0 contradict (2.8) The proof for the case ρ = 1 is complete

Finally, for ρ 6= 1, we will use the following scalings:

x = x0+ RX, t = t0+ R2T ⇒ |dx| = Rn|DX|, dt = R2dT

¯

g(X, T ) = 1

Rg(x0+ RX, t0+ R

2T ) = 1

Rg(x, t), φ(X, T ) = φ(x, t).e Therefore DXg = D¯ xg, ¯gT = Rgt, DXφ = RDe xφ, eφT = R2φt,

k¯gkV (Q 1 )= kgkV (Q R ),

ZZ

Q 1

|¯h − ¯g|2 dz = R−2

ZZ

Q R

|h − g|2 dz, and

LA(¯g, eφ, Q1) =

ZZ

Q 1

[−¯geφt+ A(DX¯g, DXφ] dz = RLe A(g, φ, QR).

From these relations, it is easy to see that the case ρ 6= 1 follows from the above proof

We then have the following parabolic version of Giaquinta’s lemma [2, Lemma A.1]

Lemma 2.2 Suppose that A satisfies the conditions of Lemma 2.1 For any ǫ > 0, there exists a constant C depending on λ, Λ, ε such that

inf







ÃZZ

Q ρ

|H − g|2 dz

!1

: H ∈ H(A, Qρ) and

ZZ

Q ρ

|DH|2 dz ≤ kgk2V (Q ρ )







≤ Cρ2∆(LA, g, Qρ) + ǫρkgkV (Q ρ )

Proof Assume first that ρ = 1 Let δ be the constant found in Lemma 2.1 We consider first the case when

∆(LA, g, Q1) ≤ δkgkV (Q 1 )

We then have

inf

(µZZ

Q 1

|H − g|2 dz

¶1

: H ∈ H(A, Q1) and

ZZ

Q 1

|DH|2 dz ≤ kgk2V (Q 1 )

)

= kgkV (Q 1 )

(µZZ

Q 1

kgkV (Q 1 )− g

kgkV (Q 1 )|2dz

¶1

: H ∈ H(A, Q1)

)

≤ ǫkgkV (Q 1 )

Otherwise, by the Poincar´e inequality, we have

inf

(µZZ

Q 1

|H − g|2 dz

¶1

: H ∈ H(A, Q1)

)

≤ µZZ

Q 1

|(g)Q 1− g|2dz

¶1/2

≤ supτ

µZZ

Q 1

|(g)Q 1− (g(x, τ))B 1|2 dz

¶1/2

+ µZZ

Q 1

|(g(x, t))B 1− g|2 dz

¶1/2

≤ CkgkV (Q 1 )≤ C

δ∆(LA, g, Q1)

Combining these estimates, we prove the lemma when ρ = 1 The case ρ 6= 1 follows from the scaling argument used in Lemma 2.1

3 Partial Regularity Theorem We now consider the system (1.1) For R > 0, let µ = supQ|u| We can assume that µ > 0 Let V0 be a constant vector in Rm

with 34µ ≤ |V0| ≤ µ We will make a change of variables

¯

x = x − x0, s = Φ2µt − t0, with Φµ= sup|u|≤µ|Φ(u)| (3.11) From now on we will work with the new variables (¯x, s) in the cylinders QR =

BR× JR, BR= {¯x : |¯x| ≤ R}, JR= (−R2, 0) The system (1.1) becomes

us= 1

Φ2 µ

Let φ be a cut-off function in Q2R That is, φ ∈ C1

c(Q2R) with φ ≡ 1 in QR and

|Dφ| ≤ 1/R, |φs| ≤ 1/R2 Testing (3.12) with (u − V0)φ and using the fact that

|a(u)Du(u − V0)Dφ| ≤ ε|Dg(u)|2+ C(ε)|Dφ|2Φ2µ|u − V0|2,

which holds due to (A.1) and (A.3), we get the following degenerate Cacciopoli’s type estimate

