Local and Global Existence of Strong Solutions to Large Cross Diffusion Systems

Dung Le

doi:10.1515/ans-2017-0013

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Local and Global Existence of Strong Solutions to Large Cross Diffusion Systems

Dung Le

Published/Copyright: April 21, 2017

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From the journal Advanced Nonlinear Studies Volume 18 Issue 1

Abstract

We study the solvability of a general class of cross diffusion systems and establish the local and global existence of their strong solutions under the weakest assumption that they are VMO. This work simplifies the setting in our previous work [15] and provides new extensions which are more verifiable in applications.

Keywords: Cross Diffusion Systems; Hölder Regularity; Global Existence

MSC 2010: 35J70; 35B65; 42B37

1 Introduction

In this paper, for T0>0 and a bounded domain Ω with smooth boundary in ℝn, n≥2, we consider the following general parabolic system of m equations (m≥2):

(1.1) { u t = div ⁡ ( A ⁢ ( u ) ⁢ D ⁢ u ) + f ^ ⁢ ( u , D ⁢ u ) , ( x , t ) ∈ Q = Ω × ( 0 , T 0 ) , u ⁢ ( x , 0 ) = U 0 ⁢ ( x ) , x ∈ Ω , u = 0 or ∂ ⁡ u ∂ ⁡ ν = 0 on ⁢ ∂ ⁡ Ω × ( 0 , T 0 ) ,

where A⁢(u) is an m×m matrix in u, and u:Ω→ℝm, f^:ℝm×ℝm⁢n→ℝm are vector-valued functions. The initial data U0 is given in W1,p0⁢(Ω,ℝm) for some p0>n, where n is the dimension of Ω. As usual, W1,p⁢(Ω,ℝm), p≥1, will denote the standard Sobolev spaces whose elements are vector-valued functions u:Ω→ℝm with finite norm

∥ u ∥ W 1 , p ⁢ ( Ω , ℝ m ) = ∥ u ∥ L p ⁢ ( Ω ) + ∥ D ⁢ u ∥ L p ⁢ ( Ω ) .

We say that u is a strong solution if u is continuous on Q¯ with D⁢u∈Lloc∞⁢(Q) and D2⁢u∈Lloc2⁢(Q).

The strongly coupled system (1.1) appears in many physical applications, for instance, in Maxwell–Stephan systems describing the diffusive transport of multicomponent mixtures, in models of reaction and diffusion in electrolysis, in flows in porous media, in diffusion of polymers and in population dynamics [7, 21, 23], among others. We refer the reader to the recent work [9] and the references therein for the models and the existence of their weak solutions.

The first fundamental problem in the study of (1.1) is the local and global existence of its solutions. One can decide to work with either weak or strong solutions. In the first case, the existence of a weak solution can be achieved via Galerkin, time discretization (see [9]) or variational methods [6], but its regularity (e.g., boundedness, Hölder continuity of the solution and its higher derivatives) is still an open issue. Several works have been done along this line to improve the early work [5] and establish partial regularity of bounded weak solutions to (1.1).

On the other hand, if strong solutions are considered, then their existence can be established via semigroup theories as in the works of Amann [3, 2]. Using the interpolation theories of Sobolev’s spaces, Amann established local and global existence of a strong solution u of (1.1) under the assumption that one can control ∥u∥W1,p⁢(Ω,ℝm) for some p>n.

In both aforementioned approaches, the assumption on the boundedness of u must be the starting point. For strongly coupled systems like (1.1), as invariant/maximum principles for cross diffusion systems are generally unavailable, the boundedness of the solutions is already a hard problem. One usually needs to use ad hoc techniques on a case by case basis to show that u is bounded (see [10, 20]). Even for bounded weak solutions, we know that they are only Hölder continuous almost everywhere (see [5]). In addition, there exist 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]).

In our recent works [12, 13, 14, 15], we chose a different approach, making use of fixed point theory and discuss the solvability of (1.1) under the weakest assumption that u is VMO (see (1.3) below) and much more general structural conditions, compared to [3, 2], on the data of (1.1). The proof in [15] relies on fixed point theories, instead of the semigroup approach in [3], and weighted Gagliardo–Nirenberg inequalities involving BMO norms.

In particular, we assumed in [15] the following conditions:

A⁢(u) is C1 in u and there exist constants λ0,C*>0 and a scalar C1 function λ⁢(u) such that for all u∈ℝm and ζ∈ℝm⁢n, we have

(1.2) λ ⁢ ( u ) ≥ λ 0 , λ ⁢ ( u ) ⁢ | ζ | 2 ≤ 〈 A ⁢ ( u ) ⁢ ζ , ζ 〉 and | A ⁢ ( u ) | ≤ C * ⁢ λ ⁢ ( u ) .

In addition, |Au|≤C|λu| and the following number is finite:

𝚲 = sup u ∈ ℝ m ⁡ | λ u ( u ) | λ ⁢ ( u ) .

With a slight abuse of notation, A⁢(u)⁢ζ, 〈A⁢(u)⁢ζ,ζ〉 in (1.2) should be understood in the following way: For A⁢(u)=[ai⁢j⁢(u)], ζ∈ℝm⁢n, we write ζ=[ζi]i=1m with ζi=(ζi,1,…,ζi,n) and

A ⁢ ( u ) ⁢ ζ = [ ∑ j = 1 m a i ⁢ j ⁢ ζ j ] i = 1 m , 〈 A ⁢ ( u ) ⁢ ζ , ζ 〉 = ∑ i , j = 1 m a i ⁢ j ⁢ 〈 ζ i , ζ j 〉 .

Also, here and throughout this paper, if B is a C1 function in u∈ℝm then we abbreviate its derivative ∂⁡B∂⁡u by Bu.

There exist a constant C and a differentiable function f:ℝm→ℝm such that for any differentiable vector-valued functions u:ℝn→ℝm and p:ℝn→ℝm⁢n, we have

| f ^ ⁢ ( u , p ) | ≤ C ⁢ λ 1 / 2 ⁢ ( u ) ⁢ | p | + f ⁢ ( u ) ,

| D ⁢ f ^ ⁢ ( u , p ) | ≤ C ⁢ λ 1 / 2 ⁢ ( u ) ⁢ | D ⁢ p | + C ⁢ | λ u ( u ) | λ 1 / 2 ⁢ ( u ) ⁢ | D ⁢ u | ⁢ | p | + | f u ⁢ ( u ) | ⁢ | D ⁢ u | ,

| f u ⁢ ( u ) | ≤ C ⁢ λ ⁢ ( u ) .

The local existence of a strong solution of (1.1) was proved in [15] under the key assumption that any strong solution u of the system satisfies the following condition: For any given μ0>0, there exists Rμ0>0 such that

(1.3) 𝚲 2 ⁢ sup x 0 ∈ Ω ¯ , t ∈ ( 0 , T 0 ) ⁡ ∥ u ⁢ ( ⋅ , t ) ∥ BMO ⁢ ( B R μ 0 ⁢ ( x 0 ) ∩ Ω ) 2 ≤ μ 0 .

This condition was referred to as condition (M’) in [15].

Here and throughout this paper, BR⁢(y) denotes a ball centered at y with radius R, and a locally integrable function U:Ω→ℝm is said to be in BMO⁢(Ω) if the following quantity is finite:

[ U ] * := sup B R ⁢ ( y ) ⊂ Ω ⨍ B R ⁢ ( y ) | U - U B R ⁢ ( y ) | d x .

We denote by UA the average of U over a measurable set A: UA=1|A|⁢∫AU⁢(x)⁢𝑑x.

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

∥ U ∥ BMO ⁢ ( Ω , ℝ m ) := [ U ] * + ∥ U ∥ L 1 ⁢ ( Ω , ℝ m ) .

We also say that U is VMO in Ω if infR>0,BR⊂Ω∥U∥BMO⁢(BR,ℝm)=0.

In this paper, for simplicity of presentation and with models in applications in mind, we consider only the following special form of the reaction terms which are linear in Du, namely, f^⁢(u,D⁢u)=B⁢(u)⁢D⁢u+f⁢(u), and study local and global existence of strong solutions. Thanks to this form of f^ the fixed point argument in [15] can be greatly simplified. Furthermore, we will provide conditions which are a bit stronger than (1.3) but verifiable in applications. In particular, we will show that a strong solution u exists globally if the norm ∥u∥W1,n⁢(Ω) does not blow up in finite time. This relaxes Amann’s conditions in [3] which required a control on ∥u∥W1,p⁢(Ω) for some p>n. Again, we are not assuming that u is bounded and our structural conditions (A) and (F) are more general than those in [3, 16]. Our results also hold for general f^⁢(u,D⁢u) with linear or quadratic growth in Du, see Remark 2.4.

We organize our paper as follows. In Section 2 we state our main results. In Section 3 we state another version of the local weighted Gagliardo–Nirenberg inequality [15, Lemma 2.4], which is one of the main ingredients of the proof in [15] and of our main theorem in this paper. Technical results and auxiliary lemmas needed for the proof of the main results for linear reaction terms are given in Section 4. We conclude the paper with an appendix presenting a full and simpler proof for the global and local weighted Gagliardo–Nirenberg inequalities in [15].

