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Advanced Nonlinear Studies

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Volume 16, Issue 1

Issues

Weighted Gagliardo–Nirenberg Inequalities Involving BMO Norms and Solvability of Strongly Coupled Parabolic Systems

Dung Le
Published Online: 2016-01-27 | DOI: https://doi.org/10.1515/ans-2015-5006

Abstract

New weighted Gagliardo–Nirenberg inequalities are introduced together with applications to the local/global existence of solutions to nonlinear strongly coupled and uniform parabolic systems. Much weaker sufficient conditions than those existing in literature for solvability of these systems will be established.

Keywords: Parabolic Systems; Hölder Regularity;

MSC 2010: 35J70; 35B65; 42B37

1 Introduction

In [14, 16], for any p1 and C2 function u on n, n2, global and local Gagliardo–Nirenberg inequalities of the form

n|Du|2p+2𝑑xC(n,p)uBMO2n|Du|2p-2|D2u|2𝑑x(1.1)

were established and applied to the solvability of scalar elliptic equations.

In this paper, we provide global and local versions of the above inequality with the Lebesgue measure dx replaced by wdx where w is some Ap weight. The purpose of such generalization becomes clear when we apply the results to the study of local/global existence of strong solutions to the following nonlinear strongly coupled and nonregular but uniform parabolic system:

{ut=div(A(x,t,u,Du))+f^(x,t,u,Du),(x,t)Q=Ω×(0,T0),u(x,0)=U0(x),xΩ,u=0on Ω×(0,T0).(1.2)

Here, and throughout this paper, Ω is a bounded domain with smooth boundary Ω in n. A typical point in n is denoted by x and a point in n×[0,) is denoted by z=(x,t). The temporal and k-order spatial derivatives of a vector-valued function

u(x,t)=(u1(x,t),,um(x,t))T,m>1

are denoted by ut and Dku, respectively. A(x,t,u,Du) is a full m×n matrix, and f^:Ω××m×nmm. The initial data U0 is given in W1,r0(Ω,m) for some r0>n. As usual, Wk,p(Ω,m), where k is an integer and p1, denotes the standard Sobolev spaces whose elements are vector-valued functions u:Ωm with finite norm

uWk,p(Ω,m)=uLp(Ω,m)+i=1kDkuLp(Ω,m).

By a strong solution of (1.2) we mean a vector-valued function u that solves (1.2) a.e. in Ω×(0,T0) and continuously assumes the initial value U0 at t=0 and boundary data on Ω×(0,T0). Moreover, for some α>0 and all t(0,T0) we have Du(,t)Cα,α2(Ω) and D2u(,t)Lp(Ω) for a.e. t(0,T0) and all p>1.

The strongly coupled system (1.2) appears in many physical applications, for instance, Maxwell–Stephan systems describing the diffusive transport of multicomponent mixtures, models in reaction and diffusion in electrolysis, flows in porous media, diffusion of polymers, or population dynamics. We refer the reader to the recent work [10] and the references therein for the models and the existence of their weak solutions. Besides the question whether a strong solution of (1.2) can exist locally near t=0, we face with a fundamental problem in the theory of PDEs to establish that this local solution exists globally. Unlike the well-established theory for scalar parabolic equations (i.e. m=1), where bounded solutions usually exist globally, there are counter examples for systems (m>1) which exhibit solutions that start smoothly and remain bounded but develop singularities in higher norms in finite times (see [8]). Even more, bounded solutions to (1.2) may not be even Hölder continuous everywhere.

We will impose the following structural conditions on (1.2). In this paper, for a vector- or matrix-valued function f(u,ζ), um and ζd, its partial derivatives will be denoted by fu,fζ.

  • (A)

    A(x,t,u,ζ) is C1 in xn, um and ζmn. Moreover, there are constants C*,C,λ0>0 and a scalar C1 function λ(x,t,u) such that, for any (x,t)n+1, um and ζ,ξnm,

    λ(x,t,u)|ζ|2Aζ(x,t,u,ζ)ξ,ξand|Aζ(x,t,u,ζ)|C*λ(x,t,u).(1.3)

    We also assume λ(x,t,u)λ0 and |Au(x,t,u,ζ)|C|λu||ζ|.

If λ(x,t,u) is also bounded from above by a constant, we say that A is regular elliptic. Otherwise, A is uniformly elliptic. The constant C* in (1.3) concerns the ratio between the largest and smallest eigenvalues of Aζ. We assume that these constants are not too far apart in the following sense.

  • (SG)

    (The spectral gap condition) (n-2)/n<C*-1.

We note that if this condition is somewhat violated then examples of blowing up in finite time can occur (see [1]).

Concerning f^, we will assume the following.

  • (F)

    There exist a constant C and a function f(x,t,u) which is C1 in x,u such that, for any C1 functions u:Ωm and p:Ωmn,

    |f^(x,t,u,p)|Cλ12(x,t,u)|p|+f(x,t,u),|Df^(x,t,u,p)|Cλ12(x,t,u)|Dp|+C|λu(x,t,u)|λ12(x,t,u)|Du||p|+|fu(x,t,u)||Du|,(1.4)|fu(x,t,u)|Cλ(x,t,u).(1.5)

For simplicity in our statements and proof, as the presence of x,t can be treated similarly, we will mostly assume that A,f^ are independent of x,t in this paper.

In the last decades, papers concerning strongly coupled parabolic systems like (1.2), with A(x,t,u,Du) being linear in Du, i.e. A(x,t,u,Du)=A(x,t,u)Du, usually relied on the results of Amann [3, 2] who showed that a solution to (4.1) exists globally if its W1,r0(Ω) norm for some r0>n, where n is the dimension of Ω, does not blow up in finite time. This requires the existence of a continuous function 𝒞 on (0,) such that

u(,t)W1,r0(Ω,m)𝒞(t)for all t(0,T0) and some r0>n.(1.6)

The verification of (1.6) is very difficult and equivalently requires Hölder continuity of the solution u. This is a very hard problem in the theory of PDEs as known techniques for the regularity of solutions to scalar equations could not be extended to systems, and counterexamples were available. Maximum or comparison principles for systems generally do not hold so that the boundedness of solutions to (4.1) is unknown. Even if the solutions are bounded, only partial regularity results are known (see [6]).

Furthermore, the assumption (1.6) gives the boundedness of u so that the ellipticity constants for the matrix A(x,t,u) are bounded. Thus A is regular elliptic. Without this assumption, one has to consider the case A(x,t,u) being uniformly elliptic when the smallest and largest eigenvalues of A(x,t,u) can be unbounded but comparable as in (1.3).

In this paper, we will replace (1.6) by a much weaker condition. Namely, we will show that it suffices to control the BMO norm of u and the uniform continuity of this norm in small balls. Roughly speaking, we will replace condition (1.6) by the following:

for any ε>0 there is Rε>0 such that supBRεΩu(,t)BMO(BRε)ε for all t(0,T0).(1.7)

By the Poincaré–Sobolev inequality, it is clear that (1.6) implies (1.7), even when r0=n.

On the other hand, since we will consider nonregular parabolic systems with A being nonlinear in Du, Amann’s results are not applicable here to give the solvability of (1.2). We will then provide an alternative approach to establish local/global existence results for (1.2) via Leray–Schauder fixed point theories. The existence results will be proven under a set of general and practical structural conditions on A,f^. Roughly speaking, we will embed (1.2) in a family of nonlinear systems which satisfy the same set of assumptions for (1.2). The strong solutions of these systems are fixed points of a family of compact vector fields in some appropriate Banach space. The key step in the argument is the establishment of a uniform bound for such solutions. We obtain the desired bound by using the local weighted Gagliardo–Nirenberg inequalities in Section 2 to deduce a decay estimate for local norms of the solutions so that an iteration argument can apply.

Though (1.7) will be required to hold uniformly for all strong solutions of the systems in the family, but, as these systems assume the same hypotheses for (4.1), we practically need only verify (1.7) for (1.2).

The techniques in this paper also give higher regularity of solutions to other systems where (1.7) yields that u is Hölder continuous. We thus devote Section 3 to the a-priori estimates and the regularity of solutions to the following system:

ut=div(A(x,t,W,Du))+f^(x,t,W,Du),

where W,u are related in some way. Later, the case W=σu for some σ(0,1] will be used in our fixed point argument to obtain local and global existence results.

We conclude this paper with Section 4 where we apply the estimates in Section 3 to study the solvability of (1.2) under the assumption (1.7).

2 Weighted Gagliardo–Nirenberg Inequalities

In this section we will establish global and local weighted Gagliardo–Nirenberg interpolation inequalities which allow us to control the Lp norm of the derivatives of the solutions in the proof of our main theorems. These inequalities generalize those in [14, 16], where no weight versions were proved (see Remark 2.2).

Here and throughout this paper, we write BR(x) for a ball centered at x with radius R and will omit x if no ambiguity can arise. In our statements and proofs, we use C,C1, to denote various constants which can change from line to line but depend only on the parameters of the hypotheses in an obvious way. We will write C(a,b,) when the dependence of a constant C on its parameters a,b, is needed to emphasize that C is bounded in terms of its parameters.

