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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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The higher integrability of weak solutions of porous medium systems

Verena BögeleinORCID iD: https://orcid.org/0000-0002-5646-2023 / Frank DuzaarORCID iD: https://orcid.org/0000-0002-6643-1634 / Riikka KorteORCID iD: https://orcid.org/0000-0002-6313-2233 / Christoph SchevenORCID iD: https://orcid.org/0000-0003-0860-5496
Published Online: 2018-06-06 | DOI: https://doi.org/10.1515/anona-2017-0270

Abstract

In this paper we establish that the gradient of weak solutions to porous medium-type systems admits the self-improving property of higher integrability.

Keywords: Porous medium-type systems; higher integrability; gradient estimates

MSC 2010: 35B65; 35K65; 35K40; 35K55

1 Introduction and results

In this article, we are interested in the self-improving property of higher integrability of weak solutions to porous medium-type systems, whose prototype is

tu-Δ(|u|m-1u)=0.

This problem has been open for some time. For non-negative solutions to porous medium-type equations it has recently been solved by Gianazza and Schwarzacher [16]. Here, we are able to treat signed solutions and the vectorial case. More precisely, we consider equations (the case N=1) or systems (the case N2) of the form

tu-div𝐀(x,t,u,D𝒖m)=divFin ΩT,(1.1)

with u:ΩTN, in a space-time cylinder ΩT:=Ω×(0,T), where Ωn is a bounded open domain, n2, T>0, and we abbreviated 𝒖m:=|u|m-1u. The assumptions on the vector field 𝐀:ΩT×N×NnNn are as follows. We assume that 𝐀 is measurable with respect to (x,t)ΩT for all (u,ξ)N×Nn, continuous with respect to (u,ξ) for a.e. (x,t)ΩT, and moreover that 𝐀 satisfies for some structural constants 0<νL< the following growth and ellipticity conditions:

{𝐀(x,t,u,ξ)ξν|ξ|2,|𝐀(x,t,u,ξ)|L|ξ|(1.2)

for a.e. (x,t)ΩT and any (u,ξ)N×Nn. Note that these assumptions are compatible with the ones in [1] and [11, Chapter 3.5] in the scalar case. For the inhomogeneity F:ΩTNn we assume that FL2(ΩT,Nn). As usual, we suppose that the solutions to (1.1) lie in a parabolic Sobolev space; the precise definition will be given below in Definition 1.1.

In the stationary elliptic case it is by now well known that weak solutions to elliptic systems of the type

-div𝐀(x,t,u,Du)=divFin Ω,

locally belong to a slightly higher Sobolev space than a priori assumed. The so-called self-improving property of higher integrability was first detected by Elcrat and Meyers [25]. Their proof is based, among other things, on a reverse Hölder-type inequality – a direct consequence of a Caccioppoli-type inequality (also called reverse Poincaré inequality) – and some adaptation of the famous Gehring Lemma [15]; the nowadays standard interior version can be retrieved from [17, Chapter 11, Theorem 1.2], for the boundary version we refer to [22] and [13, Theorem 2.4]. Originally, Gehring’s lemma was developed to establish the higher integrability of the Jacobian of quasi-conformal mappings. Over time, the self-improving property of higher integrability was first established for solutions of stationary elliptic systems [18] and later for minima of variational integrals [19] by Giaquinta and Modica. A unified treatment in the language of quasi-minima is given in [21, Theorem 6.7]. Corresponding global results for stationary elliptic problems with a Dirichlet boundary condition were established in [21, Section 6.5], [13, Section 3].

The first higher integrability result for vectorial evolutionary problems goes back to Giaquinta and Struwe [20, Theorem 2.1]. More precisely, quasilinear parabolic systems of the type

tu-div(𝐚(x,t,u)Du)=divFin ΩT,

whose coefficients 𝐚 continuously depend on (x,t,u) have been investigated. The technique of Giaquinta and Struwe does not carry over to the parabolic p-Laplacian system

tu-div(|Du|p-2Du)=divFin ΩT,

or general parabolic systems with p-growth (the growth and coercivity condition from (1.2) have to be replaced by 𝐚(x,t,u,ξ)ξν|ξ|p and |𝐚(x,t,u,ξ)|L(|ξ|p+1)). The obstruction relies in the fact that the parabolic p-Laplacian equation has a different homogeneity in the time and the diffusion term. In particular, multiples of a solution do not anymore solve the differential equation. This problem has finally been solved by Kinnunen and Lewis [23] who proved the higher integrability result for general parabolic systems with p-growth. More precisely, they have shown that weak solutions from the natural energy space C0([0,T];L2(Ω,N))Lp(0,T;W1,p(Ω,N)) have a more integrable spatial gradient, namely

DuLlocp+ε(ΩT,Nn)for some ε>0.

This shows that also in the case of parabolic systems with coefficients of p-growth and coercivity energy solutions enjoy the self-improving property of higher integrability for the gradient. The key to the result was the use of intrinsic cylinders in the sense of DiBenedetto and Friedman [8, 10, 9, 7], i.e. cylinders of the form Qϱ,λ2-pϱ2 whose space-time scaling depends on the spatial gradient of the solution via

Qϱ,λ2-pϱ2|Du|pdxdtλp.

This important result has been generalized over time in various directions. The global result with a Dirichlet boundary condition at the lateral boundary was established by Parviainen [26]. Interior higher integrability for weak solutions of higher order degenerate parabolic systems has been shown by Bögelein [3], while the corresponding global result was established in [6]. The case of parabolic equations with non-standard p(x,t)-growth was treated by Antonsev and Zhikov [2], while systems were treated by Zhikov and Pastukhova [29] and independently by Bögelein and Duzaar [4].

For the porous medium equation, the question of higher integrability of the gradient, even for non-negative solutions in the scalar case, remained an open problem for a while. The reason was that when proving regularity of the gradient the degeneracy with respect to u is much more difficult to handle. This difficulty has recently been overcome by Gianazza and Schwarzacher [16] who proved that non-negative weak solutions to porous medium equations of the type (1.1) enjoy the self-improving property of higher integrability. More precisely, this means that the integrability Dum+12Lloc2(ΩT,n) of weak solutions was improved to

Dum+12Lloc2+ε(ΩT,n)for some ε>0.

The main novelty with respect to the proof for the parabolic p-Laplacian in [23] is that Gianazza and Schwarzacher work with cylinders which are intrinsically scaled with respect to u rather than the spatial gradient Du. This means that they consider cylinders of the type Qϱ,θϱ2 whose space-time scaling is adapted to the solution u via the coupling

Qϱ,θϱ2um+1dxdtθ-m+1m-1.(1.3)

This is exactly the intrinsic scaling which is typically used in the proof of regularity of u, as for instance Hölder continuity of u, cf. [10]. At first glance it is quite surprising that this approach also yields regularity of the spatial gradient. However, these cylinders are better adapted to the equation and this is crucial for the proof. Nevertheless, the argument becomes much more involved than the one for the parabolic p-Laplacian. The overall strategy can be outlined as follows. First, one has to prove a reverse Hölder-type inequality on certain intrinsic cylinders. To achieve this, Gianazza and Schwarzacher distinguish whether a cylinder Q belongs to the non-degenerate regime in which the inequality

Q|u-(u)Q|m+1dxdtδQum+1dxdt

holds true for some particular 0<δ1, or Q belongs to the degenerate regime in which the opposite inequality is valid. In the non-degenerate regime they rely on the expansion of positivity in order to guarantee that the solution does not become too small on the cylinder. In a second step, one usually constructs a covering of super-level sets of the spatial gradient with intrinsic cylinders. However, this is not possible for the cylinders which are intrinsically scaled with respect to u. Gianazza and Schwarzacher overcame this problem by a very elegant idea. They weakened this property to the so-called sub-intrinsic cylinders for which they succeeded to prove the covering property. Thereby, they call a cylinder sub-intrinsic if (1.3) holds as an inequality, i.e. the mean value integral is bounded from above by the right-hand side.

The methods of proof of this important result are only applicable in the scalar case for non-negative solutions, because tools as the expansion of positivity are neither available in the vectorial case, nor for signed solutions.

The present paper has its origin in the effort to extend the purely scalar result to the vector-valued case. As a by-product of the vectorial case, we are able to deal also with signed solutions in the scalar case. Moreover, contrary to Gianazza and Schwarzacher, we start from the definition of weak (energy) solutions introduced in [28, Theorem 5.5], i.e. we start with solutions satisfying D𝒖mLloc2(ΩT,Nn), see (1.6). As main result, we prove that

D𝒖mLloc2+ε(ΩT,Nn)for some ε>0.

We note that starting from a vectorial version of the energy estimate used in [16], a modification of our method also applies to the definition of weak solution as considered there. The key to the higher integrability result in the vectorial case is to prove the reverse Hölder-type inequality just by the use of an energy estimate and a gluing lemma as stated in Lemmas 3.1 and 3.2. In particular, it is important to omit the use of the expansion of positivity. In fact, for the proof of the Sobolev–Poincaré-type inequality in Lemma 4.3 we only use the Gluing Lemma 3.2, the standard Sobolev inequality and some algebraic lemmas. Here, we note that contrary to (1.3) we work with differently scaled cylinders which reflect more clearly the behavior of the porous medium equation and which are adapted to the energy space (1.6) (for the heuristics see also [16, Remark 5.6]). These cylinders are given by Qϱ(θ)=Bϱ×(-θ1-mϱm+1m,θ1-mϱm+1m) with an intrinsic scaling of the form

Qϱ(θ)|𝒖m|2ϱ2dxdtθ2m,(1.4)

so that in case that the mean value of 𝒖m on the cylinder Qϱ(θ) is zero, the scaling parameter θ is comparable to |D𝒖m|. A cylinder Qϱ(θ) is called sub-intrinsic if (1.4) holds as an inequality, where the mean value integral is bounded from above by the right-hand side. Contrary to [16] we present a unified proof of the Sobolev–Poincaré-type inequalities on sub-intrinsic cylinders that works likewise in the non-degenerate and degenerate regime. These inequalities are subsequently used to derive reverse Hölder-type inequalities on intrinsic cylinders and sub-intrinsic cylinders additionally satisfying

Qϱ(θ)|D𝒖m|2dxdtθ2m.(1.5)

For the final proof of the higher integrability we cover the super-level-sets of |D𝒖m| by sub-intrinsic cylinders. Here, we rely on the construction by Gianazza and Schwarzacher. The idea is to choose with the help of the intermediate value theorem for a given center zo and radius ϱ>0 the scaling parameter θ~zo;ϱ in such a way that

Qϱ(θ~zo;ϱ)(zo)|𝒖m|2ϱ2dxdt=θ~zo;ϱ2m

is satisfied, where

Qϱ(θ~zo;ϱ)(zo)=zo+Qϱ(θ~zo;ϱ).

Unfortunately, the mapping ϱθ~zo;ϱ is not monotone. Therefore, we modify the parameter θ~zo;ϱ by a rising sun-type construction, i.e. we define

θzo;ϱ:=maxrϱθ~zo;r.

Then the mapping ϱθzo;ϱ is monotonically decreasing and furthermore one can show that the cylinders Qϱ(θzo;ϱ)(zo) are still sub-intrinsic. A crucial observation at this point is that by construction either the cylinders are intrinsic or satisfy (1.5). This allows to apply our reverse Hölder inequality. As in [16] our cylinders satisfy a Vitali covering property which allows to cover the super-level-sets of |D𝒖m| by countably many of these cylinders. In this way, we obtain a reverse Hölder inequality on the super-level-sets of |D𝒖m|. In a standard way, this implies the higher integrability by a Fubini-type argument.