2 Preliminaries and Main Results

In this section we state the main results of this paper. Our first main result concerns the local existence of strong solutions to (1.1) with f^ being linear in Du, i.e.,

(2.1) f ^ ⁢ ( u , D ⁢ u ) = B ⁢ ( u ) ⁢ D ⁢ u + f ⁢ ( u ) .

We imbed (1.1) in the following family of systems:

(2.2) { u t = div ⁡ ( A ⁢ ( σ ⁢ u ) ⁢ D ⁢ u ) + f ^ ⁢ ( σ ⁢ u , σ ⁢ D ⁢ u ) , ( x , t ) ∈ Q = Ω × ( 0 , T 0 ) , σ ∈ [ 0 , 1 ] , u ⁢ ( x , 0 ) = U 0 ⁢ ( x ) , x ∈ Ω , u = 0 or ∂ ⁡ u ∂ ⁡ ν = 0 on ⁢ ∂ ⁡ Ω × ( 0 , T 0 ) .

In [15] we assumed the spectral gap condition, which requires that the eigenvalues of the matrix A⁢(u) are not too far apart. Namely, we need that n-2n<C*-1, where C* is, in certain sense, the ratio of the largest and smallest eigenvalues of A⁢(u). One should note that there exist counterexamples for global existence of strong solutions if the eigenvalues of the matrix A⁢(u) are far apart [1, 17]. We then again assume that

(2.3) n * = 2 ⁢ C * C * - 1 > n .

Our first main result is the following.

Theorem 2.1.

Assume that (A), (F) and (2.3) hold. Assume also that the following conditions hold for any strong solution u of (2.2) with (2.1):

There exists μ0>0 sufficiently small in terms of the constants in (A) and (F) such that for some Rμ0>0, which may depends on T0, we have

𝚲 2 sup x 0 ∈ Ω ¯ , t ∈ ( 0 , T 0 ) ∥ u ( ⋅ , t ) ∥ BMO ⁢ ( B R ⁢ ( x 0 ) ∩ Ω ) 2 ≤ μ 0 .
The following quantity is finite:

C T 0 := ∬ Ω × ( 0 , T 0 ) | D u | 2 d x .
There exist constants L ⁢ ( T 0 ) and r * > n p * - n , with p * = min ⁡ { n * , p 0 } , assuming n * satisfies ( 2.3 ) and p 0 > n , such that

sup t ∈ ( 0 , T 0 ) ∥ λ ( u ( ⋅ , t ) ) ∥ L r * ⁢ ( Ω ) ≤ L ( T 0 ) .

Then (1.1) has a unique strong solution u on Ω×(0,T0). Moreover, if the above assumptions hold for all T0>0, then (1.1) has a unique strong solution u which exists globally on Ω×(0,∞).

The next results are more applicable and improve those of Amann in [3, 2]. Basically, we need only to control the W1,n⁢(Ω) norm of strong solutions while [3, 2] required that their W1,p⁢(Ω) norms do not blow up in finite time for some p>n, and thus the boundedness of the solutions is needed in his results.

Corollary 2.2.

The conclusion of Theorem 2.1 holds if (M1) and (M2) are replaced by the following assumption:

There exists a constant CT0 such that for any t∈(0,T0),

∥ u ⁢ ( ⋅ , t ) ∥ W 1 , n ⁢ ( Ω ) ≤ C T 0 .

If this condition holds for all T0>0, then u exists globally.

Finally, concerning the integrability condition of λ⁢(u) in (L), we can assume a weaker integrability of λ⁢(u) if it has a polynomial growth. We have the following result.

Corollary 2.3.

The conclusion of Corollary 2.2 holds if there exist constants Λ1,ε0>0 such that

(2.4) | λ u ⁢ ( u ) | ≤ Λ 1 ⁢ λ 1 - ε 0 ⁢ ( u ) for all ⁢ u ∈ ℝ m ,

and (L) is replaced by the following weaker one:

There exist constants L⁢(T0), s0>0 such that

sup t ∈ ( 0 , T 0 ) ∥ λ s 0 ( u ( ⋅ , t ) ) ∥ L 1 ⁢ ( Ω ) ≤ L ( T 0 ) .

It is easy to see that condition (2.4) holds if λ⁢(u) has polynomial growth in u.

Remark 2.4.

The results of this paper also hold for reaction terms with linear growth in Du. Namely, we can assume that

(2.5) | ∂ ζ ⁡ f ^ ⁢ ( u , ζ ) | ≤ C ⁢ λ 1 / 2 ⁢ ( u ) , | ∂ u ⁡ f ^ ⁢ ( u , 0 ) | ≤ C ⁢ λ ⁢ ( u ) for all ⁢ u ∈ ℝ m ⁢ and all ⁢ ζ ∈ ℝ n ⁢ m .

In fact, it is possible to obtain the same results for f^ with quadratic growth in Du. That is, we can assume that |∂ζf^(u,ζ)|≤Cλ1/2(u)(|ζ|+1), which is clearly implied by (2.5). The proof is of course more involved and will be reported in our forthcoming work.

3 Technical Results

In this section we state another version of the local weighted Gagliardo–Nirenberg inequality [15, Lemma 2.4], which is one of the main ingredients of the proof in [15] and our main theorem in this paper. 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 is an Aγ weight if the quantity

(3.1) [ w ] γ := sup B R ⁢ ( y ) ⊂ Ω ⁡ ( ⨍ B R ⁢ ( y ) w ⁢ 𝑑 x ) ⁢ ( ⨍ B R ⁢ ( y ) w 1 - γ ′ ⁢ 𝑑 x ) γ - 1 ⁢ is finite .

Here, γ′=γγ-1. For more details on these classes, we refer the reader to [19, 22].

Throughout this paper, when there exists no ambiguity C,Ci will denote universal constants that can change from line to line in our argument. If necessary, C⁢(⋯) or C(⋯) are used to denote quantities which are bounded in terms of theirs parameters in (⋯). We will also write a∼b if there exist two generic positive constants C1,C2 such that C1⁢b≤a≤C2⁢b. Furthermore, we denote by BR⁢(x0) a ball with center x0∈Ω¯. In the sequel, if the center x0 is already specified, then we simply write BR and ΩR for BR⁢(x0) and BR⁢(x0)∩Ω, respectively.

We have the following version of [15, Lemma 2.4].

Lemma 3.1.

Let u,U:Ω→Rm be vector-valued functions with u∈C1⁢(Ω), U∈C2⁢(Ω), and let Φ:Rm→R be a C1 function and Φ⁢(u)α be an Aβ+1 weight for some α>2p+2 and β<pp+2. Suppose that either U or Φ2⁢(u)⁢∂⁡U∂⁡ν vanish on the boundary ∂⁡Ω of Ω. For any ball Bt⁢(x0) with center x0∈Ω¯, we set

I 1 ⁢ ( t ) := ∫ Ω t Φ 2 ⁢ ( u ) ⁢ | D ⁢ U | 2 ⁢ p + 2 ⁢ 𝑑 x , ⁢ I ^ 1 ⁢ ( t ) := ∫ Ω t Φ 2 ⁢ ( u ) ⁢ | D ⁢ u | 2 ⁢ p + 2 ⁢ 𝑑 x ,

I ¯ 1 ( t ) := ∫ Ω t | Φ u ( u ) | 2 ( | D U | 2 ⁢ p + 2 + | D u | 2 ⁢ p + 2 ) d x , ⁢ I 2 ⁢ ( t ) := ∫ Ω t Φ 2 ⁢ ( u ) ⁢ | D ⁢ U | 2 ⁢ p - 2 ⁢ | D 2 ⁢ U | 2 ⁢ 𝑑 x .

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 exist positive constants Cε,Φ, which depend on [Φα]β+1 and Cε, such that

(3.2) I 1 ( s ) ≤ ε [ I 1 ( t ) + I ^ 1 ( t ) ] + C ε , Φ ∥ U ∥ BMO ⁢ ( Ω t ) 2 [ I ¯ 1 ( t ) + I 2 ( t ) ] + C ε ∥ U ∥ BMO ⁢ ( Ω t ) 2 sup x ∈ B t | D ψ ( x ) | 2 ∫ Ω t Φ 2 ( u ) | D U | 2 ⁢ p d x .

The only differences between the two versions are that the factor ∥U∥BMO⁢(Bt)2 in the last terms of (3.2) replaces the factor ∥U∥BMO⁢(Bt) in (2.17) of [15, Lemma 2.4] and the condition on [Φα]β+1 (both facts are not important in this paper and other applications). The two proofs differ only by the order of using Young’s inequality in the argument. Since this inequality and its global version will be very useful for other purposes, we present their proof in Appendix A. Our proofs are somehow simpler than that in [15].