For any measurable subset A of Ω and any locally integrable function U:Ωm we denote by |A| the Lebesgue measure of A and by UA the average of U over A. That is,

UA=AU(x)𝑑x=1|A|AU(x)𝑑x.

In order to state the assumption for this type of inequalities, we recall some well-known notions from Harmonic Analysis.

For γ(1,) we say that a nonnegative locally integrable function w belongs to the class Aγ or w is an Aγ weight if the quantity

[w]γ:=supBR(y)Ω(BR(y)w𝑑x)(BR(y)w1-γ𝑑x)γ-1

is finite. Here, γ=γ/(γ-1). The A and A1 classes are defined by A=γ>1Aγ and A1=γ>1Aγ. For more details on these classes we refer the reader to [4, 13, 15]. Clearly, the above implies

(BR(y)w𝑑x)(BR(y)w-1μ𝑑x)μ[w]μ+1for all μ>0.

A locally integrable function U:Ωm is said to be BMO if the quantity

[U]*:=supBR(y)ΩBR(y)|U-UBR(y)|𝑑x

is finite.

The Banach space BMO(Ω,m) consists of functions with finite norm

UBMO(Ω,m):=[U]*+UL1(Ω,m).

When no ambiguity can arise, we simply say U is BMO and omit Ω or m from the above notations.

We first have the following global weighted Gagliardo–Nirenberg inequality.

Let u,U:Ωm be vector-valued functions with uC1(Ω), UC2(Ω) and Φ:m be a C1 function. Suppose that either U or Φ2(u)Uν vanish on the boundary Ω of Ω. We set

I1:=ΩΦ2(u)|DU|2p+2dx,I^1:=ΩΦ2(u)|Du|2p+2dx,(2.1)I¯1:=Ω|Φu(u)|2(|DU|2p+2+|Du|2p+2)dx,I2:=ΩΦ2(u)|DU|2p-2|D2U|2dx.(2.2)

Suppose that

  • (GN)

    Φ(u)2p+2 belongs to the Ap/(p+2)+1 class.

Then for any ε>0 there is a constant Cε,Φ depending on ε and [Φ2p+2(u)]p/(p+2)+1 for which

I1εI^1+Cε,ΦUBMO(Ω)2[I¯1+I2].(2.3)

In the proof of this lemma we will make use of the following well-known facts from Harmonic Analysis. We first recall the definition of the centered and uncentered Hardy–Littlewood maximal operators acting on function FLloc1(Ω):

M(F)(y)=supε{Bε(y)F(x)𝑑x:ε>0 and Bε(y)Ω},M*(F)(z)=supzBε(y),ε{Bε(y)F(x)𝑑x:ε>0 and Bε(y)Ω}.

We also note here the Hardy–Littlewood theorem: for any FLq(Ω) we have

ΩM(F)q𝑑xC(q)ΩFq𝑑x,q>1.(2.4)

More generally, the Muckenhoupt theorem [12] states that if w is an Aq weight then, for any FLq(Ω),

ΩM(F)qw𝑑xΩM*(F)qw𝑑xC([w]q)ΩFqw𝑑x.(2.5)

We also make use of Hardy spaces 1. For any yΩ and ε>0, let ϕ be any function in C0(B1(y)) with |Dϕ|C1. Let ϕε(x)=ε-nϕ(xε) (then |Dϕε|C1ε-1-n). From [15], a function g is in 1(Ω) if

supε>0g*ϕεL1(Ω)andg1=gL1(Ω)+supε>0g*ϕεL1(Ω).

We are now ready to give the proof of Lemma 2.1.

Proof.

We can assume that m=1 because the proof for the vectorial case is similar. Integrating by parts, we have

I1=ΩΦ2(u)|DU|2p+2𝑑x=-ΩUdiv(Φ2(u)|DU|2pDU)dx.(2.6)

We will show that g=div(Φ2(u)|DU|2pDU) belongs to the Hardy space 1 and

g1=Ωsupε|g*ϕε|dxC([Φ2p+2(u)]p/(p+2)+1)[I¯112(I112+I^112)+I112I212].(2.7)

Once this is established, (2.6) and the Fefferman–Stein theorem on the duality of the BMO and Hardy spaces yield

I1UBMOg1C([Φ2p+2(u)]p/(p+2)+1)UBMO[I¯112(I112+I^112)+I112I212].

A simple use of Young’s inequality to the right-hand side then gives (2.3).

Therefore, in the rest of the proof we need only establish (2.7). We then write g=g1+g2 with gi=divVi, setting

V1=Φ(u)|DU|p+1(Φ(u)|DU|p-1DU-BεΦ(u)|DU|p-1DU𝑑x),V2=Φ(u)|DU|p+1BεΦ(u)|DU|p-1DU𝑑x.

Let us consider g1 first and define h=Φ(u)|DU|p-1DU. For any yΩ and Bε=Bε(y)Ω, we use integration by parts, the property of ϕε and then Hölder’s inequality for any s>1 to get

|g1*ϕε(y)|=|Bε(y)Dϕ(x-yε)(h-hBε(y))Φ(u)|DU|p+1𝑑x|C1ε|Bε(y)|h-hBε(y)|Φ(u)|DU|p+1𝑑x|C1ε(Bε(y)|h-hBε(y)|s𝑑x)1s(Bε(y)Φs(u)|DU|(p+1)s𝑑x)1s.

There is a constant C such that |Dh||Φu(u)||Du||DU|p+pΦ|DU|p-1|D2U|. Poincaré–Sobolev’s inequality, with s*=ns/(n+s), then gives

C1ε(Bε|h-hBε|s𝑑x)1sC(Bε|Dh|s*𝑑x)1s*C[Bε|Φu(u)|s*|Du|s*|DU|ps*𝑑x+BεΦs*|DU|(p-1)s*|D2U|s*𝑑x]1s*.(2.8)

Using the above estimates in (2.8), we get

C1ε(Bε|h-hBε|s𝑑x)1sC[Ψ1+Ψ2],

where

Ψ1(y)=(M(|Φu(u)|s*|DU|ps*|Du|s*)(y))1s*,Ψ2(y)=(M(Φs*|DU|(p-1)s*|D2U|s*)(y))1s*.

Setting

Ψ3(y)=(M(Φs*(u)|DU|(p+1)s*)(y))1s*

and putting these estimates together, we thus have

supε>0,|g1*ϕε|C[Ψ1+Ψ2]Ψ3.(2.9)

In the sequel, we will denote

F=Φ|DU|p+1,F¯=|Φu||DU|p+1,f=Φ|Du|p+1,f¯=|Φu||Du|p+1.(2.10)

Take s=2n/(n-1), then s*=s=2n/(n+1). With these notations and the definition of I¯1, we can use Young’s inequality and then (2.4), because 2>2n/(n+1)=s*, to get

(ΩΨ12𝑑x)12C(M(F¯s*)2s*1s*+M(f¯s*)2s*1s*)C(F¯22+f¯22)CI¯112.

Similarly,

(ΩΨ32𝑑x)12CΦ2|DU|2(p+1)2CI112.

Furthermore, (2.4) also gives

(ΩΨ22𝑑x)12=M(Φs*|DU|(p-1)s*|D2U|s*)2s*1s*CΦ2|DU|p-1|D2U|2.

Therefore, by Holder’s inequality, the above estimates and the notations (2.1) and (2.2), we get

Ωsupε|g1*ϕε|dxC[I112I¯112+I112I212].(2.11)

We now turn to g2 and note that |divV2|C(J1+J2) for some constant C and

J1:=|Φu(u)||DU|p+1|Du|J3,J2:=Φ|DU|p|D2U|J3,

with

J3:=|Bε(y)Φ|DU|p𝑑x|.

We will estimate ϕε*J1L1(Ω) and ϕε*J2L1(Ω). The calculations for these estimates are similar, we consider J1 first and denote

K=|Φu(u)||DU|p+1,L=|Du|.

We first observe that

J3(y)M*(Φ|DU|p)(x)for xBε(y).

Here, M* is the uncentered Hardy–Littlewood maximal operator. We then have

ϕε*J1(x)ϕε*K(x)L(x)M*(Φ|DU|p)(x).

Therefore,

Ω|ϕε*J1(x)|𝑑xΩK(x)L(x)M*(Φ|DU|p)(x)𝑑x=ΩK(LΦ1p+1)(M*(Φ|DU|p)Φ-1p+1)𝑑x.

We then apply Hölder’s inequality to the last integral to get

Ω|ϕε*J1|𝑑xKL2(Ω)LΦ1p+1L2(p+1)(Ω)M*(Φ|DU|p)Φ-1p+1L2(p+1)p(Ω).(2.12)

Concerning the last term on the right-hand side of (2.12), we note that w:=Φ-2p is an Aq weight for q=2(p+1)/p. Indeed, because 1-q=-p/(p+2) and q-1=(p+2)/p, we see easily that

Bw𝑑x(Bw1-q𝑑x)q-1=BΦ-2p𝑑x(BΦ2p+2𝑑x)p+2p([Φ2p+2]p/(p+2)+1)p+2p.