1.1 General setting and results

In this subsection we fix the notations, describe the general setup and present our main result. First, we define what we mean by a weak energy solution to the porous medium-type system.

Definition 1.1.

Assume that the vector field 𝐀:ΩT×N×NnNn satisfies the conditions in (1.2) and that FLloc2(ΩT,Nn). We identify a measurable map u:ΩTN in the class

uC0((0,T);Llocm+1(Ω,N))with𝒖mLloc2(0,T;Wloc1,2(Ω,N))(1.6)

as a weak solution to the porous medium-type system (1.1) if and only if the identity

ΩT[utφ-𝐀(x,t,u,D𝒖m)Dφ]dxdt=ΩTFDφdxdt(1.7)

holds true, for any testing function φC0(ΩT,N).

Existence of weak solutions can be deduced from [1] after the transformation v=|u|m-1u; see also [5] for a different approach in the case of non-negative solutions.

Throughout the paper we work with parabolic cylinders of the type

QR(zo)=BR(xo)×(to-Rm+1m,to+Rm+1m)ΩT,

whose associated parabolic dimension is

d:=n+1+1m.

Our main result reads now as follows.

Theorem 1.2.

Let m1 and σ>2. There exist constants εo=εo(n,m,ν,L)(0,1] and c=c(n,m,ν,L)1 such that the following holds true: Whenever FLlocσ(ΩT,RNn) and

uC0((0,T);Llocm+1(Ω,N))𝑤𝑖𝑡ℎ𝒖mLloc2(0,T;Wloc1,2(Ω,N))

is a weak solution of equation (1.1) under the assumptions (1.2) in the sense of Definition 1.1, there holds

D𝒖mLloc2+ε1(ΩT,Nn),

where ε1:=min{εo,σ-2}. Moreover, for every ε(0,ε1] and every cylinder Q2R(zo)ΩT, we have the quantitative local higher integrability estimate

QR(zo)|D𝒖m|2+εdxdtc[1+Q2R(zo)[|u|2mR2+|F|2]dxdt]εmm+1Q2R(zo)|D𝒖m|2dxdt+cQ2R|F|2+εdxdt.(1.8)

The quantitative local estimate (1.8) can be converted easily into an estimate on the standard parabolic cylinders CR(zo):=BR(xo)×(to-R2,to+R2). The precise statement is as follows.

Corollary 1.3.

Under the assumptions of Theorem 1.2, the estimate

CR(zo)|D𝒖m|2+εdxdtcRε[1+C2R(zo)[|u|2m+R2|F|2]dxdt]εmm+1C2R(zo)|D𝒖m|2dxdt+cC2R(zo)|F|2+εdxdt

holds true on any parabolic cylinder C2R(zo)ΩT and for every ε(0,ε1] and with a constant c=c(n,m,ν,L).

2 Preliminaries

2.1 Notations

In order not to overburden the notation, we abbreviate in the following the power of a vector (or possibly negative number) by

𝒖α:=|u|α-1ufor uN and α>0,

where we interpret 𝒖α=0 in the case u=0 and α(0,1). Throughout the paper we write zo=(xo,to)n× and use the space-time cylinders

Qϱ(θ)(zo):=Bϱ(xo)×Λϱ(θ)(to),(2.1)

where

Λϱ(θ)(to):=(to-θ1-mϱm+1m,to+θ1-mϱm+1m)

with some scaling parameter θ>0. One of the most important notions for this paper is the notion of sub-intrinsic cylinders. We call a cylinder Qϱ(θ)(zo) sub-intrinsic if and only if

Qϱ(θ)(zo)|u|2mϱ2dxdtθ2m

holds true. If the preceding inequality actually is an equality, we call the cylinder intrinsic. In the case θ=1, we simply omit the parameter in our notation and write

Qϱ(zo):=Bϱ(xo)×(to-ϱm+1m,to+ϱm+1m)

instead of Qϱ(1)(zo), and, analogously, Λϱ(to) instead of Λϱ(1)(to). If zo is the origin, we write Qϱ, Bϱ and Λϱ for Qϱ(0), Bϱ(0) and Λϱ(0). Moreover, if the center zo is clear from the context, we omit it in our notation.

For a map uL1(0,T;L1(Ω,N)) and given measurable sets AΩ and EΩ×(0,T) with positive Lebesgue measure the slicewise mean (u)A:(0,T)N of u on A is defined by

(u)A(t):=Au(t)dxfor a.e. t(0,T),(2.2)

whereas the mean value (u)EN of u on E is defined by

(u)E:=Eudxdt.

Note that if uC0((0,T);L2(Ω,N)) the slicewise means are defined for any t(0,T). If the set A is a ball Bϱ(xo), then we abbreviate (u)xo;ϱ(t):=(u)Bϱ(xo)(t) and if E is a cylinder of the form Qϱ(θ)(zo), we use the shorthand notation (u)zo;ϱ(θ):=(u)Qϱ(θ)(zo). Finally, we define the boundary term

𝔟[𝒖m,𝒂m]:=mm+1(|a|m+1-|u|m+1)-u(𝒂m-𝒖m)(2.3)

that will appear in the energy estimate from Lemma 3.1.

2.2 Auxiliary material

In order to “re-absorb” certain terms, we will use the following iteration lemma, which can be retrieved by a change of variable from [21, Lemma 6.1].

Lemma 2.1.

Let 0<ϑ<1, A,C0 and α,β>0. Then there exists a constant c=c(β,ϑ) such that there holds: For any 0<r<ϱ and any non-negative bounded function ϕ:[r,ϱ]R0 satisfying

ϕ(t)ϑϕ(s)+A(sα-tα)-β+Cfor all rt<sϱ,

we have

ϕ(r)c[A(ϱα-rα)-β+C].

Lemma 2.2.

For any α>1, there exists a constant c=c(α) such that, for all a,bRN the following assertions hold true:

  • (i)

    1c|𝒂α-𝒃α|(|a|α-1+|b|α-1)|a-b|c|𝒂α-𝒃α|,

  • (ii)

    |a-b|αc|𝒂α-𝒃α|,

  • (iii)

    |𝒂α-𝒃α|2c(𝒂2α-1-𝒃2α-1)(a-b).

The proof of (i) and (ii) can be found in [19, Lemma 2.2]. Inequality (iii) can be derived by combining the proof of [7, Chapter I, Lemma 4.4] with (i). The next lemma provides useful estimates for the boundary term 𝔟 introduced in (2.3).

Lemma 2.3.

There exists a constant c=c(m) such that for any u,aRN the following assertions hold true:

  • (i)

    1c|𝒖m+12-𝒂m+12|2𝔟[𝒖m,𝒂m]c|𝒖m+12-𝒂m+12|2,

  • (ii)

    1c|𝒖m-𝒂m|2[|u|m-1+|a|m-1]𝔟[𝒖m,𝒂m]c|𝒖m-𝒂m|2,

  • (iii)

    𝔟[𝒖m,𝒂m]c|𝒖m-𝒂m|m+1m.

Proof.

Using the auxiliary function ϕC2(N), ϕ(x)=1m+1|x|m+1, we can re-write the boundary term to

𝔟[𝒖m,𝒂m]=1m+1|u|m+1-1m+1|a|m+1-𝒂m(u-a)=ϕ(u)-ϕ(a)-ϕ(a)(u-a).

The Hessian of ϕ is given by the matrix

Hϕ(x)=|x|m-1(𝕀N+(m-1)x|x|x|x|),

whose eigenvalues are |x|m-1 and m|x|m-1. Therefore, the integral formula for the remainder in Taylor’s expansion yields

𝔟[𝒖m,𝒂m]01|a+t(u-a)|m-1(1-t)dt|u-a|2.(2.4)

Now, we distinguish between the cases |u||a| and |u|<|a|. In the first case, for any t(34,1) we have

|a+t(u-a)|t|u|-(1-t)|a|12|u|14(|u|+|a|),

from which we infer

𝔟[𝒖m,𝒂m]c341(1-t)dt(|u|+|a|)m-1|u-a|2=c(|u|+|a|)m-1|u-a|2,(2.5)

where c=c(m). In the second case |u|<|a|, we restrict ourselves to values t(0,14). Interchanging the roles of u, a and t, 1-t we end up with the same estimate for |a+t(u-a)|. In view of (2.4), this implies also in the remaining case for 𝔟[𝒖m,𝒂m] estimate (2.5). Combining this with Lemma 2.2 (i), we arrive at the first claimed estimate, since

𝔟[𝒖m,𝒂m]c(m)(|u|+|a|)m-1|u-a|2c(m)|𝒖m+12-𝒂m+12|2.

For the second asserted estimate, we apply Lagrange’s formula for the remainder in Taylor’s expansion, which yields

𝔟[𝒖m,𝒂m]12supt(0,1)(u-a)Hϕ(a+t(u-a))(u-a)m2|u-a|2supt(0,1)|a+t(u-a)|m-1c(m)(|u|+|a|)m-1|u-a|2.(2.6)

In view of Lemma 2.2 (i), this yields the second estimate from (i), since

𝔟[𝒖m,𝒂m]c(m)(|u|+|a|)m-1|u-a|2c(m)|𝒖m+12-𝒂m+12|2.

The inequalities in (ii) are a consequence of (i) and Lemma 2.2 (i) applied with u~=𝒖m+12, a~=𝒂m+12 and α=2mm+1, since

[|u|m-1+|a|m-1]𝔟[𝒖m,𝒂m]c(m)[|u|m-1+|a|m-1]|𝒖m+12-𝒂m+12|2=c(m)[|u~|2(m-1)m+1+|a~|2(m-1)m+1]|u~-a~|2c(m)|𝒖m-𝒂m|2.

The reasoning for the second bound in (ii) is similar. The inequality (iii) also follows from inequality (2.2) and Lemma 2.2 (i)–(ii), since

𝔟[𝒖m,𝒂m]c(m)|𝒖m-𝒂m||u-a|c(m)|𝒖m-𝒂m|1+1m.

The following estimate, which is known as the quasi-minimality of the mean value, can be established by Young’s and Hölder’s inequality.

Lemma 2.4.

Let α1. Then for any bounded domain ARk, kN, any uLα(A,RN), and any aRN there holds

A|u-(u)A|αdx2αA|u-a|αdx.

The following statement shows that mean values over subsets are still quasi-minimizing. This is well known for α=1. Here, we state the version for powers. As expected, the quasi-minimality constant depends on the ratio of the measures of the set and the subset.

Lemma 2.5.

Let α1. Then there exists a universal constant c=c(α) such that whenever ABRk, kN, are two bounded domains and uL2α(B,RN), there holds

B|𝒖α-(𝒖)𝑨α|2dxc|B||A|B|𝒖α-(𝒖)𝑩α|2dx.

Proof.

We start by estimating the difference |(𝒖)𝑩α-(𝒖)𝑨α|. Using Lemma 2.2 (i)–(ii), we obtain for a constant c=c(α) that

|(𝒖)𝑩α-(𝒖)𝑨α|2c[|(u)B|2α-2+|(u)A|2α-2]|(u)B-(u)A|2c[|(u)B|2α-2+|(u)A-(u)B|2α-2]|(u)B-(u)A|2c|(u)B|2α-2A|u-(u)B|2dx+c|(u)A-(u)B|2αcA|𝒖α-(𝒖)𝑩α|2dx+cA|u-(u)B|2αdxc|B||A|B|𝒖α-(𝒖)𝑩α|2dx.