We now let Φ≡1 and ψ be a cutoff function for Bs,Bt, i.e., ψ=1 in Bs and ψ=0 outside Bt and |Dψ|≤1t-s. Then Φ is an Aγ weight for all γ>1 and Φu≡0. The following version of the above lemma with u=U suffices for our purpose in this paper.

Lemma 3.2.

Let U:Ω→Rm be a vector-valued function in C2⁢(Ω). Suppose that either U or ∂⁡U∂⁡ν vanish on the boundary ∂⁡Ω of Ω. Then, for any two concentric balls Bs, Bt, with s<t, and any p≥1, ε>0, there exists Cε>0 such that

(3.3) ∫ Ω s | D U | 2 ⁢ p + 2 d x ≤ ε ∫ Ω t | D U | 2 ⁢ p + 2 d x + C ε ∥ U ∥ BMO ⁢ ( Ω t ) 2 ∫ Ω t [ | D U | 2 ⁢ p - 2 | D 2 U | 2 + ( t - s ) - 2 | D U | 2 ⁢ p ] d x .

4 The Proof of the Main Results

In this section we present the proof of Theorem 2.1 and its corollaries. The proof relies on the Leray–Schauder theorem. We obtain the existence of a strong solution u of (1.1) as a fixed point of a nonlinear map defined on an appropriate Banach space.

Let us consider the Banach space 𝐗=C⁢(Q,ℝm), where Q=Ω×(0,T0). For any given u∈𝐗 and σ∈[0,1], we consider the following linear system:

(4.1) { w t = div ⁡ ( A ⁢ ( σ ⁢ u ) ⁢ D ⁢ w ) + B ⁢ ( σ ⁢ u ) ⁢ D ⁢ w + f ⁢ ( σ ⁢ u ) , ( x , t ) ∈ Q , w ⁢ ( x , 0 ) = U 0 ⁢ ( x ) , x ∈ Ω , w = 0 or ∂ ⁡ w ∂ ⁡ ν = 0 on ⁢ ∂ ⁡ Ω × ( 0 , T 0 ) .

We then define Tσ⁢(u)=w. It is clear that a fixed point of Tσ solves (2.2). In order to apply the Leray–Schauder theorem, we need to establish the following steps:

The map Tσ:𝐗→𝐗 is well defined and compact.
There exists a constant M such that ∥u∥𝐗≤M for any fixed points of u=Tσ⁢(u).

Step 1 is fairly standard thanks to the following lemma.

Lemma 4.1.

The map Tσ:X→X is well defined and compact.

Proof.

For each u∈𝐗, A⁢(σ⁢u) satisfies the ellipticity condition (A), and the data of the linear system (4.1) are bounded and continuous. So, (4.1) satisfies the assumptions of [11, Chapter VII, Theorem 1.1], see also [4], which applies to the system

w t = div ⁡ ( 𝐚 ⁢ D ⁢ w ) + 𝐛 ⁢ D ⁢ w + 𝐠 ,

under the assumption that 𝐚,𝐛 and ∥𝐠∥q,r,Q are bounded for sufficiently large q,r such that 1r+n2⁢q=1. Here, for any vector-valued function F,

∥ F ∥ q , r , Q := ( ∫ 0 T 0 ( ∫ Ω | F ( x , t ) | q d x ) r / q d t ) 1 / r .

Theorem 1.1 in [11, Chapter VII] shows that w exists uniquely, and so Tσ⁢(u) is well defined. Moreover, as the initial condition w⁢(⋅,0)=U0⁢(x) belongs to W1,p0⁢(Ω) and then C0β⁢(Ω) for β0=1-np0>0, a combination of [11, Chapter VII, Theorems 2.1 and 3.1] shows that w belongs to Cα0,α0/2⁢(Q¯,ℝm) for some α0>0. In addition, the norm ∥w∥Cα0,α0/2⁢(Q¯) depends on β0 and ∥A⁢(σ⁢u)∥∞, ∥B⁢(σ⁢u)∥∞, ∥fσ⁢(u)∥q,r,Q. Thus, if u belongs to a bounded set K of 𝐗, then ∥u∥𝐗≤M for some M, and there exists a constant C such that

∥ w ∥ C α 0 , α 0 / 2 ⁢ ( Q ¯ ) ≤ C ⁢ ( M , ∥ U 0 ⁢ ( ⋅ , 0 ) ∥ C β 0 ⁢ ( Ω ) ) .

Hence, Tσ⁢(K) is compact in 𝐗 and Tσ:𝐗→𝐗 is a compact map. ∎

We now turn to Step 2, the hardest part of the proof, and provide a uniform estimate for the fixed points of Tσ. Such a fixed point u of Tσ satisfies (4.1) and belongs to 𝐗. Therefore, u is a bounded weak solution and continuous, and so [5, Theorems 2.1 and 3.2] apply and yield that Du is bounded in Ω×(t0,T0) for all t0>0. Thus, Du is locally bounded in Ω×(0,T0). It is then well known that D2⁢u exists in Lloc2(Ω×(0,T0) and that u is a strong solution in Ω×(0,T0).

Thus, in the rest of this section, we consider a strong solution u of (4.1). As the data of (4.1) satisfy the structural conditions (A), (F) with the same set of constants and assumptions (M1), (M2) and (L) are assumed to be uniform for all σ∈[0,1], we will only present the proof for σ=1 in the sequel.

We should also emphasize that the estimates in the rest of this section do not require the special form of f^ in (2.1) but the growth condition in (F).

For any two concentric balls Bs,Bt with s<t, we say that ψ is a cutoff function for Bs,Bt if ψ is a C1 function satisfying ψ≡1 in Bs, and ψ≡0 outside Bt and |Dψ|≤1t-s. Similarly, for T1<T2<T3, we say that η is a cutoff function for (T1,T3),(T2,T3) if η is a C1 function satisfying η⁢(t)≡0 for t≤T1, and η⁢(t)≡1 if t≥T3 and |ηt|≤1T2-T1.

We begin with the following energy estimate for Du.

Lemma 4.2.

We assume that A,f^ satisfy (A), (F). Suppose that u is a strong solution of (1.1) on Ω×(0,T0). Consider any given triple t0,T,T′ satisfying 0<t0<T<T′≤T0 and p∈[1,n*2), see the definition (2.3) of n*. Then there exists a constant C, which depends only on the parameters in (A) and (F), such that for any two concentric balls Bs,Bt with center x0∈Ω¯ and s<t, we have

sup t ∈ ( T , T ′ ) ∫ Ω s λ - 1 ( u ) | D u | 2 ⁢ p d x + ∬ Q s , t 0 | D u | 2 ⁢ p - 2 | D 2 u | 2 η d x

(4.2) ≤ C 𝚲 2 ∬ Q t , t 0 | D u | 2 ⁢ p + 2 η d x + C ( ( t - s ) - 2 ∬ Q t , t 0 | D u | 2 ⁢ p d x + t 0 - 1 ∫ T - t 0 T ∫ Ω t | D u | 2 ⁢ p d x d s ) .

Here, Qt,t0=Ωt×(T-t0,T′) and η is a cutoff function for (T-t0,T′),(T,T′).

The above lemma is a special case of the energy estimate for Du in [15, Lemma 3.2] with W=U=u and β⁢(u)=λ-1⁢(u). Roughly speaking, we differentiated the system in x to obtain

(4.3) ( D ⁢ u ) t = div ⁡ ( A ⁢ ( u ) ⁢ D 2 ⁢ v + A u ⁢ ( u ) ⁢ D ⁢ u ⁢ D ⁢ u ) + D ⁢ f ^ ⁢ ( u , D ⁢ u ) .

We then test the above with λ-1⁢(u)⁢|D⁢u|2⁢p-2⁢D⁢u⁢ψ2⁢(x)⁢η⁢(t), where ψ is a cutoff function for Bs, Bt, and η is a cutoff function for (T-t0,T′),(T,T′). Because 2⁢p<n*, from the definition (2.3) of n*, it is clear that 2⁢p-22⁢p<C*-1, and so the spectral gap condition needed in the proof of [15, Lemma 3.2] is available here. Some simple use of Hölder and Young’s inequalities gives (4.2). Here, the last integral in (4.2) comes from the integration by parts in time, and we made use of the assumption that β⁢(u)=λ-1⁢(u) is bounded from above, |η′|≤t0-1 and that |η′| is zero outside [T-t0,T].

Next, we have the following technical result.

Lemma 4.3.

In addition to the assumptions of Lemma 4.2, we suppose that the quantity

(4.4) C t 0 , T , T ′ := ∬ Ω × ( T - t 0 , T ′ ) | D u | 2 d x is finite .