Therefore, [w]q is bounded by a constant depending on [Φ2p+2]p/(p+2)+1.

By Muckenhoupt’s theorem (2.5), we can find a constant C([w]q)[w]q such that

M*(Φ|DU|p)Φ-1p+1L2(p+1)p(Ω)p2(p+1)=Ω(M*(Φ|DU|p))qw𝑑xC([w]q)ΩΦq|DU|pqw𝑑x.

Note that from the definition of q=2(p+1)/p and w:=Φ-2p, we have Φq|DU|pqw=|Φ2DU|2p+2.

Hence, using the above estimate for the last integral in (2.12), we derive

Ω|ϕε*J1|𝑑xCΦKL2(Ω)LΦ1p+1L2(p+1)(Ω)F2pp+1,(2.13)

where CΦ is a constant depending on [w]q or [Φ2p+2]p/(p+2)+1.

Recalling the definition of K=F¯ (see (2.10)) and L=|Du|, we see that

LΦ1p+1L2(p+1)(Ω)=fL2(Ω)1p+1.

Therefore, Young’s inequality yields

Ωsupε|ϕε*J1|dxCΦF¯2f21p+1F2pp+1CΦI¯112(I112+I^112).(2.14)

Next, for J2=Φ|DU|p-1|D2U||DU|J3 we repeat the calculation for J1, using

K=|Φ(u)||DU|p-1|D2U|,L=|DU|.

We then obtain an estimate similar to (2.13) for ϕε*J2L1(Ω). Now, with the new definitions of K,L, we have

KL2(Ω)=I112,LΦ1p+1L2(p+1)(Ω)=FL2(Ω)1p+1.

We then obtain the inequality

Ωsupε|ϕε*J2|dxCΦI112I212.(2.15)

Combining the estimates (2.14), (2.15), we derive

Ωsupε|g2*ϕε|dxCΦ[I¯112(I112+I^112)+I112I212].(2.16)

The above and (2.11) yield

Ωsupε|g*ϕε|dxCΦ[I¯112(I112+I^112)+I112I212].

We thus proved (2.7) and the proof of Lemma 2.1 is complete. ∎

By approximation (see [16]), Lemma 2.1 also holds for uW1,2(Ω) and UW2,2(Ω) provided that the quantities I1,I2 and I^1 defined in (2.1) and (2.2) are finite. Furthermore, if Φ is a constant then I¯1=0 on the right-hand side of (2.3). Thus if u=U and Φ is a constant then Lemma 2.1, with small ε, clearly gives

Ω|DU|2p+2𝑑xCUBMO(Ω)2Ω|DU|2p-2|D2U|2𝑑x.

This is the Gagliardo–Nirenberg inequality (1.1) established in [16].

In other applications, we may need a similar version of the lemma with the usual gradient operator D being replaced by a general differential operator 𝐃, e.g. a weighted linear combination of Dxi. One can easily see that the proof is virtually unchanged if a certain Poincaré–Sobolev inequality used in (2.8) holds. Namely,

1ε(Bε|h-hBε|s𝑑x)1sC(Bε|𝐃h|s*𝑑x)1s*

holds for some s*, depending on s, such that s,s*<2. Of course, DU,Du will be accordingly replaced by 𝐃U,𝐃u.

To study the regularity of solutions, assuming that their BMO norms in small balls are small, we have the following local version of Lemma 2.1.

Let u,U:Ωm be vector-valued functions with uC1(Ω), UC2(Ω), and let Φ:m be a C1 function such that the condition (GN) in Lemma 2.1 holds. For any ball Bt in Ω we set

I1(t):=BtΦ2(u)|DU|2p+2dx,I^1(t):=BtΦ2(u)|Du|2p+2dx,I¯1(t):=Bt|Φu(u)|2(|DU|2p+2+|Du|2p+2)dx,I2(t):=BtΦ2(u)|DU|2p-2|D2U|2dx.

Consider any ball Bs concentric with Bt, 0<s<t, and any nonnegative C1 function ψ such that ψ=1 in Bs and ψ=0 outside Bt. Then, for any ε>0 there are positive constants Cε,Φ,Cε such that

I1(s)ε[I1(t)+I^1(t)]+Cε,ΦUBMO(Bt)2[I¯1(t)+I2(t)]+CεUBMO(Bt)supxBt|Dψ(x)|2Bt|Φ|2(u)|DU|2p𝑑x.(2.17)

Proof.

We revisit the proof of Lemma 2.1. Integrating by parts, noting that ψ=0 on Ω, we have

ΩΦ2(u)ψ2|DU|2p+2𝑑x=-ΩUdiv(Φ2(u)ψ2|DU|2pDU)dx.

Again, we will show that g=div(Φ2ψ2|DU|2pDU) belongs to the Hardy space 1. We write g=g1+g2 with gi=divVi, setting

V1=Φ(u)ψ|DU|p+1(Φ(u)ψ|DU|p-1DU-BεΦ(u)ψ|DU|p-1DU𝑑x),V2=Φ(u)ψ|DU|p+1BεΦ(u)ψ|DU|p-1DU𝑑x.

In estimating V1 we follow the proof of Lemma 2.1 and replace Φ(u) by Φ(u)ψ(x). There will be some extra terms in the proof in computing D(Φ(u)ψ). In particular, in estimating Dh in the right-hand side of (2.8) we have the following term and it can be estimated as follows:

(BεΦs*(u)|Dψ|s*|DU|ps*𝑑x)1s*supxBt|Dψ|(BεΦs*(u)|DU|ps*𝑑x)1s*.

We then use the following inequality, via Young’s inequality, in the right-hand side of (2.9) (with Ω=Bt):

sup|Dψ|BtΨ1M(Φs*(u)|DU|ps*)1s*𝑑xC[BtΨ12𝑑x+supxBt|Dψ|2BtM(Φs*(u)|DU|ps*)2s*𝑑x].

The last integral can be bounded via (2.4) by

supxBt|Dψ|2BtΦ2(u)|DU|2p𝑑x.

Using the fact that |ψ|1 and taking Ω to be Bt and omitting the obvious parameter t in the sequel, the previous proof can go on and (2.11) now becomes

Btsupε|g1*ϕε|dxC[I112I¯112+I112I212]+C(ε)supBt|Dψ|2BtΦ2(u)|DU|2p𝑑x.(2.18)

Similarly, in considering g2=divV2, we will have an extra term Φ(u)|Dψ||DU|p+1J3 in J1. We then use the estimate

supε|ϕε*Φ(u)||Dψ||DU|p+1J3supBt|Dψ|M(Φ(u)|DU|p+1)M(Φ(u)|DU|p),

and, via Young’s inequality and (2.4),

Btsupε|ϕε*Φ(u)||Dψ||DU|p+1J3dxεI1(t)+C(ε)supBt|Dψ|2BtΦ2(u)|DU|2p𝑑x.

Therefore estimate (2.16) is now (2.18) with g1 being replaced by g2. Combining the estimates for g1,g2 and using Young’s inequality, we get

Btsupε|g*ϕε|dxεI1(t)+C(ε)supBt|Dψ|2Bt|Φ|2|DU|2p𝑑x+CΦ[I¯112(I112+I^112)+I112I212].

The above gives an estimate for the 1 norm of g. By the Fefferman–Stein theorem, we obtain

BtΦ2(u)ψ2|DU|2p+2𝑑xεI1(t)+CΦUBMO(Bt)[I¯112(I112+I^112)+I112I212]+C(ε)UBMO(Bt)supBt|Dψ|2BtΦ2(u)|DU|2p𝑑x.

As before, we can use Young’s inequality and then the fact that ψ=1 in Bs to obtain (2.17) and complete the proof. ∎

3 A-Priori Estimates in W1,p(Ω) for p>n

In this section we will establish the key estimate for the proof of our main theorem. As we mentioned in the Introduction, for simplicity we will assume that A,f^ are independent of x,t. The general case can be treated similarly. Throughout this section, for some fixed T0>0 we consider two vector-valued functions U,W from Ω×(0,T0) into m and solve the system

Ut=div(A(W,DU))+f^(W,DU).(3.1)

We will consider the following assumptions on U,W and (3.1):

  • (U.0)

    A,f^ satisfy (A), (F) and (SG) with u=W and ζ=DU.

  • (U.1)

    UL1((0,T0),W2,2(Ω)) and W(,t)W1,2(Ω) for a.e. t(0,T0). On the lateral boundary Ω×(0,T0), U satisfies Neumann or Dirichlet boundary conditions.

  • (U.2)

    There is a constant C such that |DW|C|DU| and |Wt|C|Ut|.

The following assumption seems to be technical but we will see in many applications that it is easy to be verified when W is a BMO function, a condition will be assumed in the main result of this section.