From this estimate we conclude

B|𝒖α-(𝒖)𝑨α|2dx2B|𝒖α-(𝒖)𝑩α|2dx+2|(𝒖)𝑩α-(𝒖)𝑨α|2c|B||A|B|𝒖α-(𝒖)𝑩α|2dx,

which proves the claim. ∎

The following lemma is from [12, Lemma 6.2]. For convenience of the reader, we nevertheless include the proof.

Lemma 2.6.

Let α>1. Then there exists a universal constant c=c(α) such that for any bounded domain ARn, any non-negative uL2α(A,RN), and any aRN there holds

A|𝒖α-(𝒖)𝑨α|2dxcA|𝒖α-𝒂α|2dx.

Proof.

Using Lemma 2.2 (iii), we obtain for a constant c=c(α) that

A|𝒖α-(𝒖)𝑨α|2dxcA(u-(u)A)(𝒖2α-1-(𝒖)𝑨2α-1)dx=cA(u-(u)A)(𝒖2α-1-𝒂2α-1)dxcA|u-(u)A||𝒖2α-1-𝒂2α-1|dx.(2.7)

In order to estimate the integrand from above, we distinguish between two cases. In the case |u|12|a|, we have

|a|α=|𝒂α-𝒖α+𝒖α||𝒂α-𝒖α|+2-α|a|α

and hence |a|α2α2α-1|𝒖α-𝒂α|. In turn, this allows us to estimate

|𝒖2α-1-𝒂2α-1|2|a|2α-1c(α)|𝒖α-𝒂α|2α-1α,

which by Lemma 2.2 (ii) implies

|u-(u)A||𝒖2α-1-𝒂2α-1|c(α)|𝒖α-(𝒖)𝑨α|1α|𝒖α-𝒂α|2α-1α.(2.8)

In the remaining case |a|<2|u|, Lemma 2.2 (i) shows

|u-(u)A||𝒖2α-1-𝒂2α-1|c(α)|u-(u)A|(|u|2α-2+|a|2α-2)|u-a|c(α)|u|2α-2|u-(u)A||u-a|=c(α)|u|α-1|u-(u)A||u|α-1|u-a|.

An application of Lemma 2.2 (i) therefore yields

|u-(u)A||𝒖2α-1-𝒂2α-1|c|𝒖α-(𝒖)𝑨α||𝒖α-𝒂α|.(2.9)

Combining (2.8) and (2.9), we infer that in any case the estimate

|u-(u)A||𝒖2α-1-𝒂2α-1|c|𝒖α-(𝒖)𝑨α|1α|𝒖α-𝒂α|2α-1α+c|𝒖α-(𝒖)𝑨α||𝒖α-𝒂α|

holds true for a constant c=c(α). We insert this into (2.2) and apply Young’s inequality twice. This leads to

A|𝒖α-(𝒖)𝑨α|2dx12A|𝒖α-(𝒖)𝑨α|2dx+cA|𝒖α-𝒂α|2dx.

Here we re-absorb the term 12A|𝒖α-(𝒖)𝑨α|2dx into the left-hand side and obtain the asserted inequality. ∎

Finally, we ensure that the mean value is also a quasi-minimizer of aA𝔟[u,a]dx.

Lemma 2.7.

There exists a universal constant c=c(m) such that for any bounded domain ARn, any non-negative uLm+1(A,RN), and any aRN there holds

A𝔟[u,(u)A]dxcA𝔟[u,a]dx.

Proof.

Due to Lemmas 2.3 (i) and 2.6 we obtain

A𝔟[u,(u)A]dxcA|𝒖m+12-(𝒖)𝑨m+12|2dxcA|𝒖m+12-𝒂m+12|2dxcA𝔟[u,a]dx.

This proves the asserted inequality. ∎

3 Energy bounds

In this section we derive an energy inequality and a gluing lemma which follow from the weak formulation (1.7) of the differential equation by testing with suitable testing functions. Later on, they will be used in order to prove Sobolev–Poincaré and reverse Hölder-type inequalities.

Lemma 3.1.

Let m1 and let u be a weak solution to (1.1) in ΩT in the sense of Definition 1.1, where the vector-field A fulfills the growth and ellipticity assumptions (1.2). Then there exists a constant c=c(m,ν,L) such that on any cylinder Qϱ(θ)(zo)ΩT with 0<ϱ1 and θ>0, and for any r[ϱ2,ϱ) and any aRN the following energy estimate:

suptΛr(θ)(to)Br(xo)θm-1𝔟[𝒖m(,t),𝒂m]ϱm+1mdx+Qr(θ)(zo)|D𝒖m|2dxdtcQϱ(θ)(zo)[|𝒖m-𝒂m|2(ϱ-r)2+θm-1𝔟[𝒖m,𝒂m]ϱm+1m-rm+1m]dxdt+cQϱ(θ)(zo)|F|2dxdt,

holds true, where b has been defined in (2.3).

Proof.

For vL1(ΩT,N), we define the following mollification in time:

[[v]]h(x,t):=1h0tes-thv(x,s)𝑑s.

From the weak form (1.7) of the differential equation we deduce the mollified version (without loss of generality we may assume that uC0([0,T);Lloc2(Ω,N)))

ΩT[t[[u]]hφ+[[𝐀(x,t,u,D𝒖m)]]hDφ]dxdt=ΩT[[F]]hDφdxdt+1hΩu(0)0Te-shφdsdx(3.1)

for any φL2(0,T;W01,2(Ω,N)). Let ηC01(Bϱ(xo),[0,1]) be the standard cut off function with η1 in Br(xo) and |Dη|2ϱ-r and ζW1,(Λϱ(θ)(to),[0,1]) defined by

ζ(t):={1for tto-θ1-mrm+1m,(t-to)θm-1+ϱm+1mϱm+1m-rm+1mfor t(to-θ1-mϱm+1m,to-θ1-mrm+1m).

Furthermore, for given ε>0 and t1Λr(θ)(to) we define the cut-off function ψεW1,(Λϱ(θ)(to),[0,1]) by

ψε(t):={1for t(to-θ1-mϱm+1m,t1],1-1ε(t-t1)for t(t1,t1+ε),0for t[t1+ε,to).

We choose

φ(x,t)=η2(x)ζ(t)ψε(t)(𝒖m(x,t)-𝒂m)

as testing function in the mollified version (3.1) of the differential equation. For the integral containing the time derivative we compute

Qϱ(θ)(zo)t[[u]]hφdxdt=Qϱ(θ)(zo)η2ζψεt[[u]]h([[𝒖]]𝒉m-𝒂m)dxdt+Qϱ(θ)(zo)η2ζψεt[[u]]h(𝒖m-[[𝒖]]𝒉m)dxdt-Qϱ(θ)(zo)η2ζψεt(1m+1|[[u]]h|m+1-𝒂m[[u]]h)dxdt=-Qϱ(θ)(zo)η2ζψεt(𝔟[[[𝒖]]𝒉m,𝒂m])dxdt=Qϱ(θ)(zo)η2(ζψε+ψεζ)𝔟[[[𝒖]]𝒉m,𝒂m]dxdt,

where we also used the identity t[[u]]h=-1h([[u]]h-u), cf. [24, Chapter 2]. Since [[u]]hu in Llocm+1(ΩT), we may pass to the limit h0 in the integral on the right-hand side and therefore find that

lim infh0Qϱ(θ)(zo)t[[u]]hφdxdtQϱ(θ)(zo)η2(ζψε+ψεζ)𝔟[𝒖m,𝒂m]dxdt=:Iε+IIε.

At this point, we pass to the limit ε0 and obtain for the first term

limε0Iε=Bϱ(xo)η2𝔟[𝒖m(,t1),𝒂m]dx,

for any t1Λr(θ)(to), whereas the term IIε can be estimated in the following way (observe that the boundary term is non-negative):

|IIε|Qϱ(θ)(zo)ζ𝔟[𝒖m,𝒂m]dxdtQϱ(θ)(zo)θm-1𝔟[𝒖m,𝒂m]ϱm+1m-rm+1mdxdt.

Next, we consider the diffusion term in (3.1). After passing to the limit h0, we use the ellipticity and growth assumption (1.2), and later on Young’s inequality. In this way, we obtain

Qϱ(θ)(zo)𝐀(x,t,u,D𝒖m)Dφdxdt=Qϱ(θ)(zo)𝐀(x,t,u,D𝒖m)[η2ζψεD𝒖m+2ηζψε(𝒖m-𝒂m)Dη]dxdtνQϱ(θ)(zo)η2ζψε|D𝒖m|2dxdt-2LQϱ(θ)(zo)η|Dη|ζψε|𝒖m-𝒂m||D𝒖m|dxdtν2Qϱ(θ)(zo)η2ζψε|D𝒖m|2dxdt-cQϱ(θ)(zo)|Dη|2ζψε|𝒖m-𝒂m|2dxdtν2Qϱ(θ)(zo)η2ζψε|D𝒖m|2dxdt-cQϱ(θ)(zo)|𝒖m-𝒂m|2(ϱ-r)2dxdt

for a constant c=c(m,ν,L). Finally, we consider the right-hand side integrals in (3.1). The second integral disappears in the limit h0, since φ(0)=0. In the integral containing the inhomogeneity F we pass to the limit h0 and subsequently apply Hölder’s inequality. In this way, we obtain

Qϱ(θ)(zo)FDφdxdt=Qϱ(θ)(zo)[η2ζψεFD𝒖m+2ηζψεF(𝒖m-𝒂m)Dη]dxdtν4Qϱ(θ)(zo)[η2ζψε|D𝒖m|2+|𝒖m-𝒂m|2(ϱ-r)2]dxdt+cQϱ(θ)(zo)|F|2dxdt.

We combine these estimates and then pass to the limit ε0. This leads to

Br(xo)𝔟[𝒖m(,t1),𝒂m]dx+ν4to-θ1-mrm+1mt1Bϱ(xo)|D𝒖m|2dxdtcQϱ(θ)(zo)[|𝒖m-𝒂m|2(ϱ-r)2+θm-1𝔟[𝒖m,𝒂m]ϱm+1m-rm+1m]dxdt+cQϱ(θ)(zo)|F|2dxdt

for any t1Λr(θ)(to), with a constant c=c(m,ν,L). In the preceding inequality we take in the first term on the left-hand side the supremum over t1Λϱ(θ)(to), and then pass to the limit t1to+θ1-mrm+1m. Finally, we take means on both sides. This procedure leads to the claimed inequality. ∎

The following lemma serves to compare the slice-wise mean values at different times. This is necessary since Poincaré’s and Sobolev’s inequality can only be applied slice-wise. Such a result, which connects means on different time slices, is termed Gluing Lemma.

Lemma 3.2.

Let m1 and let u be a weak solution to (1.1) in ΩT in the sense of Definition 1.1, where the vector-field A fulfills the growth and ellipticity assumptions (1.2). Then for any cylinder Qϱ(θ)(zo)ΩT with 0<ϱ1 and θ>0 there exists ϱ^[ϱ2,ϱ] such that for all t1,t2Λϱ(θ)(to) there holds

|(u)xo;ϱ^(t2)-(u)xo;ϱ^(t1)|cϱ1mθm-1Qϱ(θ)(zo)[|D𝒖m|+|F|]dxdt

for a constant c=c(L).

Proof.