There exists μ0>0 sufficiently small, in terms of the constants in (A) and (F), such that if for some positive Rμ0, which may depends on t0,T,T′, such that

(4.5) 𝚲 2 sup x 0 ∈ Ω ¯ , t ∈ ( T - t 0 , T ′ ) ∥ u ( ⋅ , t ) ∥ BMO ( Ω R ( x 0 ) 2 ≤ μ 0 ,

then there exist p>n2, an integer k0 and a constant C depending only on the parameters of (A) and (F), Ct0,T,T′, Rμ0, and t0,T,T′, such that

(4.6) sup t ∈ ( T , T ′ ) ⁡ ∫ Ω R λ - 1 ⁢ ( u ) ⁢ | D ⁢ u | 2 ⁢ p ⁢ 𝑑 x ≤ C for any ⁢ R < 2 - k 0 ⁢ R μ 0 .

Proof.

We follow the argument in the proof of [15, Proposition 3.1] with W=U=u. Suppose that the energy estimate (4.2) in Lemma 4.2 holds for some p≥1. We write it as

(4.7) 𝒜 ⁢ ( s ) + ℋ ⁢ ( s ) ≤ C ⁢ 𝚲 2 ⁢ ℬ ⁢ ( t ) + C ⁢ [ ( t - s ) - 2 ⁢ 𝒞 ⁢ ( t ) + t 0 - 1 ⁢ 𝒞 ^ ⁢ ( t ) ] , 0 < s < t ,

where the functions 𝒜,ℋ,ℬ,𝒞 and 𝒞^ are defined by

𝒜 ( s ) := sup t ∈ ( T , T ′ ) ∫ Ω s λ - 1 ( u ) | D u | 2 ⁢ p d x , ℋ ( s ) := ∬ Q s , t 0 | D u | 2 ⁢ p - 2 | D 2 u | 2 η d x ,

ℬ ( s ) := ∬ Q s , t 0 | D u | 2 ⁢ p + 2 η d x , 𝒞 ( s ) := ∬ Q s , t 0 | D u | 2 ⁢ p d x , 𝒞 ^ ( s ) := ∫ T - t 0 T ∫ Ω s | D u | 2 ⁢ p d x d s .

On the other hand, we apply Lemma 3.2 to estimate ℬ⁢(t), the integral of |Du|2⁢p+2, on the right-hand side of (4.7). Namely, we let U=u, multiply (3.3) by 𝚲2⁢η and integrate the result over (T-t0,T′) to get (recalling the definition of μ0 in (4.5))

𝚲 2 ⁢ ℬ ⁢ ( s ) ≤ ε ⁢ 𝚲 2 ⁢ ℬ ⁢ ( t ) + C ⁢ ( ε ) ⁢ μ 0 ⁢ ℋ ⁢ ( t ) + C ⁢ ( ε ) ⁢ μ 0 ⁢ ( t - s ) - 2 ⁢ 𝒞 ⁢ ( t ) , 0 < s < t ≤ R μ 0 .

Let us define F⁢(t):=𝚲2⁢ℬ⁢(t), G⁢(t):=ℋ⁢(t), g⁢(t):=𝒞⁢(t) and h⁢(t):=t0-1⁢𝒞^⁢(t). Then the above yields

F ⁢ ( s ) ≤ ε 0 ⁢ [ F ⁢ ( t ) + G ⁢ ( t ) ] + C ⁢ ( t - s ) - 2 ⁢ g ⁢ ( t ) ,

where ε0=𝚲2⁢ε+C⁢(ε)⁢μ0. This obviously gives

F ⁢ ( s ) ≤ ε 0 ⁢ [ F ⁢ ( t ) + G ⁢ ( t ) ] + C ⁢ ( t - s ) - 2 ⁢ g ⁢ ( t ) + C ⁢ h ⁢ ( t ) .

On the other hand, (4.7) implies

G ⁢ ( s ) ≤ C ⁢ [ F ⁢ ( t ) + ( t - s ) - 2 ⁢ g ⁢ ( t ) + h ⁢ ( t ) ] .

As ε=𝚲2⁢ε+C⁢(ε)⁢μ0, it is clear that we can choose and fix some ε sufficiently small, and then for μ0 small in terms of C,ε so that 2⁢C⁢ε0<1. Thus, if μ0 is sufficiently small in terms of the constants in (A), (F), then we can apply a simple iteration argument, see [15, Lemma 3.11], to obtain

F ⁢ ( s ) + G ⁢ ( s ) ≤ C ⁢ [ ( t - s ) - 2 ⁢ g ⁢ ( t ) + h ⁢ ( t ) ] for 0 < s < t ≤ R μ 0 .

Hence, for any R<Rμ02, we take t=2⁢R and s=R in the above to obtain

(4.8) ∬ Q R , t 0 ( | D u | 2 ⁢ p - 2 | D 2 u | 2 + | D u | 2 ⁢ p + 2 ) d x ≤ C 1 ( R - 2 ∬ Q 2 ⁢ R , t 0 | D u | 2 ⁢ p d x + t 0 - 1 ∫ T - t 0 T ∫ Ω 2 ⁢ R | D u | 2 ⁢ p d x d s ) .

The above argument shows that if there exist p≥1 and a constant C⁢(R,t0) such that the energy estimate (4.2) holds for p and

(4.9) ∬ Q 2 ⁢ R , t 0 λ - 1 ⁢ ( u ) ⁢ | D ⁢ u | 2 ⁢ p ⁢ 𝑑 x ≤ C ⁢ ( R , t 0 ) ,

then this estimate also holds for p being replaced by any q∈(p,p+1], via (4.8) and Hölder’s inequality. By assumption (4.4), (4.9) holds for p=1. It is now clear that we can repeat the argument k0 times to find a number p>n2, as long as 2⁢p<n* (so that (4.2) holds by Lemma 4.2). We then see that (4.8) and (4.9) hold for such p, and therefore estimate (4.6) follows from the energy estimate for (4.2), with t=2⁢R, s=R. The lemma is proved. ∎

Lemma 4.3 made use of a cutoff function η for the intervals [T-t0,T] and [T,T′] to avoid the dependence on the initial data at t=0. This type of result is useful when one wants to discuss the long time dynamics and global attractors of the system.

In order to establish the local and global existence results, we have to provide bounds for u in Ω×[0,T0) and allow t0=0. The next lemma considers this case.

Lemma 4.4.

Let the assumptions in Lemma 4.3 with T=t0=0 hold. That is,

(4.10) ∬ Ω × ( 0 , T ′ ) | D u | 2 d x is finite 𝑎𝑛𝑑 𝚲 2 sup x 0 ∈ Ω ¯ , t ∈ ( 0 , T ′ ) ∥ u ( ⋅ , t ) ∥ BMO ( Ω R ( x 0 ) 2 ≤ μ 0

for some positive μ0, Rμ0 sufficiently small, in terms of the constants in (A), (F) and T′. In addition, for some T1∈(0,T0) and p≥1, assume that

(4.11) u ∈ C ⁢ ( [ 0 , T 1 ) , L 2 ⁢ p ⁢ ( Ω ) ) ,

(4.12) sup t ∈ [ 0 , T 1 ) ∥ D u ( ⋅ , t ) ∥ L 2 ⁢ p ⁢ ( Ω ) is finite .

If (4.5) holds, then for the same constant C, the conclusion (4.6) now reads

(4.13) sup t ∈ ( 0 , T ′ ) ⁡ ∫ Ω R λ - 1 ⁢ ( u ) ⁢ | D ⁢ u | 2 ⁢ p ⁢ 𝑑 x ≤ C + C ⁢ ∥ D ⁢ u ⁢ ( ⋅ , 0 ) ∥ L 2 ⁢ p ⁢ ( Ω ) 2 ⁢ p for any ⁢ R < 2 - k 0 ⁢ R μ 0 .

Proof.

Thanks to assumption (4.12), we can let T,t0→0 in (4.8) to see that if ∥D⁢u∥L2⁢p⁢(Q2⁢R,0) is finite for QR,0=ΩR×(0,T′), then

(4.14) ∬ Q R , 0 ( | D ⁢ u | 2 ⁢ p - 2 ⁢ | D 2 ⁢ u | 2 + | D ⁢ u | 2 ⁢ p + 2 ) ⁢ 𝑑 x ⁢ is finite .

Using the difference quotient operator δh instead of D in (4.3) in the proof of Lemma 4.2, we obtain

( δ h ⁢ u ) t = div ⁡ ( A ⁢ ( u ) ⁢ D ⁢ ( δ h ⁢ u ) + δ h ⁢ ( A ⁢ ( u ) ) ⁢ D ⁢ u ) + δ h ⁢ f ^ ⁢ ( u , D ⁢ u ) .

We test this with λ-1⁢(u)⁢|δh⁢u|2⁢p-2⁢δh⁢u⁢ψ2⁢(x), where ψ is a cutoff function for Bs, Bt. We easily see that the energy estimate in Lemma 4.2 holds with the operator D being replaced by δh. Since u∈C⁢([0,T′),L2⁢p⁢(Ω)), by assumption (4.11), we can let T,t0→0 and obtain

sup t ∈ ( 0 , T ′ ) ∫ Ω s λ - 1 ( u ) | δ h u | 2 ⁢ p d x + ∬ Q s , 0 | δ h u | 2 ⁢ p - 2 | D δ h u | 2 d x

≤ C 𝚲 2 ∬ Q t , 0 | D u | 2 | δ h u | 2 ⁢ p d x + ( t - s ) - 2 ∬ Q t , 0 | δ h u | 2 ⁢ p ] d x + C ∫ Ω t | δ h u ( x , 0 ) | 2 ⁢ p d x .