  • (U.3)

    There is a positive C1 function β:m such that the following number is finite:

    𝚲=supW{|λW(W)|λ(W),|βW(W)|β(W)}.

    Moreover, β-1(W) and λ(W)β(W) belong to Lr(Ω) for sufficiently large r>1; (λ(W)β(W))12 is an A43 weight. Namely, there is a continuous function C on (0,T0) such that, for a.e. t(0,T0),

    β-1(W)Lr(Ω),λ(W)β(W)Lr(Ω),[(λ(W)β(W))12]43C(t).

  • (U.4)

    There is a constant C such that

    Ω×[0,T0)λ(W)β(W)|DU|2𝑑zC.(3.2)

To continue, we introduce the quantities

Γ(W)=λ(W)|βW(W)|2β(W)+β(W)|λW(W)|2λ(W).

For any fixed t0>0 we consider T(2t0,T0) and x0Ω¯. For t>0 we will denote

Qt(x0,T,t0)=Bt(x0)Ω×[T-2t0,T0).

For q1 we introduce the following quantities:

𝒜q(t,x0,T,t0)=supτ[T-t0,T0)Bt(x0)Ωβ(W)|DU|2q𝑑x,(3.3)q(t,x0,T,t0)=Qt(x0,T,t0)Γ(W)|DU|2q+2𝑑z,(3.4)𝒞q(t,x0,T,t0)=Qt(x0,T,t0)λ(W)β(W)|DU|2q+2𝑑z,(3.5)q(t,x0,T,t0)=Qt(x0,T,t0)λ(W)β(W)|DU|2q-2|D2U|2𝑑z,(3.6)𝒢q(t,x0,T,t0)=Qt(x0,T,t0)λ(W)β(W)|DU|2q𝑑z,(3.7)𝒥q(x0,T,t0)=Qt(x0,T,t0)β(W)|DU|2q𝑑z.(3.8)

We also denote, for R,t>0,

𝒟(R,t,x0):=U(,t)BMO(BR(x0)Ω).(3.9)

By (SG), there is q0>n/2 such that

2q0-22q0=δq0C*-1for some δq0(0,1).(3.10)

The main result of this section shows that if 𝒟(R,t,x0) is uniformly small for sufficiently small R, then DULp(Ω) can be controlled for some p>n.

Suppose that (U.0)(U.4) hold. Assume that there exist t0>0 and μ0(0,1), which is sufficiently small, in terms of the constants in (A), (F) such that the following holds:

  • (D)

    There is a positive Rμ0 , which may also depend on t0, T0 , for which

    𝚲2supx0Ω¯,t[T-2t0,T0)𝒟2(Rμ0,t,x0)μ0for all T[2t0,T0).

Suppose also that for x0Ω¯ and Tt0>0 the quantities (3.3)–(3.8) are finite for q[1,q0], q0 is fixed in (3.10). Then there are q>n/2 and a constant C depending on the constants in (U.0)(U.4), q, Rμ0, t0, T0 and the geometry of Ω such that

supt[t0,T0)Ω|DU|2q𝑑xC.(3.11)

The dependence of C in (3.11) on the geometry of Ω means: C depends on a number Nμ0 of balls BRμ0(xi), xiΩ¯, such that

Ω¯i=1Nμ0BRμ0(xi).(3.12)

The proof of Proposition 3.1 relies on local estimates for the integral of |DU| in finitely many balls BR(xi) with sufficiently small radius R to be determined by the geometry of Ω, namely the number Nμ0 and the continuity of the function 𝒟 defined in (3.9). We will establish local estimates for DU in these balls and then add up the results to obtain its global estimate (3.11). In the proof, we will only consider the case when BR(xi)Ω. The boundary case (xiΩ) is similar, invoking a reflection argument and using the fact that Ω is smooth to extend the function U outside Ω, see Remarks 3.7 and 3.8.

In the rest of this section, let us fix a point x0 in Ω and T2t0. We will drop x0,T,t0 in the notations (3.3)–(3.8) and (3.9).

For any s,t such that 0<s<tR let ψ be a cutoff function for two balls Bs,Bt centered at x0. That is, ψ is nonnegative, ψ1 in Bs and ψ0 outside Bt with |Dψ|1/(t-s). We also fix a cutoff function η for t for [T-2t0,T0] and [T-t0,T0]. That is η(t)=0 for tT-2t0, η(t)=1 for t(T-t0,T0] and |η(t)|1/t0 for all t.

We first have the following local energy estimate result.

Assume (U.0)(U.2). Assume that q1 satisfies condition (3.10) and that the quantities (3.3)–(3.8) are finite. There is a constant C1(q) depending also on the constants in (A) and (F) such that

𝒜q(s)+q(s)C1(q)[q(t)+1(t-s)2𝒢q(t)+1t0𝒥q(t)],0<s<tR.(3.13)

Proof.

By the assumption (U.1), we can formally differentiate (3.1) with respect to x, more precisely we can use difference quotients (see Remark 3.3), to get the weak form of

(DU)t=div(Aζ(W,DU)D2U+AW(W,DU)DWDU)+Df^(W,DU).(3.14)

For simplicity, we will assume in the proof that f^0. The presence of f^ will be discussed later in Remark 3.4. Testing (3.14) with ϕ=β(W)|DU|2q-2DUψ2η, which is legitimate since q is finite, integrating by parts in x and rearranging, we have, for Q=Ω×[T-2t0,τ] with τT,

Qϕ,(DU)tψ2η𝑑z=-QAζ(W,DU)D2U+AW(W,DU)DWDU,Dϕ𝑑z.(3.15)

Firstly, we observe that

2qϕ,(DU)tη=ddt(β(W)|DU|2qη)-|DU|2qβW(W)Wtη-β(W)|DU|2qηt,Dϕ=β(W)D(|DU|2q-2DU)ψ2+|DU|2q-2DUβWDWψ2+2β(W)|DU|2q-2DUψDψ.

Hence, we can rewrite (3.15) as

12qΩτβ(W)|DU|2qψ2𝑑x+Qβ(W)Aζ(W,DU)D2U,D(|DU|2q-2DU)ψ2η𝑑z=-Q[Aζ(W,DU)D2U,I1+AW(W,DU)DW,I2]ψ2η𝑑z+QI3𝑑z,(3.16)

where Ωτ=Ω×{τ}.

We will discuss the terms I1,I2,I3 later. Let us consider the second integral on the left-hand side. By (U.0) and the uniform ellipticity of Aζ(W,DU), we can find a constant C* such that |Aζ(W,DU)ζ|C*λ(W)|ζ|. By (3.10), α=2q-2 satisfies

α2+α=2q-22q=δqC*-1=δqλ(W)C*λ(W).

By [1, Lemma 2.1] or [11, Lemma 6.2], for such α,q there is a positive constant C(q) such that

Aζ(W,DU)D2U,D(|DU|2q-2DU)C(q)λ(W)|DU|2q-2|D2U|2.(3.17)

We then obtain from (3.16)

Ωβ(W)|DU|2qψ2𝑑x+C0(q)Qβ(W)λ(W)|DU|2q-2|D2U|2ψ2η𝑑z-Q[Aζ(W,DU)D2U,I1+AW(W,DU)DW,I2]η𝑑z+QI3𝑑z.(3.18)

The terms I1,I2 in the integrands on the right-hand side of (3.18) result from the calculation of Dϕ and they will be handled by Young’s inequality as follows, noting that the assumption (A) gives |Aζ(W,DU)|C|λ(W)| and |AW(W,DU)|C|λW(W)||DU|.

Concerning I1, for any ε>0 we can find a constant C(ε) such that

|Aζ(W,DU)D2U,|DU|2q-2DUβWDWψ2|ελ(W)β(W)|DU|2q-2|D2U|2ψ2+C(ε)λ(W)|βW(W)|2β(W)|DW|2|DU|2qψ2,|Aζ(W,DU)D2U,β(W)|DU|2q-2DUψDψ|ελ(W)β(W)|DU|2q-2|D2U|2ψ2+λ(W)β(W)|DU|2q|Dψ|2.

Similarly, for I2 we have

|AW(W,DU)DW,β(W)|DU|2q-2D2Uψ2|ελ(W)β(W)|DU|2q-2|D2U|2ψ2+β(W)|λW(W)|2λ(W)|DW|2|DU|2qψ2,|AW(W,DU)DW,|DU|2q-2DUβWDWψ2|C|λW(W)||βW(W)||DW|2|DU|2qψ2,|AW(W,DU)DW,β(W)|DU|2q-2DUψDψ|Cβ(W)|λW(W)|2λ(W)|DW|2|DU|2qψ2+Cλ(W)β(W)|DU|2q|Dψ|2.

Finally, for I3, which results from the calculation of ϕ,(DU)tη, we have

β(W)|DU|2q|ηt|1t0β(W)|DU|2q.