Φ2µ sup

s∈J R

Z

B R

|u − V0|2dx + λ

2

ZZ

Q R

|Dg(u)|2 d¯z ≤ CΦ

2 µ

R2

ZZ

Q 2R

|u − V0|2 d¯z,

(3.13) where d¯z = d¯xds By (A.2), the above yields

R2kg(u) − g(V0)k2V (QR)≤ CΦ2µ

ZZ

Q |u − V0|2 d¯z (3.14)

In order to apply Lemma 2.2, we define

A = 1

Φ2 µ

Thanks to (A.3), there is a positive constant λ such that λΦµ ≤ |Φ(V0)| ≤ Φµ This and (A.1) show that A satisfies the assumptions of Lemma 2.2

We also set IR =

ZZ

QR |u − V0|2 d¯z for each R > 0 Our first step is to show that

Lemma 3.3 For φ ∈ C1

c(QR), with |Dφ| ≤ 1/R and |φt| ≤ 1/R2, we have

R2|LA(g(u), φ, QR)| ≤ CΦµω(I2R) (I2R)1 (3.16) Proof We note that

LA(g(u), φ, QR) =

ZZ

Q R

[−(g(u) − g(V0))φs+ A(Dg(u), Dφ)] d¯z

Multiplying (3.12) by Φ(V0) and testing by φ, we get

ZZ

QR [−(Φ(V0)(u − V0)φs+ 1

Φ2 µ

Φ(V0)a(x, u)DuDφ] d¯z = 0

Hence,

LA(g(u), φ, QR) =

ZZ

QR hADg(u) − 1

Φ2 µ

Φ(V0)a(x, u)Du, Dφi d¯z

− ZZ

Q R

[g(u) − g(V0) − Φ(V0)(u − V0)]φs d¯z

Using (A.4), for u 6= 0, we estimate the first integrand on the right by

1

Φ 2

µhΦ(V0){a(V0)Φ(V0)−1− a(x, u)Φ(u)−1}Dg(u), Dφi

≤ Φ1µ|Dφ||Dg(u)||a(V0)Φ(V0)−1− a(u)Φ(u)−1| ≤ 2|Dφ||Dg(u)|ω(|u − V0|2) Similarly, the second integrand can be estimated by

¯

¯Z 1

0 (Φ(tu + (1 − t)V0) − Φ(V0))(u − V0)dt

¯

¯

¯ sup |φs| ≤ CΦRµ2ω(|u − V0|2)|u − V0|

(3.17) Thus,

|LA(g(u), φ, QR)| ≤ 2

R

ZZ

Q R

|Dg(u)ω(|u − V0|2)| d¯z +CΦµ

R 2

ZZ

Q R

ω(|u − V0|2)|u − V0| d¯z

≤ R1

µZZ

Q R

|Dg(u)|2 d¯z

¶1µZZ

Q R

ω2(|u − V0|2)| d¯z

¶1

+CΦµ

R 2

µZZ

Q R

ω2(|u − V0|2) d¯z

¶1µZZ

Q R

|u − V0|2 d¯z

¶1

This, (3.13) and the concavity of ω give the lemma

Next, we have a decay estimate for g(u) Hereafter, we will denote fR = ZZ

Q R

f d¯z

Lemma 3.4 For ǫ > 0 and σ ∈ (0, 1/4), we have

ZZ

Q σR

|g(u) − g(u)σR|2 d¯z ≤ C[σ2+ σ−n−2(ω2(IR) + ǫ2)]Φ2µIR (3.18) Proof From (A.1), we see that A satisfies the assumption of Lemma 2.2 Therefore,

we can find H ∈ H(A, QR) such that

ZZ

Q2R |DH|2 d¯z ≤ kg(u) − g(V0)k2

V (Q 2R )and ZZ

Q2R |H − g(u)|2d¯z ≤ CR2|LA(g(u), φ, Q2R)|2

+Cε2R2kg(u) − g(V0)k2

V (Q 2R )