As we now see, the integral in (4.14) is finite, and so we can let h tend to 0 and obtain a similar energy estimate (4.2) for Du with t0=0 and η≡1. Namely,

sup t ∈ ( 0 , T ′ ) ∫ Ω s λ - 1 ( u ) | D u | 2 ⁢ p d x + ∬ Q s , 0 | D u | 2 ⁢ p - 2 | D 2 u | 2 d x

≤ C 𝚲 2 ∬ Q t , 0 | D u | 2 ⁢ p + 2 d x + ( t - s ) - 2 ∬ Q t , 0 | D u | 2 ⁢ p d x + C ∫ Ω t | D u ( x , 0 ) | 2 ⁢ p d x .

Again, given the second assumption in (4.10), we can argue as in Lemma 4.3 to treat ℬ⁢(t), the integral of |Du|2⁢p+2, on the right-hand side and redefine h⁢(t):=∥D⁢u⁢(⋅,0)∥L2⁢p⁢(Ω)2⁢p, a constant in t. The same argument then yields a version of (4.8) with t0=0. In particular, we obtain

∬ Q R , 0 | D u | 2 ⁢ p + 2 d x ≤ C 1 R - 2 ∬ Q 2 ⁢ R , 0 | D u | 2 ⁢ p d x + C 1 ∥ D u ( ⋅ , 0 ) ∥ L 2 ⁢ p ⁢ ( Ω ) 2 ⁢ p .

With assumption (4.10) the iteration argument after (4.8) in the proof of Lemma 4.3 on the power p then gives (4.13). This completes the proof. ∎

We are now ready to provide the proof of the main theorem.

Proof of Theorem 2.1.

By Lemma 4.1, the map Tσ:𝐗→𝐗 defined by (4.1) is compact. In order to apply the Leray–Schauder theorem and show that there exists a fixed point u for σ=1, which is the solution of (1.1), we need only to provide a uniform bound for the fixed points of Tσ and conclude the proof. To this end, for any σ∈[0,1], we consider a fixed point u of Tσ.

Since u∈𝐗, u is a bounded weak solution and continuous, and so [5, Theorems 2.1 and 3.1] apply and yield that Du is locally bounded in Q=Ω×(0,T0). It is then well known that D2⁢u∈Lloc2⁢(Q), and thus u is a strong solution in Q.

We will apply Lemma 4.4 here to provide a uniform bound. First of all, the continuity assumption (4.11) of the lemma is clear because u∈𝐗. Next, for any q=2⁢p∈(n,p0), we show that ∥u⁢(⋅,t)∥W1,q⁢(Ω) is bounded in [0,T0) to verify (4.12). For any h>0 and any function w, we denote by w(h)=ϕh*w the mollifier/regularizer of w. For any f∈Lq′⁢(Ω), we have

∫ Ω D ⁢ ( u ⁢ ( x , t ) ( h ) ) ⁢ f ⁢ ( x ) ⁢ 𝑑 x = ∫ Ω ( D ⁢ u ⁢ ( x , t ) ) ( h ) ⁢ f ⁢ ( x ) ⁢ 𝑑 x = ∫ Ω D ⁢ u ⁢ ( x , t ) ⁢ f ( h ) ⁢ ( x ) ⁢ 𝑑 x

= ∫ Ω u ⁢ ( x , t ) ⁢ D ⁢ f ( h ) ⁢ ( x ) ⁢ 𝑑 x → ∫ Ω u ⁢ ( x , 0 ) ⁢ D ⁢ f ( h ) ⁢ ( x ) ⁢ 𝑑 x as t → 0 ,

because u∈𝐗. The last term in the above is bounded by ∥u⁢(⋅,0)∥W1,p⁢(Ω)⁢∥fh∥Lq′⁢(Ω) and u⁢(⋅,0)=U0⁢(⋅). By the uniform boundedness principle, noting that D⁢u⁢(⋅,t)∈Lq⁢(Ω) for each t>0, we see that ∥D⁢u(h)⁢(⋅,t)∥Lq⁢(Ω) is uniformly bounded with respect to h for all h>0 and t∈[0,T0). By letting h→0, we derive that supt∈[0,T0)⁡∥D⁢u⁢(⋅,t)∥L2⁢p⁢(Ω) is finite. Thus, for each fixed point u of Tσ, condition (4.12) holds.

Hence, from assumptions (M1) and (M2), the assumption (4.10) of Lemma 4.4 holds, and so the lemma can apply here to provide uniform constants C*, R1, depending only on the parameters of (A) and (F), CT0, Rμ0 and ∥D⁢U0∥Lp0⁢(Ω), such that if p<p*=12⁢min⁡{n*,p0}, then

(4.15) sup t ∈ ( 0 , T 0 ) ⁡ ∫ Ω R 1 λ - 1 ⁢ ( u ) ⁢ | D ⁢ u | 2 ⁢ p ⁢ 𝑑 x ≤ C * .

From the definition of r* it is clear that we can choose p,p1 such that n<p1<p<p* and r*=p1p-p1. As r*=p1p⁢(pp1)′, by Hölder’s inequality, we have

(4.16) ∫ Ω R 1 | D u | 2 ⁢ p 1 d x ≤ ∥ λ ( u ) ∥ L r * ⁢ ( Ω R 1 ) ( ∫ Ω R 1 λ - 1 ( u ) | D u | 2 ⁢ p d x ) p 1 / p .

From assumption (L) on λ⁢(u) and (4.15), the right-hand side of (4.16) will be bounded uniformly for all σ∈[0,1].

We then have a uniform bound for ∥u∥W1,q⁢(Ω). As q=2⁢p1>n, by Sobolev’s embedding theorem, we see that ∥u∥𝐗≤M for some constant M and all σ∈[0,1]. The Leray–Schauder theory then applies to provide a fixed point u=T1⁢(u). This fixed point is the unique strong solution of system (1.1). ∎

Proof of Corollary 2.2.

We just need to show that assumption (D) implies (M1) and (M2). It is clear that (D) yields (M2). To verify (M1), we argue by contradiction. If this is not the case, then there exist sequences {xn}⊂Ω¯, {σn}⊂[0,1], {tn}⊂(0,T0), {rn}, rn→0, and a sequence of strong solutions {uσn} such that for Un⁢(⋅)=uσn⁢(⋅,tn),

∥ U n ∥ BMO ⁢ ( B r n ⁢ ( x n ) ∩ Ω ) > ε 0 for some ε 0 > 0 .

By (D), we see that the sequence {Un} is bounded in W1,n⁢(Ω). We can then assume that Un converges weakly to some U in W1,2⁢(Ω) and strongly in L2⁢(Ω). We then have ∥Un∥BMO⁢(BR∩Ω)→∥U∥BMO⁢(BR∩Ω) for any given ball BR. It is easy to see that U∈W1,n⁢(Ω) and, by Poincaré’s inequality, U is VMO and ∥U∥BMO⁢(BR∩Ω)<ε02 if R is sufficiently small. The number R is independent of λ0≥1 because ∥U∥W1,n⁢(Ω) is independent of λ0. Furthermore, we can assume also that xn converges to some x∈Ω¯. Thus, for large n, we have rn<R2 and xn∈BR/2⁢(x). Then, for large n, Brn⁢(xn)⊂BR⁢(x) and

∥ U n ∥ BMO ⁢ ( B r n ⁢ ( x n ) ∩ Ω ) ≤ ∥ U n ∥ BMO ⁢ ( B R ⁢ ( x ) ∩ Ω ) ≤ ∥ U ∥ BMO ⁢ ( B R ⁢ ( x ) ∩ Ω ) + ε 0 2 < ε 0 .

We obtain a contradiction. Thus, (M1) holds and the proof is complete. ∎

Proof of Corollary 2.3.

We need only to show that (D) and (L′) together imply (L). Let u be any strong solution of (2.2) and λ⁢(u) satisfy (2.4). There exist s0,C0>0 such that

(4.17) ∥ λ s 0 ⁢ ( u ) ∥ L 1 ⁢ ( Ω ) ≤ C 0 ⁢ ( T 0 ) .

We will show that for any r>1, there exists a constant C, depending on C0,s0,r,|Ω|,T0 and ∥u∥W1,n⁢(Ω), such that ∥λ⁢(u)∥Lr⁢(Ω)≤C.

We choose and fix s>0 and p∈(1,n) such that s⁢p*=s0, where p*=n⁢pn-p. Then (4.17) implies

(4.18) ∥ λ s ⁢ ( u ) ∥ L p * ⁢ ( Ω ) ≤ C 0 1 / p * ⁢ ( T 0 ) .