As we assume that |Wt|C|Ut|, we have, from the equation of U,

|Wt|C|Aζ(W,DU)||D2U|+C|AW(W,DU)||DW|.

Hence,

|DU|2q|βW(W)||Wt|ηελ(W)β(W)|DU|2q-2|D2U|2+C(ε)λ(W)|βW(W)|2β(W)|DU|2q+2+Cλ(W)|βW(W)|2β(W)|DU|2q|DW|2+Cβ(W)|λW(W)|2λ(W)|DU|2q.

We then use the fact that |DW|C|DU|, choose ε sufficiently small and put the above estimates for the terms in I1,I2,I3 in (3.18) to obtain a number C1 depending on q,C(q) (see (3.17)) such that, for Bsτ=Bs×{τ} and τ[T-t0,T0),

Bsτβ(W)|DU|2qη𝑑x+Qsλ(W)β(W)|DU|2q-2|D2U|2η𝑑zC1QtΓ(W)|DU|2q+2ψ2η𝑑z+C1𝚲Qtβ(W)[1(t-s)2λ(W)|DU|2q+1t0|DU|2q]𝑑z.

Here, we used the definition of ψ and Γ(W). As the above holds for all τ[T-t0,T0), from the notations (3.3) and (3.4), the above gives the lemma. ∎

For i=1,,n and h0 we denote by δi,h the difference quotient operator

δi,hu=h-1(u(x+hei)-u(x)),

with ei being the unit vector of the i-th axis in n. We then apply δi,h to the system for U and then test the result with |δi,hU|2q-2δi,hUψ2. The proof then continues to give the desired energy estimate by letting h tend to 0.

If f^0 then there is an extra term |Df^(W,DU)||DU|2q-1ψ2 in (3.18). This term will give rise to similar terms in the proof. Indeed, by (1.4) in (F) with u=W and p=DU,

|Df^(W,DU)|Cλ12(W)|D2U|+C|λW(W)|λ12(W)|DW||DU|+|fW(W)||DW|.

Therefore, by Young’s inequality and (1.5), |fW(W)|Cλ(W), we get

|Df^(W,DU)|β(W)|DU|2q-1C[λ12(W)|D2U|+C|λW(W)|λ12(W)|DW||DU|+|fW(W)||DW|]β(W)|DU|2q-1ελ(W)β(W)|DU|2q-2|D2U|2+C(ε)λ(W)β(W)|DU|2q+CΓ(W)|DU|2q+2+Cβ(W)|DU|2q+Cλ(W)β(W)|DU|2q.

Choosing ε>0 sufficiently small, we then see that the proof can continue to obtain the energy estimate (3.13).

The energy estimate of the lemma can be established by the same argument if A and f^ depend on x and t. We can assume that |Ax(t,x,u,Du)| and |f^x(t,x,u,Du)| satisfy the same growth as |A| and |f^|.

Inspecting our proof here and the proof of [11, Lemma 6.2], we can see that the constant C(q) in (3.17) is decreasing in q and hence C1(q) is increasing in q. Note also that this is the only place we need (3.10).

We discuss the case when the centers of Bρ,BR are on the boundary Ω. We assume that U satisfies the Neumann boundary condition on Ω. By flattening the boundary we can assume that BRΩ is the set

B+={x:x=(x1,,xn) with xn0 and |x|<R}.

For any point x=(x1,,xn) we denote by x¯ its reflection across the plane xn=0, i.e., x¯=(x1,,-xn). Accordingly, we denote by B- the reflection of B+. For a function u given on B+×(0,T) we denote its even reflection by u¯(x,t)=u(x¯,t) for xB-. We then consider the even extension of u^ in B=B+B-:

u^(x,t)={u(x,t)if xB+,u¯(x,t)if xB-.

With these notations, for xB+ we observe that

Ut=U¯t,divx(DxU)=divx¯(Dx¯U¯),DxWDxU=Dx¯W¯Dx¯U¯.

Therefore, it is easy to see that U^ satisfies in B a system similar to the one for U in B+. Thus, the proof can apply to U^ to obtain the same energy estimate near the boundary.

For the Dirichlet boundary condition we make use of the odd reflection u¯(x,t)=-u(x¯,t) and then define u^ as in Remark 3.7. Since DxiU=0 on Ω if in, we can test the system (3.14), obtained by differentiating the system of U with respect to xi, with |DxiU|2q-2DxiUψ2 and the proof goes as before because no boundary integral terms appear in the calculation. We need only consider the case i=n. We observe that DxnU^ is the even extension of DxnU in B therefore U^ satisfies a system similar to (3.14). The proof then continues.

We now apply the local Gagliardo–Nirenberg inequality in the previous section to the functions W,U.

Let Bs,Bt be two concentric balls in Ω with radii t>s>0 and ψ be a C1 cutoff function for two balls Bs,Bt. Let WC1(Ω) and UC2(Ω) such that there is a constant C such that |DW|C|DU|. Furthermore, assume that [λ(W)β(W)]1p+2 belongs to the Ap/(p+2)+1 class. There is a constant Cλ,β depending on [(λ(W)β(W))1p+2]p/(p+2)+1 such that

p(s)+𝚲2𝒞p(s)ε𝚲2𝒞p(t)+C(ε)𝚲2Cλ,βsupτ[T-2t0,T0)U(,τ)BMO(Bt)2[p(t)+p(t)+1(t-s)2𝒢p(t)].(3.19)

Proof.

Let u=W and the function Φ(W) in Lemma 2.4 be [λ(W)β(W)]12. The assumption that

[Φ(W)]2p+2=[λ(W)β(W)]1p+2

belongs to the Ap/(p+2)+1 class makes the lemma applicable here.

We now redefine

I1(t):=BtΦ2(W)|DU|2p+2𝑑x,I¯1(t):=Bt|ΦW(W)|2|DU|2p+2𝑑x,

and note that, since |DW|C|DU|, the quantities I^1(t),I¯1(t) in Lemma 2.4 are majorized respectively by the above I1(t),I¯1(t). Hence, we can choose ε sufficiently small in Lemma 2.4 to obtain a constant CΦCλ,β such that

I1(s)εI1(t)+C(ε)CΦUBMO(Bt)2[I¯1(t)+I2(t)]+C(ε)CΦUBMO(Bt)supxBt|Dψ(x)|2Btλ(W)|DU|2p𝑑x.

It is clear that |ΦW(W)|2Γ(W) so that

I¯1(t)+I2(t)Bt(λ(W)β(W)|DU|2p-2|D2U|2+|Γ(W)||DU|2p+2)𝑑x.

We then have

BsΦ2(W)|DU|2p+2𝑑xεBtΦ2(W)|DU|2p+2𝑑x+𝚲2Cλ,βUBMO(Bt)2Bt(λ(W)β(W)|DU|2p-2|D2U|2+|Γ(W)||DU|2p+2)𝑑x+𝚲2Cλ,βUBMO(Bt)supxBt|Dψ(x)|2Btλ(W)β(W)|DU|2p𝑑x.

Multiplying the above inequality with 𝚲2η, integrating the result over [T-2t0,T0) and using the notations (3.3)–(3.8) (with Qt=Bt×[T-2t0,T0)), we see that the above implies

𝚲2𝒞p(s)ε𝚲2𝒞p(t)+C(ε)𝚲2Cλ,βsupτ[T-2t0,T0)U(,τ)BMO(Bt)2[p(t)+p(t)+1(t-s)2𝒢p(t)].

Because Γ(W)𝚲2Φ2(W), we have p(s)𝚲2𝒞p(s). We see that the above gives (3.19). ∎

Let us recall the following elementary iteration result (e.g., see [7, Lemma 6.1, p.192]).

Let f,g,h be bounded nonnegative functions in the interval [ρ,R] with g,h being increasing. Assume that for ρs<tR we have

f(s)ε0f(t)+[(t-s)-αg(t)+h(t)]

with α>0 and 0ε0<1. Then

f(ρ)c(α,ε0)[(R-ρ)-αg(R)+h(R)].

The constant c(α,ε0) can be taken to be (1-ν)-α(1-ν-αν0)-1 for any ν satisfying ν-αν0<1.

We then have another lemma for the main proof of this section.

Let F,G,g,h be bounded nonnegative functions in the interval [ρ,R] with g,h being increasing. Assume that for ρs<tR we have

F(s)ε[F(t)+G(t)]+[(t-s)-αg(t)+h(t)],(3.20)G(s)C[F(t)+(t-s)-αg(t)+h(t)](3.21)

with C0, α,ε>0. If 2Cε<1 then there is constant c(C,α,ε) such that

F(s)+G(s)c(C,α,ε)[(t-s)-αg(t)+h(t)],ρs<tR.(3.22)

Proof.

Let ε0=2Cε. We obtain from (3.20)

CF(s)ε02[F(t)+G(t)]+C[(t-s)-αg(t)+h(t)].(3.23)

Let t1=(s+t)/2 and use (3.23) with s being t1 and (3.21) with t being t1 to obtain a constant C1 such that

G(s)ε02[F(t)+G(t)]+C1[(t-s)-αg(t)+h(t)].(3.24)

Of course, we can assume that C1 so that (3.23) and (3.24) give

F(s)+G(s)ε0[F(t)+G(t)]+C1[(t-s)-αg(t)+h(t)].