≤ CΦ2

µ[ω2(I4R)I4R+ ǫ2I4R], using (3.14), (3.16) For σ ∈ (0, 1), using a decay result in [1] for the function H,

we have

ZZ

Q σR

|g(u) − g(u)σR|2 d¯z ≤

ZZ

Q σR

|g(u) − HσR|2 d¯z

≤ ZZ

Q σR

|g(u) − H|2 d¯z +

ZZ

Q σR

|H − HσR|2d¯z

≤ ZZ

Q σR

|g(u) − H|2 d¯z + σ2

ZZ

Q R

|H − HR|2d¯z

By the general Poincar´e inequality [8, Lemma 3], we also have

ZZ

Q R

|H − HR|2 d¯z ≤ CR2

ZZ

Q 2R

|DH|2 d¯z ≤ CR2kg(u) − g(V0)k2V 2R≤ CΦ2µI4R2 Combining these estimates and the fact that

ZZ

Q σR

|g(u) − H|2d¯z ≤ Cσ−n−2

ZZ

Q 2R

|g(u) − H|2d¯z,

we obtain the lemma

Finally, we will show that u is still averagely far away from the singular point in smaller cylinders so that the above argument can be repeated More importantly, this will allow us to derive the decay estimate for u from that of g(u)

Lemma 3.5 Assume that

3

4µ ≤ |V0| ≤ µ, IR≤ ε2µ2 (3.19) For any given θ ∈ (0, 1), there exist ε, σ > 0 sufficiently small such that there is

a sequence of vectors {Vi} satisfying

i): Vi= (u)Qσi R and |Vi| ≥1

2µ for all i > 1

Q |u − Vi+1|2 d¯z ≤ θ2

ZZ

Q |u − Vi|2 d¯z, (3.20)

|Vi+1− Vi|2≤ (σ−n−2+ θ)2

ZZ

Qσi R |u − Vi|2 d¯z (3.21) Proof Given any θ ∈ (0, 1) By (3.18), (3.19) and the continuity of ω, we can find

σ, ǫ, ε, in that order, sufficiently small such that

ZZ

QσR |g(u) − g(u)σR|2d¯z ≤ [θεΦµµ]2 (3.22)

We will fix such σ and make ε even smaller later on Let ¯V be a vector such that g( ¯V ) = g(u)σR Thanks to (A.2) and by choosing ε sufficiently small, we have

|g( ¯V ) − g(V0)|2≤

ZZ

QσR (|g(u) − g( ¯V )|2+ |g(u) − g(V0)|2) d¯z ≤ (C + θ2)[εΦµµ]2

We also have |g( ¯V ) − g(V0)| ≥ CΦµ| ¯V − V0| Therefore, | ¯V − V0| ≤ (C + θ)εµ, so that | ¯V | ≥ |V0| − (C + θ)εµ ≥ 1

2µ, if ε is sufficiently small

By (A.2) again, we have |g(u) − g( ¯V )| ≥ CΦµ|u − ¯V | for some universal constant

C If ε ≤ C, we derive from (3.22) that

Φ2µ

ZZ

Q σR

|u − uσR|2 d¯z ≤ Φ2µ

ZZ

Q σR

|u − ¯V |2d¯z ≤ θ2Φ2µIR, which implies

ZZ

Q σR

|u − uσR|2 d¯z ≤ θ2

ZZ

Q R

|u − V0|2 d¯z (3.23) Hence,

|V0− uσR|2≤

ZZ

Q σR

(|u − uσR|2+ |u − V0|2) d¯z ≤ (σ−n−2+ θ2)

ZZ

Q R

|u − V0|2 d¯z (3.24)

We obtain ii) for i = 0 If we can establish that |Vi| ≥ 1

2µ, then it is clear that the above argument could be repeated with V0 being replaced by Vi = uσ i R