We define g⁢(⋅)=λs+ε0⁢(u⁢(⋅,t)). The definition of Λ1 in (2.4) gives

| D ⁢ g | ≤ C ⁢ ( s ) ⁢ | λ u | λ 1 - ε 0 ⁢ ( u ) ⁢ λ s ⁢ ( u ) ⁢ | D ⁢ u | ≤ C ⁢ ( s ) ⁢ Λ 1 ⁢ λ s ⁢ ( u ) ⁢ | D ⁢ u | .

Hence, by Hölder’s inequality, ∥D⁢g∥Lp⁢(Ω)≤C⁢∥λs⁢(u)∥Lp*⁢(Ω)⁢∥D⁢u∥Ln⁢(Ω). This and (4.18) and (D) provide some C⁢(T0) such that ∥D⁢g∥Lp⁢(Ω)≤C⁢(T0). Using Hölder’s inequality, we have

∥ g ∥ L 1 ⁢ ( Ω ) ≤ C ⁢ ∥ λ s ⁢ ( u ) ∥ L p * ⁢ ( Ω ) ( s + ε 0 ) / s ≤ C ⁢ C 0 1 + ε 0 / ( p * ⁢ s ) ⁢ ( T 0 ) .

Hence,

∥ g ∥ W 1 , p ⁢ ( Ω ) ≤ C ⁢ ( T 0 ) + C ⁢ C 0 1 + ε 0 / ( p * ⁢ s ) ⁢ ( T 0 ) .

By Sobolev’s embedding theorem, ∥g∥Lp*⁢(Ω) is bounded. From the definition of g, we can find a constant C¯⁢(T0) such that ∥λs+ε0⁢(u)∥Lp*⁢(Ω)≤C¯⁢(T0). Thus, there exists a constant C1⁢(T0) such that

∥ λ s 0 + p * ⁢ ε 0 ⁢ ( u ) ∥ L 1 ⁢ ( Ω ) ≤ C 1 ⁢ ( T 0 ) .

This shows that if (4.17) holds for some s0, then it also holds for s0 being s0+p*⁢ε0 and a new constant C1⁢(T0). It is then clear that we can repeat this argument to see that ∥λs0+k⁢p*⁢ε0⁢(u)∥L1⁢(Ω)≤Ck⁢(T0) for all integers k and some Ck⁢(T0). This fact and a simple use of Hölder’s inequality show that (L) holds. The proof is complete. ∎

Remark 4.5.

By (4.16), u is Hölder in x. We can show that u is also Hölder continuous in x,t. Indeed, (4.8) with p=1 shows that |D2u|2,|Du|4 are in L1⁢(Q). From the system of u and a simple use of Hölder’s inequality, we obtain that

∥ u t ∥ L 1 ⁢ ( Q ) ≤ ∥ A ⁢ ( u ) ∥ L 2 ⁢ ( Q ) ⁢ ∥ D 2 ⁢ u ∥ L 2 ⁢ ( Q ) + ∥ A u ⁢ ( u ) ∥ L 2 ⁢ ( Q ) ⁢ ∥ | D ⁢ u | 2 ∥ L 2 ⁢ ( Q ) + ∥ f ^ ∥ L 1 ⁢ ( Q ) .

Since |A(u)|,|Au(u)|≤λ(u) and f^ has linear growth in Du, the right-hand side is finite and bounded by a constant independent of λ0 (using (4.8) with p=1 and then (4.16) to see that ∥D2⁢u∥L2⁢(Q)≤C⁢λ0-1). Thus, ut belongs to L1⁢(Q). It is well known that if u is Hölder continuous in x, and ut is in L1⁢(Q), then u is Hölder in x,t (see [18, Lemma 4]).

Communicated by Shair Ahmad

A Appendix

In this section we provide the details of the key 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. The proof somehow simplifies that in [15], as we will not use the Muckenhoupt’s inequality for the uncentered maximal operator but simple Hölder’s inequality.

Again, we write BR⁢(x) for a ball centered at x with radius R and will omit x if no ambiguity arises. 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:Ω→ℝm, we denote by |A| the Lebesgue measure of A and by UA the average of U over A. That is,

U A = ⨍ A U ⁢ ( x ) ⁢ 𝑑 x = 1 | A | ⁢ ∫ A U ⁢ ( x ) ⁢ 𝑑 x .

From the definition (3.1) of Aγ weights, we clearly have

(A.1) ( ⨍ B R ⁢ ( y ) w ⁢ 𝑑 x ) ⁢ ( ⨍ B R ⁢ ( y ) w - 1 / μ ⁢ 𝑑 x ) μ ≤ [ w ] μ + 1 for all μ > 0 .

A simple use of Hölder’s inequality also gives

(A.2) [ w δ ] γ ≤ [ w ] γ δ for all ⁢ δ ∈ ( 0 , 1 ) .

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

Lemma A.1.

Let u,U:Ω→Rm be vector-valued functions with u∈C1⁢(Ω), U∈C2⁢(Ω), and let Φ:Rm→R be a C1 function. Suppose that either U or Φ2⁢(u)⁢∂⁡U∂⁡ν vanish on the boundary ∂⁡Ω of Ω. We set

(A.3) I 1 := ∫ Ω Φ 2 ⁢ ( u ) ⁢ | D ⁢ U | 2 ⁢ p + 2 ⁢ 𝑑 x , I ^ 1 := ∫ Ω Φ 2 ⁢ ( u ) ⁢ | D ⁢ u | 2 ⁢ p + 2 ⁢ 𝑑 x ,

(A.4) I ¯ 1 := ∫ Ω | Φ u ( u ) | 2 ( | D U | 2 ⁢ p + 2 + | D u | 2 ⁢ p + 2 ) d x ,

(A.5) I 2 := ∫ Ω Φ 2 ⁢ ( u ) ⁢ | D ⁢ U | 2 ⁢ p - 2 ⁢ | D 2 ⁢ U | 2 ⁢ 𝑑 x .

Suppose that the following holds:

Φα⁢(u) belongs to the Aβ+1 class for some α>2p+2 and β<pp+2.

Then, for any ε>0, there exists a constant Cε,Φ depending on ε and [Φα⁢(u)]β+1 such that

(A.6) I 1 ≤ ε ⁢ I ^ 1 + C ε , Φ ⁢ ∥ U ∥ BMO ⁢ ( Ω ) 2 ⁢ [ I ¯ 1 + I 2 ] .

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 Hardy–Littlewood maximal operator acting on function F∈Lloc1⁢(Ω):

M ( ( F ) ( y ) = sup ε { ∫ B ε ⁢ ( y ) F ( x ) d x : ε > 0 and B ε ( y ) ⊂ Ω } .

We also recall the Hardy–Littlewood theorem: For any F∈Lq⁢(Ω), we have

(A.7) ∫ Ω M ⁢ ( F ) q ⁢ 𝑑 x ≤ C ⁢ ( q ) ⁢ ∫ Ω F q ⁢ 𝑑 x , q > 1 .

We also make use of the Hardy space ℋ1. For any y∈Ω and ε>0, let ϕ be a function in C0∞⁢(B1⁢(y)) with |Dϕ|≤C1. Let ϕε⁢(x)=ε-n⁢ϕ⁢(xε) (then |Dϕε|≤C1ε-1-n). From [22], a function g is in ℋ1⁢(Ω) if

sup ε > 0 g * ϕ ε ∈ L 1 ( Ω ) and ∥ g ∥ ℋ 1 = ∥ g ∥ L 1 ⁢ ( Ω ) + ∥ sup ε > 0 g * ϕ ε ∥ L 1 ⁢ ( Ω ) .

We are now ready to give the proof of Lemma A.1.

Proof of Lemma A.1.

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

(A.8) I 1 = ∫ Ω Φ 2 ⁢ ( u ) ⁢ | D ⁢ U | 2 ⁢ p + 2 ⁢ 𝑑 x = - ∫ Ω U ⁢ div ⁡ ( Φ 2 ⁢ ( u ) ⁢ | D ⁢ U | 2 ⁢ p ⁢ D ⁢ U ) ⁢ 𝑑 x .

We will show that g=div⁡(Φ2⁢(u)⁢|D⁢U|2⁢p⁢D⁢U) belongs to the Hardy space ℋ1 by showing that there exists a constant C such that

(A.9) ∫ Ω sup ε | g * ϕ ε | d x ≤ C [ I ¯ 1 1 / 2 + I 2 1 / 2 ] I 1 1 / 2 + C ( [ Φ α ( u ) ] β + 1 ) [ I ¯ 1 1 / 2 ( I 1 1 / 2 + I ^ 1 1 / 2 ) + I 1 1 / 2 I 2 1 / 2 ] ,

(A.10) ∥ g ∥ L 1 ⁢ ( Ω ) ≤ C ⁢ [ I ¯ 1 1 / 2 + I 2 1 / 2 ] ⁢ I 1 1 / 2 + C ⁢ [ I ¯ 1 1 / 2 ⁢ ( I 1 1 / 2 + I ^ 1 1 / 2 ) + I 1 1 / 2 ⁢ I 2 1 / 2 ] .