Thus, if ε0<1 or 2Cε<1 then Lemma 3.10 applies with f(t)=F(t)+G(t) to give

F(ρ)+G(ρ)c(α,ε)[(R-ρ)-αg(R)+h(R)].

Obviously, the above argument holds if we replace the interval [ρ,R] by any subinterval [s,t]. The above inequality then gives (3.22). ∎

Proof of Proposition 3.1.

For any R>0 we denote (Cλ,β is defined in Lemma 3.9)

ε0(R)=𝚲2Cλ,βsupτ[t0,T0)U(,τ)BMO(BR)2.(3.25)

Fix some q0>n/2 as in the proposition and let μ0,R0:=R0>0 in (D) be such that

C1(q0)ε0(R0)=C1(q0)Cλ,β𝚲2supτ[t0,T0)U(,τ)BMO(BR0)2<12,(3.26)

where C1(q0) is the constant in (3.13). We recall that (see Remark 3.6) C1(q) is increasing in q so that if (3.26) holds then there is μ*(0,1) such that

C1(q)ε0(R)<μ*2<12,1qq0,R(0,R0].

By (3.19) and the notation (3.25), we have, for any T2t0>0,

p(s)+𝚲2𝒞p(s)ε0(R0)(q(t)+q(t)+1(t-s)2𝒢q(t))

for all s,t such that 0<s<tR0.

On the other hand, if q satisfies (3.10), then (3.13) gives (from now on C1=C1(q))

q(s)C1q(t)+C1(t-s)2𝒢q(t)+C11t0𝒥q(t),0<s<tR0.

It is clear that the above two estimates imply (3.20) and (3.21) of Lemma 3.11 with F(t)=p(t)+𝚲2𝒞p(t), G(t)=q(t), g(t)=𝒢q(t) and h(t)=t0-1𝒥q(t). Thus, the assumption (3.26) on ε0 and (3.22) of Lemma 3.11 provide a constant C2 depending on μ*,C1 such that

q(s)+p(s)+𝚲2𝒞p(s)C2(t-s)2𝒢q(t)+C21t0𝒥q(t),0<s<tR0,

or

q(s)+p(s)C2(t-s)2𝒢q(t)+C21t0𝒥q(t),0<s<tR0.

For t=2s the above gives (if q satisfies (3.10))

q(s)+q(s)C3Q2s(1s2λ(W)|DU|2q+1t0|DU|2q)𝑑z,0<sR02.

Using this estimate for q(t) in (3.13), with s=R0/4 and t=R0/2, respectively, we derive

𝒜q(R04)+q(R04)C4QR0/2(1R02λ(W)|DU|2q+1t0|DU|2q)𝑑z.(3.27)

Now, we will argue by induction to obtain a bound for 𝒜q for some q>n/2. If some q with q1 satisfies (3.10), then we can find a constant Cq and tqt0 such that

Ω×[T-2tq,T0)(λ(W)β(W)|DU|2q+β(W)|DU|2q)𝑑zCq(3.28)

and that (3.26) holds. Then (3.27) implies a similar bound for 𝒜q(R1),q(R1), R1=R04. We now can cover Ω by NR1 balls BR1, see (3.12), and add up the estimates for 𝒜q(R1),q(R1) to obtain (tq is t0)

supt[T-tq,T0)Ωβ(W)|DU|2q𝑑x+Ω×[T-2tq,T0)λ(W)β(W)|DU|2q-2|D2U|2𝑑zC.(3.29)

For some α(0,1) to be determined later let p=αq. By Young’s inequality and (3.29), we obtain a constant C such that

Ω×[T-2tq,T0)|DU|2p-2|D2U|2𝑑zC(α)Ω×[T-2tq,T0)(1+|DU|2q-2)|D2U|2𝑑zC.(3.30)

Here, we have used the fact that λ(W)β(W) is bounded from below so that the second integral in the left-hand side of (3.30) is bounded by the second integral on the left-hand side of (3.29), which also holds for q=1 thanks to our assumption (3.2).

Using Hölder’s inequality, the assumption that β(W) belongs to Lr(Ω) for r sufficiently large and the bound in (3.29), we obtain

Ω|DU|2p𝑑x(Ωβ(W)-α1-α𝑑x)1-α(Ωβ(W)|DU|2q𝑑x)αC.

By Sobolev’s inequality, setting Qq=Ω×[T-tq,T0), we get

Qq|DU|2p(1+2n)𝑑z(supt[T-tq,T0)BR1|DU|2p𝑑x)2nQq|DU|2p-2|D2U|2𝑑z.

We derive from the above three estimates that

Qq|DU|2p(1+2n)𝑑zC.

For p*(1,1+2n) we write p*=γ+(1-γ)(1+2n) and use Hölder’s inequality to get

Qqλ(W)β(W)|DU|2pp*𝑑zλ(W)β(W)L1γ(Qq)(Qq|DU|2p(1+2n)𝑑z)1-γ.(3.31)

Since λ(W) is bounded from below and λ(W)β(W) belongs to Lr(Ω) for r sufficiently large, the above estimate yields

Qqβ(W)|DU|2pp*𝑑zC.(3.32)

We now choose and fix α,γ such that pp*=qq* for some q*>1. This is the case if α is close to 1 and γ  is close to 0, so that

q*=α[γ+(1-γ)(1+2n)]>1.

From (3.31) and (3.32), we see that (3.28) holds again with the exponent q and the interval [T-2tq,T0) being qq* and [T-tq,T0), respectively.

By our assumption (3.2), estimate (3.28) holds for q=1. For integers k=0,1,2, we define Lk=q*k and repeat the argument finitely many times, with the same choice of α,γ, as long as Lk satisfies (3.10), qkq0 and (λ(W)β(W))1/(Lk+2) is an Ap weight with p=LkLk+2+1. The last condition holds because Lk1 and because of the assumption that (λ(W)β(W))13 is an A43 weight (see Remark 3.12). We then find an integer k0 such that

supt[T-2-ktq,T0)Ωβ(W)|DU|2Lk𝑑xC(Cq,R0,t0,NR1),k=0,,k0.

Obviously, we can choose α,γ such that Lk0(n2,q0]. Now, let p0(n2,Lk0) and write p0=α0Lk0 for some α0(0,1). By Hölder’s inequality and the above estimate, if τT-2-k0tq, then

Ω|DU(x,τ)|2p0𝑑x(Ωβ(W)-α01-α0𝑑x)1-α0(Ωβ(W)|DU|2Lk0𝑑x)α0C.

It is clear from the proof that the integer k0 does not depend on t0 so that we can divide the interval [T-2t0,T-t0] into k0 equal length subintervals and repeat the argument to see that the above estimate holds for τT-t0 and T2t0. This gives (3.11) and the proof is complete. ∎

By Hölder’s inequality and the definition of Ap weights it is easy to see that if w is an Ap weight for some p>1 then wδ is an Aq weight for any δ(0,1) and q(1,p).

4 Local and Global Existence of Strong Solutions

In this section, we consider the system

{ut=div(A(x,t,u,Du))+f^(x,t,u,Du)in Q=Ω×(0,T0),u(x,0)=U0(x)in Ω,(4.1)

and u satisfies homogeneous Dirichlet or Neumann boundary conditions on Ω×(0,T0).

Throughout this section we will assume that A,f^ satisfy (A), (F) and (SG).

We first apply our estimates in the previous section to show that Amann’s conditions in [3, 2] can be weakened under some mild extra assumptions which naturally occur in applications. We consider the case when A(t,x,u,Du) is linear in Du and (4.1) satisfies the assumptions in Amann’s works (we refer the reader to [3, 2] for the precise statements) so that local existence results hold. We have the following global existence result.

Assume that A(t,x,u,Du)=A(t,x,u)Du for some full m×m matrix A(t,x,u) satisfying the assumptions in [3, 2]. Let (0,T0) be the maximal existence time interval for the solution u of (4.1). Assume further that there is a positive C1 function β:m such that the following number is finite:

𝚲=sup(x,t)Ω×(0,T0){|λW(x,t,W)|λ(x,t,W),|βW(W)|β(W)}.(4.2)

Suppose that there is a sufficiently large r>1 and a continuous function C(t) such that for a.e. t(0,T0) and u as a function in x the following estimates hold:

β-1(u)Lr(Ω),λ(x,t,u)β(u)Lr(Ω),[(λ(x,t,u)β(u))12]43C(t).(4.3)

In addition, we assume that

  • (M)

    for any given μ0>0 there is a positive Rμ0 , which may also depend on t0,T0 , for which

    𝚲2supx0Ω¯,t(0,T0)u(,t)BMO(Bx0)2μ0.

Then u exists globally, i.e. T0=.

Proof.