In particular, we take A = 1

Φ 2

µa(Vi) in (3.15) and replace V0 with Vi in the proof

of Lemma 3.3 Using (A.3) we easily see that (3.18) continues to hold Thus, let assume that i) and ii) are true up to i − 1 The above argument applies to give (3.20) We then have

ÃZZ

Qσi R |u − Vi|2 d¯z

!1/2

≤ θiεµ, |Vi− Vi−1| ≤pσ−(n+2)+ θ2θi−1εµ Hence,

|Vi| ≥ |V0| −

∞

X

i=0

p

σ−(n+2)+ θ2θiεµ ≥ 34µ −

√

σ−(n+2)+ θ2

µ

2,

if ε is sufficiently small Our proof is complete by induction

Combining Lemma 3.5, Lemma 3.4 and (A.2), we obtain the following decay estimate for u in a sequence of nested cylinders

Proposition 1 Suppose that there exist R > 0 and V0 such that 34µ ≤ |V0| ≤ µ and

ZZ

QR |u − V0|2 d¯z ≤ ε2µ2 Let ǫ > 0 be given If ε is sufficiently small, then there exist σ0 ∈ (0, 1) and a constant C such that for ρ = σiR and Vi = uQ ρ, with 0 < σ < σ0 and i ≥ 1, there holds

ZZ

Q σρ

|u − Vi|2 d¯z ≤ CK(σ, ǫ)

ZZ

Q ρ

|u − Vi−1|2 d¯z, (3.25)

where K(σ, ǫ) = σ2+ σ−n−2[ω2

ÃZZ

Q ρ

|u − Vi−1|2 d¯z

! + ǫ2]

This decay estimate allows us to give the proof of our main partial regularity result

Proof of Theorem 1 We consider a point (x0, t0) 6∈ Σ1S

Σ2 For some β > 0 we have

βµ ≤ |(u)Q R (x 0 ,t 0 )| ≤ µ for all R > 0, and lim inf

R→0

ZZ

Q R

|u − uR|2dz = 0 Let V0= (u)Q R (x 0 ,t 0 ) It is clear that we can replace the condition 34µ ≤ |V0| ≤ µ

in Proposition 1 by βµ ≤ |V0| ≤ µ, where β can be any real in (0, 1) Hence, in terms of the new variables (¯x, s) defined in (3.11), we will verify that the conditions

of Proposition 1 are fulfilled and (3.25) holds

Denote QR,µ= BR(x0) × [t0− 1

Φ 2

µR2, t0] We see that the conditions of Proposi-tion 1 are satisfied if we can show that

ZZ

Q R

|u − V0|2 d¯z is small To see this, let

k = min{1, Φµ} and ¯R = kR By the choice of k, we see that QR,µ¯ ⊆ QR Hence, ZZ

Q R ¯

|u − V0|2d¯z =

ZZ

Q R,µ ¯

|u − V0|2 dz ≤ Φ

2 µ

kn+2

ZZ

Q R

|u − V0|2 dz

Therefore, lim infR→0

ZZ

Q R

|u − uR|2d¯z = 0 and the conditions of Proposition 1 are verified It is then standard to follow the argument of [5] and iterate (3.25)

to assert that u is H¨older continuous in a neighborhood of (x0, t0) The H¨older exponent may depend on (x0, t0) and β, µ Going back to the original variables (x, t), we see that u is H¨older continuous in a neighborhood of (x0, t0)

To see that the singular set Σ2 is small We simply use the following general Poincar´e inequality by Struwe [8, Lemma 3]

ZZ

Q R

|u − uR|2 dz ≤ CR2

ZZ

Q 2R

|Du|2 dz

We remark that this inequality was proven in [8] using only the fact that u satisfies (1.1) with |a(u)Du| ≤ C|Du| for some constant C This is true in our situation because of (A.1) and the fact that u is bounded Thus, Σ2is a subset of

Σ∗2= {(x, t) ∈ Ω × (0, T ) : lim inf

R→0

1

Rn

ZZ

Q |Du|2 dz > 0},