Once this is established, (A.8) and the Fefferman–Stein theorem on the duality of the BMO and Hardy spaces yield I1≤∥U∥BMO⁢∥g∥ℋ1 (see [22]), and so

I 1 ≤ C ⁢ ( [ Φ α ⁢ ( u ) ] β + 1 ) ⁢ ∥ U ∥ BMO ⁢ [ I 1 1 / 2 ⁢ ( I ¯ 1 1 / 2 + I 2 1 / 2 ) + I ¯ 1 1 / 2 ⁢ ( I 1 1 / 2 + I ^ 1 1 / 2 ) + I 1 1 / 2 ⁢ I 2 1 / 2 ] .

A simple use of Young’s inequality to the right-hand side then gives (A.6).

Therefore, in the rest of the proof we need only to establish (A.9). We then write g=g1+g2 with gi=div⁡Vi, where

V 1 = Φ ⁢ ( u ) ⁢ | D ⁢ U | p + 1 ⁢ ( Φ ⁢ ( u ) ⁢ | D ⁢ U | p - 1 ⁢ D ⁢ U - ⨍ B ε Φ ⁢ ( u ) ⁢ | D ⁢ U | p - 1 ⁢ D ⁢ U ⁢ 𝑑 x )

and

V 2 = Φ ⁢ ( u ) ⁢ | D ⁢ U | p + 1 ⁢ ⨍ B ε Φ ⁢ ( u ) ⁢ | D ⁢ U | p - 1 ⁢ D ⁢ U ⁢ 𝑑 x .

Let us consider g1 first. We define h=Φ⁢(u)⁢|D⁢U|p-1⁢D⁢U. For any y∈Ω 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

| g 1 * ϕ ε ( y ) | = | ∫ B ε ⁢ ( y ) D ϕ ( x - y ε ) ( h - h B ε ⁢ ( y ) ) Φ ( u ) | D U | p + 1 d x |

≤ C 1 ε | ⨍ B ε ⁢ ( y ) | h - h B ε ⁢ ( y ) | Φ ( u ) | D U | p + 1 d x |

(A.11) ≤ C 1 ε ( ⨍ B ε ⁢ ( y ) | h - h B ε ⁢ ( y ) | s d x ) 1 / s ( ⨍ B ε ⁢ ( y ) Φ s ′ ( u ) | D U | ( p + 1 ) ⁢ s ′ d x ) 1 / s ′ .

There exists a constant C such that |Dh|≤|Φu(u)||Du||DU|p+pΦ|DU|p-1|D2U|. The Poincaré–Sobolev inequality, with s*=n⁢sn+s, then gives

C 1 ε ( ⨍ B ε | h - h B ε | s d x ) 1 / s ≤ C ( ⨍ B ε | D h | s * d x ) 1 / s *

≤ C [ ⨍ B ε | Φ u ( u ) | s * | D u | s * | D U | p ⁢ s * d x + ⨍ B ε Φ s * | D U | ( p - 1 ) ⁢ s * | D 2 U | s * d x ] 1 / s * .

Using the above estimate in (A.11), we get

(A.12) sup ϕ ε | ∫ Ω ϕ ε g 1 d x | ≤ C [ Ψ 1 ( y ) + Ψ 2 ( y ) ] Ψ 3 ( y ) ,

where Ψi⁢(y)=(M⁢(Fiqi⁢(y)))1/qi with q1=q2=s*, q3=s′ and

F 1 = Φ u ⁢ ( u ) ⁢ | D ⁢ u | ⁢ | D ⁢ U | p , F 2 = Φ ⁢ ( u ) ⁢ | D ⁢ U | p - 1 ⁢ | D 2 ⁢ U | , F 3 = Φ ⁢ ( u ) ⁢ | D ⁢ U | p + 1 .

Take s=2⁢nn-1, then s*=s′=2⁢nn+1. We see that qi<2. Hence, by (A.7),

( ∫ Ω Ψ i 2 ⁢ 𝑑 x ) 1 / 2 = ( ∫ Ω M ⁢ ( F i q * ) 2 / q * ⁢ 𝑑 x ) 1 / 2 ≤ ( ∫ Ω F i 2 ⁢ 𝑑 x ) 1 / 2 .

Therefore, by Holder’s inequality, the above estimates and the notations (A.3) and (A.5), we get

∫ Ω sup ε | g 1 * ϕ ε | d x ≤ C [ I 1 1 / 2 I ¯ 1 1 / 2 + I 1 1 / 2 I 2 1 / 2 ] .

We consider g2 and note that |divV2|≤C(J1+J2) for some constant C and

J 1 := | Φ u ⁢ ( u ) | ⁢ | D ⁢ u | ⁢ | D ⁢ U | p + 1 ⁢ J 3 , J 2 := Φ ⁢ ( u ) ⁢ | D ⁢ u | p ⁢ | D 2 ⁢ U | ⁢ J 3 ,

with

J 3 ⁢ ( y ) := | ⨍ B ε ⁢ ( y ) Φ ⁢ ( u ) ⁢ | D ⁢ U | p ⁢ 𝑑 x | .

In the sequel, for any r>1p+1, we define r*=1-1r⁢(p+1), f=Φ⁢(u)⁢|D⁢U|p+1 and f^=Φ⁢(u)⁢|D⁢u|p+1.

We consider J3. If r1>1p+1, we use Hölder’s inequality to have

| ⨍ B ε Φ 1 / ( p + 1 ) ⁢ Φ p / ( p + 1 ) ⁢ | D ⁢ U | p ⁢ 𝑑 x | ≤ ( ⨍ B ε ( Φ 1 / ( p + 1 ) ) 1 / r 1 * ⁢ 𝑑 x ) r 1 * ⁢ ( ⨍ B ε f p ⁢ r 1 ⁢ 𝑑 x ) 1 / [ r 1 ⁢ ( p + 1 ) ] .

This gives the following estimate for J3:

(A.13) J 3 ≤ ( ⨍ B ε Φ ⁢ ( u ) 1 / [ r 1 * ⁢ ( p + 1 ) ] ⁢ 𝑑 x ) r 1 * ⁢ ( ⨍ B ε f p ⁢ r 1 ⁢ 𝑑 x ) 1 / [ r 1 ⁢ ( p + 1 ) ] .

For J1, we write J1=K⁢L⁢J3, with K=|Φu(u)||DU|p+1| and L=|D⁢u|. We have

| ϕ ε * J 1 | ≤ ( ⨍ B ε K s ′ ⁢ 𝑑 x ) 1 / s ′ ⁢ ( ⨍ B ε L s ⁢ 𝑑 x ) 1 / s ⁢ J 3 .

We write Ls=Φ-s/(p+1)⁢Φs/(p+1)⁢|D⁢u|s and use Hölder’s inequality to have, for any r>1p+1,

( ⨍ B ε L s ⁢ 𝑑 x ) 1 / s ≤ ( ⨍ B ε Φ - s / [ r * ⁢ ( p + 1 ) ] ⁢ 𝑑 x ) r * / s ⁢ ( ⨍ B ε f ^ s ⁢ r ⁢ 𝑑 x ) 1 / [ r ⁢ s ⁢ ( p + 1 ) ] .

Combining these estimates with (A.13), we have

(A.14) sup ε | ϕ ε * J 1 | ≤ C 1 M ( K s ′ ) 1 / s ′ M ( f ^ s ⁢ r ) 1 / [ r ⁢ s ⁢ ( p + 1 ) ] M ( f p ⁢ r 1 ) 1 / [ r 1 ⁢ ( p + 1 ) ] ,

where

C 1 = ( ⨍ B ε Φ 1 / [ r 1 * ⁢ ( p + 1 ) ] ⁢ 𝑑 x ) r 1 * ⁢ ( ⨍ B ε Φ - s / [ r * ⁢ ( p + 1 ) ] ⁢ 𝑑 x ) r * / s .

We rewrite

C 1 = [ sup ε ( ⨍ B ε Φ 1 / [ r 1 * ⁢ ( p + 1 ) ] d x ) ( ⨍ B ε Φ - s / [ r * ⁢ ( p + 1 ) ] d x ) r * / ( r 1 * ⁢ s ) ] r 1 * .

We now choose s,r,r1 such that s′=s⁢r=p⁢r1 and s⁢r<2. In this case if r<1 and s=r+1r, then s′=r+1 and r1=r+1p>1p+1. Let

α ⁢ ( r ) = 1 r 1 * ⁢ ( p + 1 ) and β ⁢ ( r ) = r * r 1 * ⁢ s .

We see that

α ⁢ ( r ) β ⁢ ( r ) = s r * ⁢ ( p + 1 ) .

By the definition of weights in (3.1), w=Φα⁢(r) and (A.1) with μ=β⁢(r), it is clear that

C 1 ≤ C ⁢ [ Φ α ⁢ ( r ) ] β ⁢ ( r ) + 1 r 1 * .