It is clear that the assumptions of the theorem imply those of Proposition 3.1. The bound (3.11) then shows that the W1,q norm, with some q>n, of u(,t) does not blow up in (0,T0) so that Amann’s results can apply here to give the global existence of u. ∎

On the other hand, if A is nonlinear in Du, Amann’s results can not apply here and we can alternatively establish local and global existence results for (4.1) using fixed point theories. To this end, we embed the systems (4.1) in the following family of systems with σ[0,1]:

{Ut=div(A^σ(x,t,U,DU))+F^σ(x,t,U,DU)in Q=Ω×(0,T0),U(x,0)=U0(x)in Ω,U satisfies homogeneous Dirichlet or Neumann BC on Ω×(0,T0).(4.4)

We will introduce a family of maps 𝒯(σ,), σ[0,1], acting in some suitable Banach space 𝒳 such that strong solutions to (4.4) are their fixed points.

In order to define the maps 𝒯(σ,), we will use the notations 1g(x,t,u,ζ), 2g(x,t,u,ζ) to denote the partial derivatives of a function g(x,t,u,ζ) with respect to its variables u,ζ.

To begin, let Q=Ω×(0,T0) and u0 be the strong solution to the linear parabolic system

{(u0)t=div(2A(x,t,0,0)Du0)+2f^(x,t,0,0)Du0+1f^(x,t,0,0)u0in Q,u0(x,0)=U0(x)in Ω,u0 satisfies homogeneous Dirichlet or Neumann BC on Ω×(0,T0).

It is well known that u0 is in C((0,T0),C2(Ω)) and (u0)tC(Q). Furthermore, for any τ0>0 and α(0,1), u0 is Hölder continuous in t and Du0 is Hölder continuous in x with any exponent α(0,1) in Ω×(τ0,T0) for any τ0>0.

Fixing some α0(0,1) and τ0>0 as in (M), we consider the Banach spaces

𝒳1=L2((0,T0),W1,2(Ω)),𝒳2=C(Ω×[0,τ0]),𝒳3=Cα0,α02(Ω×(τ0,T0)),

and

𝒳={v:v𝒳1𝒳2,Dv𝒳3}

with norm

v𝒳=v𝒳1+v𝒳2+Dv𝒳3.

Since the dependence of A,f^ on x,t is not important in what follows, we will omit them in the notations and calculation below for the simplicity of our presentation.

For each v𝒳 and σ[0,1], we denote w=v+u0 and define

Aσ(w)=012A(σw,tσDw)𝑑t,F1,σ(w)=011f^(tσw,0)𝑑t,F2,σ(w)=012f^(σw,tσDw)𝑑t.

For any given v𝒳 and w=v+u0 let u=𝒯(σ,v) be the weak solution u to the linear parabolic system

ut=div(Aσ(w)D(u+u0))+F2,σ(w)D(u+u0)+F1,σ(w)(u+u0)-(u0)t(4.5)

in Q=Ω×(0,T0) and u satisfies the initial and boundary condition

u=0on Ω×[0,T0).

Clearly, if u(σ) is a fixed point of 𝒯(σ,) for some σ(0,1], i.e. u(σ)=𝒯(σ,u(σ)), then U=u(σ)+u0 solves

Ut=div(Aσ(U)DU)+F2,σ(U)DU+F1,σ(U)U.(4.6)

We will assume that A,f^ satisfy (A) and (F) so that A(σU,0)=0 and f^(0,0)=0. Hence,

Aσ(U)DU=012A(σU,tσDU)𝑑tDU=σ-1A(σU,σDU)(4.7)

and

F2,σ(U)DU+F1,σ(U)U=012f^(σU,tσDU)𝑑tDU+011f^(tσU,0)𝑑tU=σ-1f^(σU,σDU).(4.8)

Therefore, for σ(0,1] we will define

A^σ(U,ζ)=σ-1A(σU,σζ),F^σ(U,ζ)=σ-1f^(σU,σζ),A^0(U,ζ)=2A(0,0)ζ,F^0(U,ζ)=2f^(0,0)ζ+1f^(0,0)U.(4.9)

We then consider the following family of systems for σ[0,1]:

{Ut=div(A^σ(U,DU))+F^σ(U,DU)in Q=Ω×(0,T0),U(x,0)=U0(x)in Ω,U(x,t)=0in Ω×(0,T0).(4.10)

By (4.9), we can see that u0 solves (4.10) for σ=0.

We assume that A,f^ satisfy (A), (F) and (SG). For some T0>0 we assume that there is t0(0,T0) such that λ(t,u) is bounded for t(0,t0). As in Theorem 4.1, we suppose that the conditions (4.2), (4.3) and (M) uniformly hold for σ[0,1] with u being a solution U of (4.10). Namely, there is a sufficiently large r>1 and a continuous function C(t) such that for a.e. t(0,T0) and U as a function in x the following hold:

β-1(U)Lr(Ω),λ(x,t,U)β(U)Lr(Ω),[(λ(x,t,U)β(U))12]43C(t),

and

  • (M’)

    for any given μ0>0 there is a positive Rμ0 , which may also depend on T0 , for which

    𝚲2supx0Ω¯,t(0,T0)U(,t)BMO(Bx0)2μ0.(4.11)

Proof.

We will use Leray–Schauder’s fixed point index theory to establish the existence of a fixed point of 𝒯(1,), which is a strong solution to (4.1) and the above theorem then follows. The main ingredient of the proof is to establish a uniform estimate for the fixed points of 𝒯(σ,) in 𝒳. To this end, we need a crucial Hölder regularity for these fixed points in Ω×(0,T0). We will make use of Proposition 3.1 which provides such regularity for these fixed points. However, this estimate holds for tt0 if we have some information on their spatial derivatives in early time, namely [t0/2,t0], so that the quantities in the energy estimate of Lemma 3.2 are finite. This is the main reason for the assumption that λ(x,t,u) is bounded when t is near 0 which together with (M’) and the results in [6] will give the needed boundedness of the spatial derivatives near t=0.

We will establish the following facts:

  • (i)

    𝒯(σ,):𝒳𝒳 is compact for σ(0,1].

  • (ii)

    𝒯(0,):𝒳𝒳 is a constant map.

  • (iii)

    A fixed point u=𝒯(σ,u) is a solution to (4.10). For σ=1, such fixed points are solutions to (4.1).

  • (iv)

    There is M>0 such that any fixed point u(σ)𝒳 of 𝒯(σ,), σ[0,1], satisfies u(σ)𝒳<M.

Once (i)–(iv) are established, the theorem follows from the Leray–Schauder index theory. Indeed, we let 𝐁 be the ball centered at 0 with radius M of 𝒳 and consider the Leray–Schauder indices

i(𝒯(σ,),𝐁,𝒳):=deg(Id-𝒯(σ,),𝐁,0),

where the right-hand side denotes the Leray–Schauder degree with respect to zero of the vector field Id-𝒯(σ,). This degree is well defined on the closure of the open set 𝐁𝒳 because 𝒯(σ,) is compact (see (i)) and Id-𝒯(σ,) does not have zero on 𝐁 (see (iv)).

By the homotopy invariance of the indices and (ii), we have

i(𝒯(σ,),𝐁,𝒳)=i(𝒯(0,),𝐁,𝒳)=1.

Thus, 𝒯(σ,) has a fixed point in 𝐁 for all σ[0,1]. Our theorem then follows from (iii).

Using regularity properties of solutions to linear parabolic systems with continuous coefficients (see, e.g., [5]), we see that (i) holds. Checking (ii) and (iii) is fairly standard and straightforward.

To check (iv), let u(σ)𝒳 be a fixed point of 𝒯(σ,), σ[0,1]. We need only consider the case σ>0. We now denote W=σ(u(σ)+u0) and U=u(σ)+u0 and need to show that U𝒳 is uniformly bounded for σ[0,1]. First of all, the uniform boundedness for U𝒳1, or equivalently DUL2(Q), is fairly standard. We multiply the systems (4.6) with U and integrate over Q. A simple use of integration by parts and Young’s inequality shows that DUL2(Q) can be estimated by the integrals over Q of f(W)|U|. By (F), |fu(u)|Cλ(u) so that f(W)|U|Cλ(W)|U|2. By our assumptions, λ(W)β(W) and β-1(W) are in Lr(Ω) for some large r1 with its norms being uniformly bounded, a simple use of Hölder’s inequality then shows that λ(W) satisfies the same properties. Similarly, U is BMO so that it is in Lq(Ω) for all q1. Hölder’s inequality then gives a uniform bound for the integral of λ(W)|U|2 and then of DUL2(Q).

Next, as we assume that λ(t,W) is bounded and U is VMO in Ω×(0,τ0], the argument in [6] applies here to show that U(x,t) is uniformly Hölder continuous in Ω×(0,τ0]. Therefore, U𝒳2 is uniformly bounded.

Concerning U𝒳3, we will show that Proposition 3.1 can be applied to the systems (4.10). As U=u(σ)+u0 and W=σU, with u(σ)𝒳, the conditions (U.1) and (U.2) are clearly verified.