Let t1,t2Λϱ(θ)(to) with t1<t2 and assume that r[ϱ2,ϱ]. For δ>0 and 0<ε1, we define ξεW01,(t1-ε,t2+ε) by

ξε(t):={0for to-θ1-mϱm+1mtt1-ε,t-t1+εεfor t1-ε<t<t1,1for t1tt2,t2+ε-tεfor t2<t<t2+ε,0for t2+εtto,

and a radial function ΨδW01,(Br+δ(xo)) by Ψδ(x):=ψδ(|x-xo|), where

ψδ(s):={1for 0sr,r+δ-sδfor r<s<r+δ,0for r+δsϱ,

for s[0,ϱ]. For fixed i{1,,N} we choose φε,δ=ξεΨδei as testing function in the weak formulation (1.7), where ei denotes the i-th canonical basis vector in N. In the limit ε,δ0 we obtain

Br(xo)[u(,t2)-u(,t1)]eidx=t1t2Br(xo)[𝐀(x,t,u,D𝒖m)+F]eix-xo|x-xo|dn-1(x)dt.

We multiply the preceding inequality by ei and sum over i=1,,N. This yields

Br(xo)[u(,t2)-u(,t1)]dx=t1t2Br(xo)[𝐀(x,t,u,D𝒖m)+F]x-xo|x-xo|dn-1(x)dt.

Here, we use the growth condition (1.2)2 and immediately get for any t1,t2Λϱ(θ)(to) and any r[ϱ2,ϱ] that there holds

|Br(xo)[u(,t2)-u(,t1)]dx|t1t2Br(xo)[L|D𝒖m|+|F|]dn-1dt.

Since

t1t2Bϱ(xo)[L|D𝒖m|+|F|]dxdt=0ϱt1t2Br(xo)[L|D𝒖m|+|F|]dn-1dtdrϱ2ϱt1t2Br(xo)[L|D𝒖m|+|F|]dn-1dtdr,

there exists a radius ϱ^[ϱ2,ϱ) with

t1t2Bϱ^(xo)[L|D𝒖m|+|F|]dn-1dt2ϱt1t2Bϱ(xo)[L|D𝒖m|+|F|]dxdt.

Therefore, we choose in the above inequality r=ϱ^ and then take means on both sides of the resulting inequality. This implies

|(u)xo;ϱ^(t2)-(u)xo;ϱ^(t1)|cϱΛϱ(θ)(to)Bϱ(xo)[|D𝒖m|+|F|]dxdt=cϱ1mθm-1Qϱ(θ)(zo)[|D𝒖m|+|F|]dxdt

for any t1,t2Λϱ(θ)(to) and with a constant c=c(L). ∎

4 Parabolic Sobolev–Poincaré-type inequalities

Throughout this section we consider so-called sub-intrinsic cylinders. These cylinders are characterized as follows: On the scaled cylinder Qϱ(θ)(zo)ΩT with 0<ϱ1 and θ>0 the following coupling between the mean of |u|2mϱ2 on Qϱ(θ)(zo) and θ holds true:

Qϱ(θ)(zo)|u|2mϱ2dxdt2d+2θ2m.(4.1)

The following lemma is the first step towards a Poincaré-type inequality for weak solutions to the porous medium system. This is necessary because the standard Poincaré inequality in n× cannot be applied directly, since weak solutions u a priori do not possess the necessary regularity with respect to time; note that we only assume for the spatial derivative D𝒖mLloc2(ΩT,Nn), while no regularity assumption with respect to time is incorporated in the definition of weak solutions. Nevertheless, we are able to prove some sort of Poincaré inequality. This is achieved by considering the space and time direction separately. In x-direction we can apply the Poincaré inequality on n, while in t-direction the needed regularity is gained from the gluing lemma.

Lemma 4.1.

Let m1 and let u be a weak solution to (1.1) in ΩT in the sense of Definition 1.1, where the vector-field A fulfills the growth and ellipticity assumptions (1.2). Then on any cylinder Qϱ(θ)(zo)ΩT satisfying the sub-intrinsic coupling (4.1) for some 0<ϱ1 and some θ>0, the inequality

Qϱ(θ)(zo)|𝒖m-(𝒖m)zo;ϱ(θ)|2ϱ2dxdtcQϱ(θ)(zo)|𝒖m-(𝒖m)zo;ϱ(t)|2ϱ2dxdt+c[Qϱ(θ)(zo)[|D𝒖m|+|F|]dxdt]2(4.2)

holds true with a universal constant c=c(n,m,L).

Proof.

In the following we shall again omit for simplification the reference point zo in our notation. Moreover, we let ϱ^[ϱ2,ϱ] be the radius from Lemma 3.2. By adding and subtracting the slice-wise means (𝒖)ϱ^m(t) as defined in (2.2), we obtain the inequality

Qϱ(θ)|𝒖m-(𝒖m)ϱ(θ)|2ϱ2dxdt3[Qϱ(θ)|𝒖m-(𝒖)ϱ^m(t)|2ϱ2dxdt+1ϱ2Λϱ(θ)|Λϱ(θ)[(𝒖)ϱ^m(t)-(𝒖)ϱ^m(τ)]dτ|2dt+1ϱ2|Λϱ(θ)(𝒖)ϱ^m(τ)dτ-(𝒖m)ϱ(θ)|2]=:3[I+II+III],(4.3)

with the obvious meaning of I,II,III. In the following, we treat the terms of the right side in order. We start with the term I. Using the fact that ϱ^[ϱ2,ϱ], we can first replace the slice-wise means (𝒖)ϱ^m(t) by (𝒖)ϱm(t) with the help of Lemma 2.5, and afterwards apply Lemma 2.6, to obtain

IcQϱ(θ)|𝒖m-(𝒖)ϱm(t)|2ϱ2dxdtcQϱ(θ)|𝒖m-(𝒖m)ϱ(t)|2ϱ2dxdt,

where c=c(m,n). Since IIII, it remains to treat the term II. In turn, we apply Lemma 2.2 (i) and Lemma 3.2 to infer that for any t,τΛϱ(θ) there holds

|(𝒖)ϱ^m(t)-(𝒖)ϱ^m(τ)|c[|(u)ϱ^(t)|m-1+|(u)ϱ^(τ)|m-1]|(u)ϱ^(t)-(u)ϱ^(τ)|cϱ1mθm-1Qϱ(θ)[|D𝒖m|+|F|]dxdt[|(u)ϱ^(t)|m-1+|(u)ϱ^(τ)|m-1],

where c=c(m,L). Taking squares on both sides, integrating with respect to t and τ over Λϱ(θ) and applying Hölder’s inequality and the sub-intrinsic coupling (4.1), we infer

IIcϱ2(m-1)mθ2(m-1)[Qϱ(θ)|u|2mdxdt]m-1m[Qϱ(θ)[|D𝒖m|+|F|]dxdt]2c[Qϱ(θ)[|D𝒖m|+|F|]dxdt]2

for a constant c depending only on n,m, and L. At this point, we use the estimates for I – III in (4) and obtain the claimed inequality. ∎

With the help of Lemma 4.1 we can now easily deduce a Poincaré-type inequality. Later on, Lemma 4.1 will also be the starting point for the proof of a Sobolev–Poincaré-type inequality; see Lemma 4.3.

Lemma 4.2.

Let m1 and let u be a weak solution to (1.1) in ΩT in the sense of Definition 1.1, where the vector-field A fulfills the growth and ellipticity assumptions (1.2). Then on any cylinder Qϱ(θ)(zo)ΩT satisfying the sub-intrinsic coupling (4.1) for some 0<ϱ1 and some θ>0, the Poincaré-type inequality

Qϱ(θ)(zo)|𝒖m-(𝒖m)zo;ϱ(θ)|2ϱ2dxdtcQϱ(θ)(zo)[|D𝒖m|2+|F|2]dxdt

holds true with a universal constant c=c(n,m,L).

Proof.

In the following we shall again omit for simplification the reference point zo in our notation. We will take estimate (4.2) from Lemma 4.1 as starting point for our considerations. To the first integral on the right-hand side, we apply Poincaré’s inequality slice wise for a.e. tΛϱ(θ). In this way, we obtain

Qϱ(θ)|𝒖m-(𝒖m)ϱ(t)|2ϱ2dxdtcQϱ(θ)|D𝒖m|2dxdt,

where c=c(n,m). Applying Hölder’s inequality to the second integral on the right-hand side of (4.2) yields the claimed Poincaré-type inequality on sub-intrinsic cylinders. ∎

The next statement can be interpreted as some sort of Sobolev–Poincaré inequality for the L2-deviation of 𝒖m from its mean value on the sub-intrinsic cylinder Qϱ(θ)(zo). Later on, we shall use this inequality to estimate the right-hand side in the energy inequality from Lemma 3.1. As usual, this leads to a reduction in the integration exponent of the energy term of the right-hand side, i.e. the integral containing D𝒖m. Similar to Lemma 4.2, we take Lemma 4.1 as starting point in the proof. Then the idea is to extract a part of the integration exponent from the L2-oscillation integral by the sup-term (occurring in the left-hand side of the energy estimate) and then to apply Sobolev’s inequality to the remainder.

Lemma 4.3.

Let m1 and let u be a weak solution to (1.1) in ΩT in the sense of Definition 1.1, where the vector-field A fulfills the growth and ellipticity assumptions (1.2). Then on any sub-cylinder Qϱ(θ)(zo)ΩT as in (4.1) for some 0<ϱ1 and some θ>0, and for any given ε(0,1] the following Sobolev-type inequality holds:

Qϱ(θ)(zo)|𝒖m-(𝒖m)zo;ϱ(θ)|2ϱ2dxdtεsuptΛϱ(θ)(to)Bϱ(xo)θm-1𝔟[𝒖m(,t),(𝒖m)zo;ϱ(θ)]ϱm+1mdx+cε2n[Qϱ(θ)(zo)|D𝒖m|2qdxdt]1q+cQϱ(θ)(zo)|F|2dxdt

for a universal constant c=c(n,m,L) and q:=nd<1.

Proof.

In the following, we shall again omit the reference point zo in our notation. As in the proof of Lemma 4.2 we take inequality (4.2) from Lemma 4.1 as starting point. Moreover, we abbreviate (𝒖m)ϱ(t) by (𝒖m)ϱ. From the context, it is clear that (𝒖m)ϱ is to be interpreted as a function of t. To the first integral on the right-hand side, we apply the lower bound for the boundary term from Lemma 2.3 (ii) and Hölder’s inequality with exponents m(n+2)m-1, n+2d. In this way, we obtain

Qϱ(θ)|𝒖m-(𝒖m)ϱ(t)|2ϱ2dxdt=cϱ2Qϱ(θ)|𝒖m-(𝒖m)ϱ|4n+2|𝒖m-(𝒖m)ϱ|2nn+2dxdtcϱ2Qϱ(θ)[|𝒖m|1m+|(𝒖m)ϱ|1m]2(m-1)n+2𝔟[𝒖m,(𝒖m)ϱ]2n+2|𝒖m-(𝒖m)ϱ|2nn+2dxdtcϱ2[Qϱ(θ)[|𝒖m|1m+|(𝒖m)ϱ|1m]2mdxdt]m-1m(n+2)[Qϱ(θ)𝔟[𝒖m,(𝒖m)ϱ]2d|𝒖m-(𝒖m)ϱ|2nddxdt]dn+2cϱ2[Qϱ(θ)|u|2mdxdt]m-1m(n+2)[Qϱ(θ)𝔟[𝒖m,(𝒖m)ϱ]2d|𝒖m-(𝒖m)ϱ|2nddxdt]dn+2.