From (4.7), (4.8) and the assumption that A,f^ satisfy (A) and (F) we see that (U.0) is verified. Indeed, we will show that A^σ(U,ζ) and F^σ(U,ζ) satisfy the structural conditions (A) and (F). Firstly,

A^σ(U,ζ),ζ=σ-1A(σU,σζ),ζ=σ-2A(σU,σζ),σζλ(σU)|ζ|2,A^σ(U,ζ)=σ-1A(σU,σζ)C*λ(σU)|ζ|,UA^σ(U,ζ)=1A(σU,ζ)λ(σU)|ζ|.

Therefore A^σ satisfies (A) with u=σU.

Secondly,

|F^σ(U,ζ)|σ-1(λ12(σU)|σζ|+f(σU))λ(σU)|ζ|+σ-1f(σU),

and |U(σ-1f(σU))|=|fu(σU)|λ(σU). Here, fu(u) denotes the derivative of f(u) with respect to its variable u. Also,

|DF^σ(U,ζ)|Cσ-1(λ12(σU)|D(σζ)|+λu(σU)λ12(σU)|D(σU)||σζ|+|fu(σU)||D(σU)|)C(λ12(σU)|Dζ|+λu(σU)λ12(σU)|DU||ζ|+|fu(σU)||DU|).

Hence, F^σ(U,ζ) satisfies (F). We see that (U.0) is verified for U and W=σU.

In addition, since u(σ)𝒳, u(σ) is bounded and VMO near t=0, the results in [6] show that u(σ) is Hölder continuous in Q and Du(σ)Cβ,β2(Ω×(t1,T0)) for any t1>0 and some β(0,1). Hence, u(σ)=𝒯(σ,u(σ)) is the solution to the linear system (4.5), whose coefficients with v being u(σ) are in Cloc1(Ω×(0,T0)), so that U=u(σ)+u0 belongs to Wloc2,2(Q). Therefore, because W(,t),U(,t) belong to C1(Ω) for t>0, the quantities in the energy estimate of Lemma 3.2 are finite for all q1.

Finally, it is clear that (4.11) in the assumption (M’) of our theorem gives the condition (D) of the proposition. More importantly, the uniform bound in (4.11) then gives some positive constants μ0,R(μ0) such that the proposition applies to all W,U.

Therefore, Proposition 3.1 applies to W=σ(u(σ)+u0),U=u(σ)+u0 and gives a uniform estimate for u(σ)(,t)W1,2q(Ω) for some q>n/2 and all t(t0,T0) and σ[0,1]. By Sobolev’s imbedding theorems this shows that U is Hölder continuous with its norm uniformly bounded with respect to σ[0,1]. Again, the results in [6] imply that Du(σ)Cα,α2(Ω×(τ0,T0)) for any α(0,1) and its norm is uniformly bounded. We then obtain a uniform estimate for u(σ)𝒳3 and (iv) is verified. The proof of Theorem 4.2 is complete. ∎

We applied Proposition 3.1 to strong solutions in the space 𝒳 so that U,DU are bounded and the key quantities , are finite. However, the bound provided by the proposition did not involve the supremum norms of U,DU but the BMO norm of U in (M’) and the constants in (A) and (F).

We conclude this paper by considering the case when λ(x,t,u)(λ0+|u|)M for some λ0,M>0 when t is large. We will first show that the conditions, with the exception of (M’), are easily verifiable if the solutions are uniformly BMO. Condition (M’) will be discussed in Remark 4.5.

We then recall the following result from [9, Theorem 6] on the connection between BMO functions and weights: Let Ψ be a positive function such that Ψ,Ψ-1 are BMO. Then Ψ belongs to γ>1Aγ and [Ψ]γ is bounded by a constant depending on [Ψ]BMO and [Ψ-1]BMO.

Assume that λ(x,t,u) is bounded in (0,t0] for some t0>0 and λ(x,t,u)(λ0+|u|)M for tt0 and some λ0,M>0. Suppose that U(,t)BMO(Ω) is bounded on (0,T0) and for any ε>0 there is Rε>0 such that

λ0-1U(,t)BMO(BRε)εfor all BRεΩ and t(0,T0).

Then there is a strong solution in Ω×(0,T0).

Proof.

As in the proof of Theorem 4.2, we need only to show that Proposition 3.1 can apply for tt0. For W=σU with σ[0,1] assumptions (U.0)(U.2) are clearly satisfied. We will show that the condition (U.3) holds here. To this end, we choose β(W)(λ0+|W|)-M+2ε0 with ε0(0,1). If M2, we can take β(W)1. It is clear that the constant 𝚲λ0-1.

Let w:=(λ(W)β(W))12. Since w(λ0+|W|)ε0, with ε0(0,1), the assumption that W is BMO implies that w is BMO. Also, w-1 is BMO because w is bounded from below. By the aforementioned result in [9], w is an Ap weight for all p>1. Therefore, w=(λ(W)β(W))12 is in A43 class. On the other hand, it is well known that if W belongs to the BMO space then it belongs to Lp(Ω) for any p>1. Here, β-1(W) and λ(W)β(W) have polynomial growth in W so that they also belong to Lp(Ω) for any p>1. We have shown that (U.3) is verified.

Moreover, it is easy to see that (3.28) holds for q=1 by testing the system with U and then using the fact that β(W) is bounded from above. The proof of Theorem 4.4 is complete. ∎

To establish the uniform continuity condition (M’) one can try to establish a uniform boundedness of DULn(Ω) and apply Poincaré’s inequality to see that U is VMO. If this can be done then one can argue by contradiction to obtain (M’). We sketch the idea of the proof here. If (M’) is not true then along a sequence σn,tn,rn, rn>0, Un()=U(,tn) converge weakly to some U in W1,2(Ω) and strongly in L2(Ω) but UnBMO(Brn)>ε0 for some r,ε0>0. We then have UnBRUBR for any given R>0. It is not difficult to see that DULn so that U satisfies (M’). Furthermore, if rn<R then UnBMO(Br)UnBMO(BR). Choosing R sufficiently small and letting n tend to infinity, we obtain a contradiction.

References

  • [1]

    Ahmad S. and Le D., Global and blow up solutions to cross diffusion systems, Adv. Nonlinear Anal. 4 (2015), no. 3, 209–219.  Google Scholar

  • [2]

    Amann H., Dynamic theory of quasilinear parabolic systems III. Global existence, Math Z. 202 (1989), 219–250.  Google Scholar

  • [3]

    Amann H., Dynamic theory of quasilinear parabolic equations II. Reaction-diffusion systems, Differential Integral Equations 3 (1990), 13–75.  Google Scholar

  • [4]

    Franchi B., Perez C. and Wheeden R. L., Self-improving properties of John Nirenberg and Poincaré inequalities on spaces of homogeneous type, J. Functional Analysis 153 (1998), 108–146.  Google Scholar

  • [5]

    Friedman A., Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.  Google Scholar

  • [6]

    Giaquinta M. and Struwe M., On the partial regularity of weak solutions of nonlinear parabolic systems, Math. Z. 179 (1982), 437–451.  Google Scholar

  • [7]

    Giusti E., Direct Methods in the Calculus of Variations, World Scientific, Singapore, 2003.  Google Scholar

  • [8]

    John O. and Stara J., On the regularity of weak solutions to parabolic systems in two spatial dimensions, Comm. Partial Differential Equations 27 (1998), 1159–1170.  Google Scholar

  • [9]

    Johnson R. L. and Neugebauer C. J., Properties of BMO functions whose reciprocals are also BMO, Z. Anal. Anwend. 12 (1993), 3–11.  Google Scholar

  • [10]

    Juengel A., The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity, to appear.  Google Scholar

  • [11]

    Le D., Regularity of BMO weak solutions to nonlinear parabolic systems via homotopy, Trans. Amer. Math. Soc. 365 (2013), 2723–2753.  Google Scholar

  • [12]

    Muckenhoupt B., Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226.  Google Scholar

  • [13]

    Orobitg J. and Pérez C., Ap weights for nondoubling measures in n and applications, Trans. Amer. Math. Soc. 354 (2002), 2013–2033.  Google Scholar

  • [14]

    Riviere T. and Strzelecki P., A sharp nonlinear Gagliardo Nirenberg type estimate and applications the regularity of elliptic systems, Comm. Partial Differential Equations 30 (2005), 589–604.  Google Scholar

  • [15]

    Stein E. M., Harmonic Analysis, Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, 1993.  Google Scholar

  • [16]

    Strzelecki P., Gagliardo Nirenberg inequalities with a BMO term, Bull. London Math. Soc. 38 (2006), 294–300.  Google Scholar

About the article

Received: 2015-06-11

Accepted: 2015-09-15

Published Online: 2016-01-27

Published in Print: 2016-02-01


Citation Information: Advanced Nonlinear Studies, Volume 16, Issue 1, Pages 125–146, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2015-5006.

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