Now, we use the sub-intrinsic coupling (4.1), Hölder’s inequality with exponents d2, dd-2 and for a.e. tΛϱ(θ) Sobolev’s inequality slicewise (note that 2nd1, since n2). This yields

Qϱ(θ)|𝒖m-(𝒖m)ϱ(t)|2ϱ2dxdtcθ2(m-1)n+2ϱ2dn+2[Qϱ(θ)𝔟[𝒖m,(𝒖m)ϱ]2d|𝒖m-(𝒖m)ϱ|2nddxdt]dn+2=c[Qϱ(θ)[θm-1𝔟[𝒖m,(𝒖m)ϱ]ϱm+1m]2d|𝒖m-(𝒖m)ϱ|2ndϱ2nddxdt]dn+2c[Λϱ(θ)[Bϱθm-1𝔟[𝒖m,(𝒖m)ϱ]ϱm+1mdx]2d[Bϱ|𝒖m-(𝒖m)ϱ|2nd-2ϱ2nd-2dx]d-2ddt]dn+2csuptΛϱ(θ)[Bϱθm-1𝔟[𝒖m(,t),(𝒖m)ϱ(t)]ϱm+1mdx]2n+2[Λϱ(θ)[Bϱ|𝒖m-(𝒖m)ϱ|2nd-2ϱ2nd-2dx]d-2ddt]dn+2csuptΛϱ(θ)[Bϱθm-1𝔟[𝒖m(,t),(𝒖m)ϱ(θ)]ϱm+1mdx]2n+2[Qϱ(θ)|D𝒖m|2nddxdt]dn+2,

with a universal constant c=c(n,m). In the last line we have used Lemma 2.7 in order to replace in the boundary term 𝔟 the slice wise mean (𝒖m)ϱ(t) by the mean (𝒖m)ϱ(θ). Inserting this inequality into (4.2) and applying Young’s and Hölder’s inequality, this results for any ε(0,1] in

Qϱ(θ)|𝒖m-(𝒖m)ϱ(θ)|2ϱ2dxdtcsuptΛϱ(θ)[Bϱθm-1𝔟[𝒖m(,t),(𝒖m)ϱ(θ)]ϱm+1mdx]2n+2[Qϱ(θ)|D𝒖m|2nddxdt]dn+2+c[Qϱ(θ)[|D𝒖m|+|F|]dxdt]2εsuptΛϱ(θ)Bϱθm-1𝔟[𝒖m(t),(𝒖m)ϱ(θ)]ϱm+1mdx+cε2n[Qϱ(θ)|D𝒖m|2nddxdt]dn+cQϱ(θ)|F|2dxdt.

This completes the proof of the Sobolev–Poincaré-type inequality. ∎

5 Reverse Hölder inequality

As it is well known, the core of each higher-integrability result is a so-called reverse Hölder inequality for the quantity in question, which in our case is the gradient D𝒖m. These reverse Hölder inequalities result in a certain way from the previously established Caccioppoli-type estimate and Sobolev–Poincaré-type inequalities. In principle, the right-hand side integrals of the Caccioppoli inequality are estimated by applying the Sobolev–Poincaré inequalities. However, the proof turns out to be more subtle than originally expected. The assumption that a sub-intrinsic coupling assumption must be imposed for the cylinder Q2ϱ(θ)(zo) is obvious, since this was presupposed in Lemma 4.3. However, this is not sufficient because the factor θm-1 in the energy estimate has to be converted into an Lm+1-oscillation integral of u. This is done by a super-intrinsic coupling on the cylinder Qϱ(θ)(zo); see the assumption (5.1)2. Both assumptions together, i.e. (5.1)1 and (5.1)2, mean that the cylinder Q2ϱ(θ)(zo) is intrinsic in some sense. On such an intrinsic cylinder the oscillations of u are small compared to the mean value of u. This case could be called the non-degenerate case.

Proposition 5.1.

Let m1 and let u be a weak solution to (1.1) in ΩT in the sense of Definition 1.1, where the vector-field A fulfills the structural assumptions (1.2). Then on any cylinder Q2ϱ(θ)(zo)ΩT with an intrinsic coupling of the form

Q2ϱ(θ)(zo)|u|2m(2ϱ)2dxdtθ2mQϱ(θ)(zo)|u|2mϱ2dxdt(5.1)

for some 0<ϱ1 and θ>0, the following reverse Hölder-type inequality holds true:

Qϱ(θ)(zo)|D𝒖m|2dxdtc[Q2ϱ(θ)(zo)|D𝒖m|2qdxdt]1q+cQ2ϱ(θ)(zo)|F|2dxdt

for some universal constant c=c(n,m,ν,L) and where q:=nd<1.

Proof.

Once again, we omit the reference to the center zo in the notation. We consider radii r,s with ϱr<s2ϱ. From the energy estimate in Lemma 3.1, we obtain

suptΛr(θ)Brθm-1𝔟[𝒖m(,t),(𝒖m)r(θ)]rm+1mdx+Qr(θ)|D𝒖m|2dxdtcQs(θ)|𝒖m-(𝒖m)r(θ)|2(s-r)2dxdt+cQs(θ)θm-1𝔟[𝒖m,(𝒖m)r(θ)]sm+1m-rm+1mdxdt+cQs(θ)|F|2dxdt=:I+II+III,(5.2)

with the obvious meaning of I, II, III. We abbreviate

r,s:=sm+12msm+12m-rm+12m,(5.3)

and observe that

sm+12m-rm+12m(s-r)m+12m.

This together with Lemma 2.5 yields for the first term

Icr,s4mm+1Qs(θ)|𝒖m-(𝒖m)s(θ)|2s2dxdt.(5.4)

For the second term we use the intrinsic coupling (5.1)2, Lemma 2.3 (ii)–(iii), Hölder’s inequality and Lemma 2.5 to infer that

IIcr,s2Qs(θ)θm-1𝔟[𝒖m,(𝒖m)r(θ)]sm+1mdxdtcr,s2[Qs(θ)|𝒖m-(𝒖m)r(θ)|2s2dxdt]m-12mQs(θ)𝔟[𝒖m,(𝒖m)r(θ)]sm+1mdxdt+cr,s2|(𝒖m)r(θ)|m-1mQs(θ)𝔟[𝒖m,(𝒖m)r(θ)]s2dxdtcr,s2Qs(θ)|𝒖m-(𝒖m)r(θ)|2s2dxdtcr,s2Qs(θ)|𝒖m-(𝒖m)s(θ)|2s2dxdt.

Inserting the estimates for I and II above and applying Lemma 4.3, we find for any ε(0,1] that

suptΛr(θ)Brθm-1𝔟[𝒖m(,t),(𝒖m)r(θ)]rm+1mdx+Qr(θ)|D𝒖m|2dxdt  cr,s4mm+1[εsuptΛs(θ)Bsθm-1𝔟[𝒖m(,t),(𝒖m)s(θ)]sm+1mdx+1ε2n[Qs(θ)|D𝒖m|2qdxdt]1q+Qs(θ)|F|2dxdt].

With the choice ε=12cr,s4mm+1, this yields

suptΛr(θ)Brθm-1𝔟[𝒖m(,t),(𝒖m)r(θ)]rm+1mdx+Qr(θ)|D𝒖m|2dxdt  12suptΛs(θ)Bsθm-1𝔟[𝒖m(,t),(𝒖m)s(θ)]sm+1mdx+cr,s4m(n+2)n(m+1)[Q2ϱ(θ)|D𝒖m|2qdxdt]1q+cr,s4mm+1Q2ϱ(θ)|F|2dxdt,

for a constant c=c(n,m,ν,L). To re-absorb the term 12[] from the right-hand side into the left-hand side, we apply the Iteration Lemma 2.1. This leads to the claimed reverse Hölder-type inequality, i.e. to

suptΛϱ(θ)Bϱθm-1𝔟[𝒖m(,t),(𝒖m)ϱ(θ)]ϱm+1mdx+Qϱ(θ)|D𝒖m|2dxdtc[Q2ϱ(θ)|D𝒖m|2qdxdt]1q+cQ2ϱ(θ)|F|2dxdt.

This finishes the proof of Proposition 5.1. ∎

The next lemma deals with the degenerate case which is characterized by the fact that u is small compared to the oscillations of u. In terms of integral quantities this means that on the one hand Q2ϱ(θ)(zo) is sub-intrinsic, and on the other hand the scaling parameter θ2m is smaller than the mean of |D𝒖m|2 on Qϱ(θ)(zo). As in the non-degenerate case, we need the assumption (5.5)1, i.e. that Q2ϱ(θ)(zo) is sub-intrinsic, as a prerequisite for the application of Lemma 4.3, which serves to deal with some of the right-hand side integrals of the Caccioppoli-type estimate. However, during this procedure, a term of the order of magnitude δθ2m appears, and it is precisely there where we need assumption (5.5)2, which converts this term into the oscillation term that can be re-absorbed into the left-hand side of Caccioppoli’s inequality.

Proposition 5.2.

Let m1 and let u be a weak solution to (1.1) in ΩT in the sense of Definition 1.1, where the vector-field A fulfills the structure assumptions (1.2). Then on any cylinder Q2ϱ(θ)(zo)ΩT satisfying a coupling of the form

Q2ϱ(θ)(zo)|u|2m(2ϱ)2dxdtθ2mKQϱ(θ)(zo)[|D𝒖m|2+|F|2]dxdt(5.5)

for some scaling parameter θ>0 and some constant K1, the following reverse Hölder-type inequality holds true:

Qϱ(θ)(zo)|D𝒖m|2dxdtc[Q2ϱ(θ)(zo)|D𝒖m|2qdxdt]1q+cQ2ϱ(θ)(zo)|F|2dxdt

with a constant c=c(n,m,ν,L)K(n+2)(m-1)n(m+1) and q:=nd<1.

Proof.

We omit in our notation the reference to the center zo. Furthermore, we consider radii r,s with ϱr<s2ϱ. As in the proof of Proposition 5.1 we start from inequality (5.2) which follows from the energy estimate in Lemma 3.1 and we recall the abbreviation (5.3). Estimate (5.4) for I is the same as in the proof of Proposition 5.1. This is clear, since we did not use hypothesis (5.1)2 for their proof. Therefore, it remains to consider the term II. Applying Young’s inequality, Lemma 2.3 (iii), and Lemma 2.5, we infer for any δ(0,1] that

IIr,s2Qs(θ)θm-1𝔟[𝒖m,(𝒖m)r(θ)]sm+1mdxdtδθ2m+r,s4mm+1δm-1m+1Qs(θ)𝔟[𝒖m,(𝒖m)r(θ)]2mm+1s2dxdtδθ2m+cr,s4mm+1δm-1m+1Qs(θ)|𝒖m-(𝒖m)r(θ)|2s2dxdtδθ2m+cr,s4mm+1δm-1m+1Qs(θ)|𝒖m-(𝒖m)s(θ)|2s2dxdt.

From (5.4), the preceding estimate and Lemma 4.3 we obtain for δ,ε(0,1] that

I+IIδθ2m+cr,s4mm+1δm-1m+1[εsuptΛs(θ)Bsθm-1𝔟[𝒖m(,t),(𝒖m)s(θ)]sm+1mdx+1ε2n[Qs(θ)|D𝒖m|2qdxdt]1q+Qs(θ)|F|2dxdt].

Moreover, from the coupling (5.5)2 we infer that

θ2m2dKQr(θ)[|D𝒖m|2+|F|2]dxdt.

We insert the estimates for I and II into (5.2) and choose δ=2-(d+1)K-1. This allows us to re-absorb the integral of |D𝒖m|2 into the left-hand side. Proceeding in this way, we obtain

suptΛr(θ)Brθm-1𝔟[𝒖m(,t),(𝒖m)r(θ)]rm+1mdx+Qr(θ)|D𝒖m|2dxdtcεKm-1m+1r,s4mm+1suptΛs(θ)Bsθm-1𝔟[𝒖m(,t),(𝒖m)s(θ)]sm+1mdx   +cKm-1m+1r,s4mm+1[1ε2n[Qs(θ)|D𝒖m|2qdxdt]1q+Qs(θ)|F|2dxdt].

At this stage the choice

ε=12cKm-1m+1r,s4mm+1

yields

suptΛr(θ)Brθm-1𝔟[𝒖m(,t),(𝒖m)r(θ)]rm+1mdx+Qr(θ)|D𝒖m|2dxdt12suptΛs(θ)Bsθm-1𝔟[𝒖m(,t),(𝒖m)s(θ)]sm+1mdx   +cK(m-1)(n+2)(m+1)nr,s4m(n+2)(m+1)n[[Q2ϱ(θ)|D𝒖m|2qdxdt]1q+Q2ϱ(θ)|F|2dxdt].

Now, we apply the Iteration Lemma 2.1 to re-absorb the sup-term from the right-hand side into the left. This leads us to

suptΛϱ(θ)Bϱθm-1𝔟[𝒖m(,t),(𝒖m)ϱ(θ)]ϱm+1mdx+Qϱ(θ)|D𝒖m|2dxdtc[Q2ϱ(θ)|D𝒖m|2qdxdt]1q+cQ2ϱ(θ)|F|2dxdt,

where the constant c is of the form c(n,m,ν,L)K(m-1)(n+2)(m+1)n. This finishes the proof of the proposition. ∎

6 Proof of the higher integrability

As we have seen in the last section, one can establish reverse Hölder inequalities in both the degenerate and the non-degenerate regime. It should be recalled, however, that the cylinders on which these reverse Hölder inequalities are valid, are essentially scaled by the solution u. More precisely, the relationship between Qϱ(θ)(zo)|u|2mϱ2dxdt, the scaling parameter θ and Qϱ(θ)(zo)|D𝒖m|2dxdt plays the decisive role. Therefore, the main objective in the proof of the higher integrability theorem is to find parabolic cylinders covering the super-level set of the spatial gradient of 𝒖m in the sense of a Vitali-type covering, such that on each cylinder either a coupling in the form of (5.1) or in the form of (5.5) holds true. These cylinders will be constructed by some sort of stopping time argument, combined with a rising sun-type construction. This very nice idea, which has already been explained in the introduction, goes back to [16]. Once the covering has been constructed by means of such cylinders, the application of the reverse Hölder inequalities leads to a quantitative estimate of |D𝒖m|2 on the super-level sets in terms of |D𝒖m|2q for q=nd<1. The decay in terms of the super-level sets can then be converted into the higher integrability of D𝒖m.

Before we start the construction of the system of non-uniform cylinders reflecting the character of the porous medium system as explained above, we fix the setup. We consider a fixed cylinder

Q8R(yo,τo)B8R(yo)×(τo-(8R)m+1m,τo+(8R)m+1m)ΩT

with R(0,1]. In the following, we abbreviate Qϱ:=Qϱ(yo,τo) for ϱ(0,8R] and define

λo:=1+[Q4R[|u|2m(4R)2+|D𝒖m|2+|F|2]dxdt]1m+1.

At this point, we recall the notation for space-time cylinders Qϱ(θ)(zo) from (2.1), which will be used in the following construction. Moreover, we observe that

Qϱ(θ)(zo)Q4R

whenever zoQ2R, ϱ(0,R] and θ1.

6.1 Construction of a non-uniform system of cylinders

The following construction of a non-uniform system of cylinders is similar to the one in [16, 27]. Let zoQ2R. For a radius ϱ(0,R] we define

θ~ϱθ~zo;ϱ:=inf{θ[λo,):1|Qϱ|Qϱ(θ)(zo)|u|2mϱ2dxdtθm+1}.

Note that θ~ϱ is well defined, since the set of those θλo for which the integral condition is satisfied, is non-empty. In fact, in the limit θ the integral on the left-hand side converges to zero, while the right-hand side blows up with speed θm+1. Note also that the condition in the infimum above can be rewritten as

Qϱ(θ)(zo)|u|2mϱ2dxdtθ2m.

Therefore, we either have that

θ~ϱ=λoandQϱ(θ~ϱ)(zo)|u|2mϱ2dxdtθ~ϱ2m=λo2m,

or that

θ~ϱ>λoandQϱ(θ~ϱ)(zo)|u|2mϱ2dxdt=θ~ϱ2m(6.1)

holds true. In any case we have θ~Rλo1. On the other hand, if λo<θ~R, then (again by definition and the fact that QR(θ~R)(zo)Q4R), we have

θ~Rm+1=1|QR|QR(θ~R)(zo)|u|2mR2dxdt42|QR|Q4R|u|2m(4R)2dxdt4d+2λom+1.

Therefore, we end up with the bound

θ~R4d+2m+1λo.(6.2)

Next, we establish that the mapping (0,R]ϱθ~ϱ is continuous. To this end, consider ϱ(0,R] and ε>0, and define θ+:=θ~ϱ+ε. Then there exists δ=δ(ε,ϱ)>0 such that

1|Qr|Qr(θ+)(zo)|u|2mr2dxdt<θ+m+1

for all radii r(0,R] with |r-ϱ|<δ. Indeed, the preceding strict inequality holds by the very definition of θ~ϱ with r=ϱ, since the integral on the left-hand side decreases with the replacement of θ~ϱ by θ+ (note that the domain of integration shrinks), while the right-hand side strictly increases. The claim now follows, since both, i.e. the integral on the right- and the left-hand side, are continuous with respect to the radius. With other words, we have shown that θ~rθ+=θ~ϱ+ε for r sufficiently close to ϱ. Therefore, it remains to prove θ~rθ-:=θ~ϱ-ε for r close to ϱ. This is clear from the construction if θ-λo, since θ~rλo for any r. In the other case, after diminishing δ=δ(ε,ϱ)>0 if necessary, we get

1|Qr|Qr(θ-)(zo)|u|2mr2dxdt>θ-m+1

for all r(0,R] with |r-ϱ|<δ. For r=ϱ, this is a direct consequence of the definition of θ~ϱ, since otherwise, we would have θ~ϱθ-, which is a contradiction. For r with |r-ϱ|<δ the claim follows from the continuity of both sides as a function of r. By definition of θ~r, the preceding inequality implies θ~rθ-=θ~ϱ-ε, as claimed. This completes the proof of the continuity of (0,R]ϱθ~ϱ.

Unfortunately, the mapping (0,R]ϱθ~ϱ might not be monotone. For this reason we modify θ~ϱ in a way, such that the modification – denoted by θϱ – becomes monotone. The precise construction is as follows: We define

θϱθzo;ϱ:=maxr[ϱ,R]θ~zo;r.

This construction can be viewed as a rising sun construction, because on those intervals (ϱ,r¯) on which θ~r<θ~r¯, for r(ϱ,r¯), one replaces θ~r by θ~r¯. Then by construction the mapping (0,R]ϱθϱ is continuous and monotonically decreasing; see Figure 1 for an illustration of the construction.

Illustration of the rising sun construction.
Figure 1

Illustration of the rising sun construction.

Moreover, the cylinders Qs(θϱ)(zo) are sub-intrinsic whenever ϱs. More specifically, we have

Qs(θϱ)(zo)|u|2ms2dxdtθϱ2mfor any 0<ϱsR.(6.3)

In fact, the definition of θs and its monotonicity imply θ~sθsθϱ, so that Qs(θϱ)(zo)Qs(θ~s)(zo). Therefore, we have

Qs(θϱ)(zo)|u|2ms2dxdt(θϱθ~s)m-1Qs(θ~s)(zo)|u|2ms2dxdt(θϱθ~s)m-1θ~s2m=θϱm-1θ~sm+1θϱ2m.

We now define

ϱ~:={Rif θϱ=λo,min{s[ϱ,R]:θs=θ~s}if θϱ>λo.(6.4)

In particular, we have θr=θ~ϱ~ for any r[ϱ,ϱ~]; see again Figure 1. Next, we claim that

θϱ(sϱ)d+2m+1θsfor any s(ϱ,R].(6.5)

In the case that θϱ=λo we know that also θs=λo, so that (6.5) trivially holds. Therefore, it remains to consider the case θϱ>λo. If s(ϱ,ϱ~], then θϱ=θs, and the claim (6.5) follows again. Finally, if s(ϱ~,R], then the monotonicity of ϱθϱ, (6.1) and (6.3) imply

θϱ=θ~ϱ~=[1|Qϱ~|Qϱ~(θϱ~)(zo)|u|2mϱ~2dxdt]1m+1(sϱ~)d+2m+1[1|Qs|Qs(θs)(zo)|u|2ms2dxdt]1m+1(sϱ)d+2m+1θs.

We now apply (6.5) with s=R. Since θR=θ~R the estimate (6.2) for θ~R yields

θϱ(Rϱ)d+2m+1θR(4Rϱ)d+2m+1λo.(6.6)

In the following, we consider the system of concentric cylinders Qϱ(θzo;ϱ)(zo) with radii ϱ(0,R] and zoQ2R. Note that the cylinders are nested in the sense that

Qr(θzo;r)(zo)Qs(θzo;s)(zo)whenever 0<r<sR.

The inclusion holds true due to the monotonicity of the mapping ϱθzo;ϱ. The disadvantage of this system of nested cylinders is, that in general the cylinders only fulfill a sub-intrinsic coupling condition.

6.2 Covering property

Here, we will prove a Vitali-type covering property for the cylinders constructed in the last subsection. The precise result is the following:

Lemma 6.1.

There exists a constant c^=c^(n,m)20 such that the following holds true: Let F be any collection of cylinders Q4r(θz;r)(z), where Qr(θz;r)(z) is a cylinder of the form constructed in Section 6.1 with radius r(0,Rc^). Then there exists a countable subfamily G of disjoint cylinders in F such that

QQQ𝒢Q^,(6.7)

where Q^ denotes the c^4-times enlarged cylinder Q, i.e. if Q=Q4r(θz;r)(z), then Q^=Qc^r(θz;r)(z).

Proof.

For j we consider the sub-collection

j:={Q4r(θz;r)(z):R2jc^<rR2j-1c^}

and choose 𝒢jj as follows: We let 𝒢1 be any maximal disjoint collection of cylinders in 1. Note that 𝒢1 is finite, since by (6.6) and the definition of 1 the n+1-measure of each cylinder Q𝒢1 is bounded from below. Now, assume that 𝒢1,𝒢2,,𝒢k-1 have already been selected for some integer k2. Then we choose 𝒢k to be any maximal disjoint subcollection of

{Qk:QQ= for any Qj=1k-1𝒢j}.

Note again that also 𝒢k is finite. Finally, we define

𝒢:=j=1𝒢j.

Then 𝒢 is a countable collection of disjoint cylinders and 𝒢. At this point it remains to prove that for each Q there exists a cylinder Q𝒢 with QQ, and that this implies QQ^.

To this end, fix Q=Q4r(θz;r)(z). Then there exists j such that Qj. By the maximality of 𝒢j, there exists a cylinder

Q=Q4r(θz;r)(z)i=1j𝒢i

with QQ. We know that rR2j-1c^ and r>R2jc^, so that r2r. This ensures that B4r(x)B20r(x). In the following, we shall prove

θz;r64d+2m+1θz;r.(6.8)

By r~[r,R] we denote the radius from (6.4) associated to the cylinder Qr(θz;r)(z). Recall that either Qr~(θz;r)(z) is intrinsic or r~=R and θz;r=λo. In the latter case we have due to the definition of θz;r that

θz;r=λoθz;r.

Therefore, we may assume that Qr~(θz;r)(z) is intrinsic, which means

θz;rm+1=1|Qr~|Qr~(θz;r)(z)|u|2mr~2dydτ.(6.9)

In the following, we distinguish between the cases r~Rη and r~>Rμ, where μ:=16. In the latter case we exploit (6.9) and the definition of λo and θz;r to obtain

θz;rm+1(4Rr~)21|Qr~|Q4R|u|2m(4R)2dydτ(4Rr~)d+2λom+1(4μ)d+2θz;rm+1.

This shows that

θz;r(4μ)d+2m+1θz;r.

Therefore, it suffices to consider the case r~Rμ. Since r~r and |x-x|<4r+4r12r, we know that B4r~(x)Bμr~(x). In addition, we have

|t-t|θz;r1-m(4r)m+1m+θz;r1-m(4r)m+1m.(6.10)

Without restriction one can now assume θz;rθz;r, because otherwise (6.8) trivially holds. Now, the monotonicity of ϱθz;ϱ and r2r2r~μr~ yield

θz;rθz;rθz;μr~,

so that

θz;r1-m(4r~)m+1m+|t-t|2θz;r1-m(4r~)m+1m+θz;r1-m(4r)m+1m28m+1mθz;μr~1-mr~m+1mθz;μr~1-m(μr~)m+1m.

But this means

Λ4r~(θz;r)(t)Λμr~(θz;μr~)(t).

Therefore, from (6.9) and (6.3) with ϱ=s=μr~, we obtain

θz;rm+1μ2|Qr~|Qμr~(θz;μr~)(z)|u|2m(μr~)2dydτμd+2θz;rm+1.

This implies that

θz;rμd+2m+1θz;r.

This finishes the proof of (6.8). With (6.10), r2r, and (6.8) we conclude

θz;r1-m(4r)m+1m+|t-t|2θz;r1-m(4r)m+1m+θz;r1-m(4r)m+1m4m+1m[1+22m+1m64(m-1)(d+2)m+1]θz;r1-mrm+1mθz;r1-m(c^r)m+1m

for a constant c^=c^(n,m)>4. This yields the inclusion Λ4r(θz;r)(t)Λc^r(θz;r)(t). After possibly enlarging c^, so that c^20, this implies QQ^=Qc^r(θz;r)(z). This establishes (6.7) and completes the proof of the Vitali covering-type lemma. ∎

6.3 Stopping time argument

For λ>λo and r(0,2R], we define the super-level set of the function |D𝒖m| by

𝑬(r,λ):={zQr:z is a Lebesgue point of |D𝒖m| and |D𝒖m|(z)>λm}.

The Lebesgue points are to be understood with regard to the cylinders constructed in Section 6.1. Note that n+1 a.e. point is a Lebesgue point with respect to these cylinders; cf. [14, Section 2.9.1] and the Vitali-type covering Lemma 6.1. For fixed radii RR1<R22R, we consider the concentric parabolic cylinders

QRQR1QR2Q2R.

Note that the inclusion

Qϱ(κ)(zo)=Bϱ(xo)×(to-κ1-mϱm+1m,to+κ1-mϱm+1m)QR2

holds true whenever zoQR1, κ[λo,) and ϱ(0,R2-R1]. We fix zo𝑬(R1,λ) and abbreviate θsθzo;s for s(0,R] throughout this section. By Lebesgue’s differentiation theorem, cf. [14, Section 2.9.1] we have that

lims0Qs(θs)(zo)[|D𝒖m|2+|F|2]dxdt|D𝒖m|2(zo)>λ2m.(6.11)

In the following, we consider values of λ satisfying

λ>Bλo,where B:=(4c^RR2-R1)n+2m+1>1,(6.12)

where c^=c^(n,m) denotes the constant from the Vitali-type covering Lemma 6.1. For radii s with

R2-R1c^sR(6.13)

we have, by the definition of λo, for any s as in (6.13) that

Qs(θs)(zo)[|D𝒖m|2+|F|2]dxdt|Q4R||Qs(θs)|Q4R[|D𝒖m|2+|F|2]dxdt|Q4R||Qs|θsm-1λom+1(4Rs)d+(d+2)(m-1)m+1λo2m(4c^RR2-R1)d+(d+2)(m-1)m+1λo2m=B2mλo2m<λ2m.

In the last chain of inequalities we used (6.6), (6.13) and d+(d+2)(m-1)m+1=2m(n+2)m+1. On the other hand, on behalf of (6.11) we find a sufficiently small radius 0<s<R2-R1c^ such that the above integral with Qs(θs)(zo) as domain of integration, possesses a value larger than λ2m. Consequently, by the absolute continuity of the integral there exists a maximal radius 0<ϱzo<R2-R1c^ such that

Qϱzo(θϱzo)(zo)[|D𝒖m|2+|F|2]dxdt=λ2m.(6.14)

The maximality of the radius ϱzo implies in particular that

Qs(θs)(zo)[|D𝒖m|2+|F|2]dxdt<λ2mfor any s(ϱzo,R].(6.15)

Finally, we know from the construction that Qc^ϱzo(θϱzo)(zo) is contained in Qc^ϱzo(zo), which in turn is contained in QR2.

6.4 A reverse Hölder inequality

As before, we consider zo𝑬(r1,λ) with λ as in (6.12) and abbreviate θϱzoθzo;ϱzo. As in (6.4) we construct the radius ϱ~zo[ϱzo,R]. Exactly at this point, we pass from the possibly sub-intrinsic cylinder Qϱzo(θϱzo)(zo) to the intrinsic cylinder Qϱ~zo(θϱzo)(zo). Observe that θs=θϱzo for any s[ϱzo,ϱ~zo], and, in particular, θϱ~zo=θϱzo. Our aim now is to prove the following reverse Hölder inequality:

Qϱzo(θϱzo)(zo)|D𝒖m|2dxdtc[Q4ϱzo(θϱzo)(zo)|D𝒖m|2qdxdt]1q+cQ4ϱzo(θϱzo)(zo)|F|2dxdt(6.16)

with q:=nd<1 and c=c(n,m,ν,L). We distinguish between the cases in which ϱ~zo2ϱzo or ϱ~zo>2ϱzo. In the case ϱ~zo2ϱzo we apply Proposition 5.1 on the intrinsic cylinder Qϱ~zo(θϱzo)(zo) (note that Qϱ~zo(θϱzo)(zo) is intrinsic and, thanks to (6.3), Q2ϱ~zo(θϱzo)(zo) is sub-intrinsic) and obtain

Qϱzo(θϱzo)(zo)|D𝒖m|2dxdt2dQϱ~zo(θϱzo)(zo)|D𝒖m|2dxdtc[Q2ϱ~zo(θϱzo)(zo)|D𝒖m|2qdxdt]1q+cQ2ϱ~zo(θϱzo)(zo)|F|2dxdtc[Q4ϱzo(θϱzo)(zo)|D𝒖m|2qdxdt]1q+cQ4ϱzo(θϱzo)(zo)|F|2dxdt,

where c=c(n,m,ν,L). In the other case ϱ~zo>2ϱzo, we want to apply Proposition 5.2 on the cylinder Qϱzo(θϱzo)(zo). However, this is only permitted if the hypothesis (5.5) is satisfied. First, we notice that (5.5)1 is an immediate consequence of (6.3), and therefore we only need to verify (5.5)2. To this end, we consider two cases. If θϱzo=λo, we obtain (5.5)2 by the following computation:

θϱzo2m=λo2m<λ2m=Qϱzo(θϱzo)(zo)[|D𝒖m|2+|F|2]dxdt.

Here we used (6.14) for the last identity. If θϱzo>λo, then by construction Qϱ~zo(θϱzo)(zo) is intrinsic. Moreover, since 12ϱ~zo>ϱzo, we can apply (6.3) with (ϱ,s) replaced by (ϱzo,12ϱ~zo). This together with Lemma 4.2 and (6.15) (applied with s=ϱ~zo(ϱzo,R]) ensures that

θϱzo=[Qϱ~zo(θϱzo)(zo)|u|2mϱ~zo2dxdt]12m[Qϱ~zo(θϱzo)(zo)|𝒖m-(𝒖m)zo;12ϱ~zo(θϱzo)|2ϱ~zo2dxdt]12m+|(𝒖m)zo;12ϱ~zo(θϱzo)|1mϱ~zo1mc[Qϱ~zo(θϱzo)(zo)|𝒖m-(𝒖m)zo;ϱ~zo(θϱzo)|2ϱ~zo2dxdt]12m+2-1m[Q12ϱ~zo(θϱzo)(zo)|u|2m(12ϱ~zo)2dxdt]12mc[Qϱ~zo(θϱzo)(zo)[|D𝒖m|2+|F|2]dxdt]12m+2-1mθϱzocλ+2-1mθϱzo

for a constant c=c(n,m,L). Re-absorbing 2-1mθϱzo into the left-hand side and using (6.14), we find that

θϱzocλ=c[Qϱzo(θϱzo)(zo)[|D𝒖m|2+|F|2]dxdt]12m

for a constant c=c(n,m,L)1. This yields (5.5)2 in the second case with K=c2m1. Therefore, we are allowed to apply Proposition 5.2 on the cylinder Qϱzo(θϱzo)(zo), thereby obtaining that

Qϱzo(θϱzo)(zo)|D𝒖m|2dxdtc[Q2ϱzo(θϱzo)(zo)|D𝒖m|2qdxdt]1q+cQ2ϱzo(θϱzo)(zo)|F|2dxdt.

In conclusion, we have shown that in any case the claimed reverse Hölder inequality (6.16) holds true.

6.5 Estimate on super-level sets

So far we have shown that if λ satisfies (6.12), then for every zo𝑬(R1,λ) there exists a cylinder Qϱzo(θzo;ϱzo)(zo) with Qc^ϱzo(θzo;ϱzo)(zo)QR2 such that (6.14), (6.15) and (6.16) hold true on this specific cylinder. As before, we abbreviate θϱzoθzo;ϱzo. We define the super-level set of the inhomogeneity F by

𝑭(r,λ):={zQr:z is a Lebesgue point of F and |F|>λm}.

As for the super-level set 𝑬(r,λ) the Lebesgue points have to be understood with regard to the cylinders constructed in Section 6.1. Using (6.14) and (6.16), we obtain for η(0,1] (to be specified later in a universal way) that

λ2m=Qϱzo(θϱzo)(zo)[|D𝒖m|2+|F|2]dxdtc[Q4ϱzo(θϱzo)(zo)|D𝒖m|2qdxdt]1q+cQ4ϱzo(θϱzo)(zo)|F|2dxdtcη2mλ2m+c[1|Q4ϱzo(θϱzo)(zo)|Q4ϱzo(θϱzo)(zo)𝑬(R2,ηλ)|D𝒖m|2qdxdt]1q+c|Q4ϱzo(θϱzo)(zo)|Q4ϱzo(θϱzo)(zo)𝑭(R2,ηλ)|F|2dxdt

for a constant c=c(n,m,ν,L). In the preceding inequality we choose the η in the form η2m=12c. This choice allows the re-absorption of 12λ2m into the left-hand side. Furthermore, we use Hölder’s inequality and (6.15) to estimate

[1|Q4ϱzo(θϱzo)(zo)|Q4ϱzo(θϱzo)(zo)𝑬(R2,ηλ)|D𝒖m|2qdxdt]1q-1[Q4ϱzo(θϱzo)(zo)|D𝒖m|2dxdt]1-qλ2m(1-q).

We insert this above, and multiply the result, i.e. the inequality where we already fixed η and re-absorbed 12λ2m, by |Q4ϱzo(θϱzo)(zo)|. This leads to the inequality

λ2m|Q4ϱzo(θϱzo)(zo)|cQ4ϱzo(θϱzo)(zo)𝑬(R2,ηλ)λ2m(1-q)|D𝒖m|2qdxdt+cQ4ϱzo(θϱzo)(zo)𝑭(R2,ηλ)|F|2dxdt

again with c=c(n,m,ν,L). Now, (6.15) with the choice s=c^ϱzo allows us to estimate λ2m from below. The precise argument is as follows: Using in turn (6.15), the monotonicity of ϱθϱ and (6.5), i.e. that θc^ϱzoθϱzoc^d+2m+1θc^ϱzo, we obtain that

λ2m>Qc^ϱzo(θc^ϱzo)(zo)|D𝒖m|2dxdt1c^(m-1)(d+2)m+1Qc^ϱzo(θϱzo)(zo)|D𝒖m|2dxdt.

Inserting this above and keeping in mind that c^ depends only on n and m, we deduce

Qc^ϱzo(θϱzo)(zo)|D𝒖m|2dxdtcQ4ϱzo(θϱzo)(zo)𝑬(R2,ηλ)λ2m(1-q)|D𝒖m|2qdxdt+cQ4ϱzo(θϱzo)(zo)𝑭(R2,ηλ)|F|2dxdt(6.17)

with c=c(n,m,ν,L).

So far, we showed that for any value λ>Bλo the super-level set 𝑬(R1,λ) can be covered by a family {Q4ϱzo(θzo;ϱzo)(zo)} of parabolic cylinders with center zo𝑬(R1,λ), which are contained in QR2, and such that on each cylinder estimate (6.5) holds true. At this point, we use the Vitali-type Covering Lemma 6.1 and gain a countable subfamily

{Q4ϱzi(θzi;ϱzi)(zi)}i

consisting of pairwise disjoint cylinders, such that the c^4-times enlarged cylinders Qc^ϱzi(θzi;ϱzi)(zi) are contained in QR2 and cover the super-level set 𝑬(R1,λ), i.e.

𝑬(R1,λ)i=1Qc^ϱzi(θzi;ϱzi)(zi)QR2.

Since the cylinders Q4ϱzi(θzi;ϱzi)(zi) are pairwise disjoint, we obtain from (6.5) that

𝑬(R1,λ)|D𝒖m|2dxdti=1Qc^ϱzi(θzi;ϱzi)(zi)|D𝒖m|2dxdtci=1Q4ϱzi(θzi;ϱzi)(zi)𝑬(R2,ηλ)λ2m(1-q)|D𝒖m|2qdxdt+ci=1Q4ϱzi(θzi;ϱzi)(zi)𝑭(R2,ηλ)|F|2dxdtc𝑬(R2,ηλ)λ2m(1-q)|D𝒖m|2qdxdt+c𝑭(R2,ηλ)|F|2dxdt,

where the constant c depends only on n, m, ν, and L. On 𝑬(R1,ηλ)𝑬(R1,λ) we have the pointwise bound |D𝒖m|2λ2m and therefore

𝑬(R1,ηλ)𝑬(R1,λ)|D𝒖m|2dxdt𝑬(R2,ηλ)λ2m(1-q)|D𝒖m|2qdxdt.

We combine the last two inequalities and get the following reverse Hölder inequality on super-level sets:

𝑬(R1,ηλ)|D𝒖m|2dxdtc𝑬(R2,ηλ)λ2m(1-q)|D𝒖m|2qdxdt+c𝑭(R2,ηλ)|F|2dxdt.

Here, we replace ηλ by λ and recall that η<1 depends only on n,m,ν, and L. With this replacement we obtain for any ληBλo=:λ1 that

𝑬(R1,λ)|D𝒖m|2dxdtc𝑬(R2,λ)λ2m(1-q)|D𝒖m|2qdxdt+c𝑭(R2,λ)|F|2dxdt(6.18)

holds true with a constant c=c(n,m,ν,L). This is the desired estimate on super-level sets.

6.6 Proof of the gradient estimate

For k>λ1 we define the truncation of |D𝒖m| by

|D𝒖m|k:=min{|D𝒖m|,km},

and for r(0,2R] the corresponding super-level set

𝑬k(r,λ):={zQr:|D𝒖m|k>λm}.

Note that |D𝒖m|k|D𝒖m| a.e., as well as 𝑬k(r,λ)= for kλ and 𝑬k(r,λ)=𝑬(r,λ) for k>λ. Therefore, it follows from (6.18) that

𝑬k(R1,λ)|D𝒖m|k2-2q|D𝒖m|2qdxdtc𝑬k(R2,λ)λ2m(1-q)|D𝒖m|2qdxdt+c𝑭(R2,λ)|F|2dxdt

whenever k>λλ1. Since 𝑬k(r,λ)= for kλ, the last inequality also holds in this case. Now, we multiply the preceding inequality by λεm-1, where ε(0,1] will be chosen later in a universal way, and integrate the result with respect to λ over the interval (λ1,). This gives

λ1λεm-1[𝑬k(R1,λ)|D𝒖m|k2-2q|D𝒖m|2qdxdt]dλcλ1λm(2-2q+ε)-1[𝑬k(R2,λ)|D𝒖m|2qdxdt]dλ+cλ1λεm-1[𝑭(R2,λ)|F|2dxdt]dλ.(6.19)

Here we exchange the order of integration with the help of Fubini’s theorem. For the integral on the left-hand side Fubini’s theorem implies

λ1λεm-1[𝑬k(R1,λ)|D𝒖m|k2-2q|D𝒖m|2qdxdt]dλ=𝑬k(R1,λ1)|D𝒖m|k2-2q|D𝒖m|2q[λ1|D𝒖m|k1mλεm-1dλ]dxdt=1εm𝑬k(R1,λ1)[|D𝒖m|k2-2q+ε|D𝒖m|2q-λ1εm|D𝒖m|k2-2q|D𝒖m|2q]dxdt,

while for the first integral on the right-hand side we find that

λ1λm(2-2q+ε)-1[𝑬k(R2,λ)|D𝒖m|2qdxdt]dλ=𝑬k(R2,λ1)|D𝒖m|2q[λ1|D𝒖m|k1mλm(2-2q+ε)-1dλ]dxdt1m(2-2q+ε)𝑬k(R2,λ1)|D𝒖m|k2-2q+ε|D𝒖m|2qdxdt12m(1-q)𝑬k(R2,λ1)|D𝒖m|k2-2q+ε|D𝒖m|2qdxdt.

Finally, for the last integral in (6.19) we obtain

λ1λεm-1[𝑭(R2,λ)|F|2dxdt]dλ=𝑭(R2,λ1)|F|2[λ1|F|1mλεm-1dλ]dxdt1εm𝑭(R2,λ1)|F|2+εdxdt1εmQ2R|F|2+εdxdt.

We insert these estimates into (6.19) and multiply by εm. This leads to

𝑬k(R1,λ1)|D𝒖m|k2-2q+ε|D𝒖m|2qdxdtλ1εm𝑬k(R1,λ1)|D𝒖m|k2-2q|D𝒖m|2qdxdt+cε1-q𝑬k(R2,λ1)|D𝒖m|k2-2q+ε|D𝒖m|2qdxdt+cQ2R|F|2+εdxdt.

The last inequality is now combined with the corresponding inequality on the complement QR1𝑬k(R1,λ1), i.e. with the inequality

QR1𝑬k(R1,λ1)|D𝒖m|k2-2q+ε|D𝒖m|2qdxdtλ1εmQR1𝑬k(R1,λ1)|D𝒖m|k2-2q|D𝒖m|2qdxdt.

We also take into account that |D𝒖m|k|D𝒖m|. All together this gives the inequality

QR1|D𝒖m|k2-2q+ε|D𝒖m|2qdxdtcε1-qQR2|D𝒖m|k2-2q+ε|D𝒖m|2qdxdt+λ1εmQ2R|D𝒖m|2dxdt+cQ2R|F|2+εdxdt,

where c=c(n,m,ν,L)1. Now, we choose

0<εmin{εo,σ-2},where εo:=1-q2c<1.

Note that εo depends only on n,m,ν, and L. Moreover, observe that λ1ε(ηBλo)εBλoε, since η1, B1 and 0<ε1. Therefore, from the previous inequality we conclude that for any pair of radii R1, R2 with RR1<R22R there holds

QR1|D𝒖m|k2-2q+ε|D𝒖m|2qdxdt12QR2|D𝒖m|k2-2q+ε|D𝒖m|2qdxdt+c(RR2-R1)m(n+2)m+1λoεmQ2R|D𝒖m|2dxdt+cQ2R|F|2+εdxdt.

We can now apply the Iteration Lemma 2.1 to the last inequality, which yields

QR|D𝒖m|k2-2q+ε|D𝒖m|2qdxdtcλoεmQ2R|D𝒖m|2dxdt+cQ2R|F|2+εdxdt.

On the left side we apply Fatou’s lemma and pass to the limit k. In the result, we go over to means on both sides. This gives

QR|D𝒖m|2+εdxdtcλoεmQ2R|D𝒖m|2dxdt+cQ2R|F|2+εdxdt.

At this point, we estimate λo with the help of the energy estimate from Lemma 3.1 applied with θ=1 and a=0 and Hölder’s inequality. This leads to the bound

λoc[1+Q8R[|u|2mR2+|F|2]dxdt]1m+1,

where c=c(m,ν,L). Inserting this above, we deduce

QR|D𝒖m|2+εdxdtc[1+Q8R[|u|2mR2+|F|2]dxdt]εmm+1Q2R|D𝒖m|2dxdt+cQ2R|F|2+εdxdt,

where c=c(n,m,ν,L). The claimed estimate (1.8) involving the cylinders QR and Q2R now follows by a covering argument. This completes the proof of Theorem 1.2. ∎

6.7 Proof of Corollary 1.3

It remains to deduce a corresponding estimate on a standard parabolic cylinder

C2R(zo):=B2R(xo)×(to-(2R)2,to+(2R)2)ΩT.

To this end, we rescale the solution u, the vector-field 𝐀, and the right-hand side F via

{v(x,t):=u(xo+Rx,to+R2t),𝐁(x,t,u,ξ):=R𝐀(xo+Rx,to+R2t,u,1Rξ),G(x,t):=RF(xo+Rx,to+R2t)

whenever (x,t)C2 and (u,ξ)N×Nn. Then v is a weak solution of the differential equation

tv-div𝐁(x,t,v,D𝒗m)=divGin Q2C2,

in the sense of Definition 1.1. Moreover, assumptions (1.2) are satisfied for the rescaled vector-field 𝐁 in place of 𝐀. Therefore, estimate (1.8) is applicable to v on the cylinder Q2, which yields

Q1|D𝒗m|2+εdxdtc[1+Q2[|v|2m+|G|2]dxdt]εmm+1Q2|D𝒗m|2dxdt+cQ2|G|2+εdxdt

for every ε(0,εo], with a constant c=c(n,m,ν,L). Scaling back and recalling that Q2C2, we arrive at the estimate

R2+εCR(zo)|D𝒖m|2+εdxdtcR2[1+C2R(zo)[|u|2m+R2|F|2]dxdt]εmm+1C2R(zo)|D𝒖m|2dxdt+cR2+εC2R(zo)|F|2+εdxdt.

Dividing both sides by R2+ε yields the assertion of Corollary 1.3. ∎

Acknowledgements

We would like to thank Juha Kinnunen for many constructive discussions on the subject and his persuasive work to pursue this topic further. Without his motivational efforts this work would probably not have come about.

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About the article

Received: 2017-12-06

Accepted: 2018-01-05

Published Online: 2018-06-06


Funding Source: Suomen Akatemia

Award identifier / Grant number: Project 308063

The third author was supported by the Academy of Finland, project 308063.


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 1004–1034, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0270.

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