We see that

α ⁢ ( r ) = r + 1 r ⁢ p + r + 1 and β ⁢ ( r ) = r ⁢ ( p + 1 ) - 1 r ⁢ ( p + 1 ) + 1 .

Clearly, α⁢(r) decreases to 2p+2 and β⁢(r) increases to pp+2 as r→1-. Thus, if α>2p+2 and β<pp+2, then for r close to 1, we have α⁢(r)<α and β⁢(r)>β, and so [Φα⁢(r)]β⁢(r)+1≤C⁢([Φα]β+1), see (A.2). Hence, for r<1 and r being near 1, from (A.14) and the above estimates, we obtain

sup ε | ϕ ε * J 1 | ≤ C ( [ Φ α ] β + 1 ) M ( K s ⁢ r ) 1 / ( r ⁢ s ) M ( f ^ s ⁢ r ) 1 / [ r ⁢ s ⁢ ( p + 1 ) ] M ( f s ⁢ r ) p / [ r ⁢ s ⁢ ( p + 1 ) ] .

Integrating the above over Ω and applying Hölder’s inequality and then (A.7) (because r⁢s<2) to the right-hand side, we obtain

∫ Ω sup ε | ϕ ε * J 1 | d x ≤ C ( [ Φ α ] β + 1 ) ∥ K ∥ L 2 ⁢ ( Ω ) ∥ f ^ ∥ L 2 ⁢ ( p + 1 ) ⁢ ( Ω ) ∥ f ∥ 2 p / ( p + 1 ) .

Using Young’s inequality for the last two factors and the notations (A.3) and (A.4), we obtain

∫ Ω sup ε | ϕ ε * J 1 | d x ≤ C ( [ Φ α ] β + 1 ) I ¯ 1 1 / 2 ( I ^ 1 1 / 2 + I 1 1 / 2 ) .

Next, we write J2=Φ⁢|D⁢U|p-1⁢|D2⁢U|⁢|D⁢U|⁢J3=K⁢L⁢J3 with K=Φ⁢|D⁢U|p-1⁢|D2⁢U| and L=|D⁢U|.

We repeat the same argument, and estimate (A.14) now reads

sup ε | ϕ ε * J 2 | ≤ C 2 M ( K s ′ ) 1 / s ′ M ( f s ⁢ r ) 1 / [ r ⁢ s ⁢ ( p + 1 ) ] M ( f p ⁢ r 1 ) 1 / [ r 1 ⁢ ( p + 1 ) ] ,

where C2 also satisfies C2≤C⁢[Φα]β+1r1*. We then use the previous arguments, starting from (A.14), to have

∫ Ω sup ε | ϕ ε * J 2 | d x ≤ C ( [ Φ α ] β + 1 ) I 2 1 / 2 I 1 1 / 2 .

Combining the estimates, we obtain

∫ Ω sup ε | g * ϕ ε | d x ≤ C [ I ¯ 1 1 / 2 + I 2 1 / 2 ] I 1 1 / 2 + C ( [ Φ α ( u ) ] β + 1 ) [ I ¯ 1 1 / 2 ( I 1 1 / 2 + I ^ 1 1 / 2 ) + I 1 1 / 2 I 2 1 / 2 ] .

This proves (A.10). It is not difficult to establish (A.10) estimating ∥g∥L1⁢(Ω). We simply use Hölder inequality in a similar way in treating J1,J2, and replacing J3 by Φ⁢(u)⁢|D⁢U|p. We leave the details to the readers. The proof is then complete. ∎

We now give the proof of Lemma 3.1. 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. Recall the following notations Ωt=Ω∩Bt and

I 1 ⁢ ( t ) := ∫ Ω t Φ 2 ⁢ ( u ) ⁢ | D ⁢ U | 2 ⁢ p + 2 ⁢ 𝑑 x , ⁢ I ^ 1 ⁢ ( t ) := ∫ Ω t Φ 2 ⁢ ( u ) ⁢ | D ⁢ u | 2 ⁢ p + 2 ⁢ 𝑑 x ,

I ¯ 1 ( t ) := ∫ Ω t | Φ u ( u ) | 2 ( | D U | 2 ⁢ p + 2 + | D u | 2 ⁢ p + 2 ) d x , ⁢ I 2 ⁢ ( t ) := ∫ Ω t Φ 2 ⁢ ( u ) ⁢ | D ⁢ U | 2 ⁢ p - 2 ⁢ | D 2 ⁢ U | 2 ⁢ 𝑑 x .

Proof of Lemma 3.1.

We revisit the proof of Lemma A.1. By integrating by parts and noting that ψ=0 on ∂⁡Ω, we have

∫ Ω Φ 2 ⁢ ( u ) ⁢ ψ 2 ⁢ | D ⁢ U | 2 ⁢ p + 2 ⁢ 𝑑 x = - ∫ Ω U ⁢ div ⁡ ( Φ 2 ⁢ ( u ) ⁢ ψ 2 ⁢ | D ⁢ U | 2 ⁢ p ⁢ D ⁢ U ) ⁢ 𝑑 x .

Again, we will show that g=div⁡(Φ2⁢ψ2⁢|D⁢U|2⁢p⁢D⁢U) belongs to the Hardy space ℋ1. We write g=g1+g2 with gi=div⁡Vi, where

V 1 = Φ ⁢ ( u ) ⁢ ψ ⁢ | D ⁢ U | p + 1 ⁢ ( Φ ⁢ ( u ) ⁢ ψ ⁢ | D ⁢ U | p - 1 ⁢ D ⁢ U - ⨍ Ω ε Φ ⁢ ( u ) ⁢ ψ ⁢ | D ⁢ U | p - 1 ⁢ D ⁢ U ⁢ 𝑑 x ) ,

V 2 = Φ ⁢ ( u ) ⁢ ψ ⁢ | D ⁢ U | p + 1 ⁢ ⨍ Ω ε Φ ⁢ ( u ) ⁢ ψ ⁢ | D ⁢ U | p - 1 ⁢ D ⁢ U ⁢ 𝑑 x .

In estimating V1, we follow the proof of Lemma A.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 (A.11), we have an extra term, which can be estimated as follows:

( ⨍ Ω ε Φ s * ( u ) | D ψ | s * | D U | p ⁢ s * d x ) 1 / s * ≤ sup x ∈ B t | D ψ | ( ⨍ Ω ε Φ s * ( u ) | D U | p ⁢ s * d x ) 1 / s * .

Accordingly, in the right-hand side of (A.12), we have the following term:

sup | D ψ | Ψ 3 M ( Φ s * ( u ) | D U | p ⁢ s * ) 1 / s * .

Using Hölder’s inequality, we have

∫ Ω t Ψ 3 ⁢ M ⁢ ( Φ s * ⁢ ( u ) ⁢ | D ⁢ U | p ⁢ s * ) 1 / s * ⁢ 𝑑 x ≤ I 1 1 / 2 ⁢ ( ∫ Ω t M ⁢ ( Φ s * ⁢ ( u ) ⁢ | D ⁢ U | p ⁢ s * ) 2 / s * ⁢ 𝑑 x ) 1 / 2 .

The last integral can be bounded via (A.7) by

(A.15) I * := ∫ Ω t Φ 2 ⁢ ( u ) ⁢ | D ⁢ U | 2 ⁢ p ⁢ 𝑑 x .

Using the fact that |ψ|≤1 and Ω=Bt, the proof can continue and give

(A.16) ∫ Ω t sup ε | g 1 * ϕ ε | d x ≤ C [ I 1 1 / 2 I ¯ 1 1 / 2 + I 1 1 / 2 I 2 1 / 2 + sup B t | D ψ | I 1 1 / 2 I * 1 / 2 ] .

Similarly, in considering g2=div⁢V2, we will have the following extra term in the definition of J1:

| D ⁢ ψ | ⁢ Φ ⁢ ( u ) ⁢ | D ⁢ U | p + 1 ⁢ ⨍ Ω ε Φ ⁢ ( u ) ⁢ | D ⁢ U | p ⁢ 𝑑 x ,

which can be estimated by

sup B t | D ψ | M ( Φ ( u ) | D U | p + 1 ) M ( Φ ( u ) | D U | p ) .

Again, using Hölder’s inequality and the Hardy–Littlewood inequality (A.7), the integral over Bt of this quantity is bounded by supBt|Dψ|I11/2I*1/2.

Therefore, (A.16) holds true with g1 being replaced by g2. Combining these estimates for g1,g2, we get

∫ Ω t sup ε | g * ϕ ε | d x ≤ sup B t | D ψ | I 1 1 / 2 I * 1 / 2 + C Φ [ I ¯ 1 1 / 2 ( I 1 1 / 2 + I ^ 1 1 / 2 ) + I 1 1 / 2 I 2 1 / 2 ] .

The above gives an estimate for the ℋ1 norm of g. By the Fefferman–Stein duality theorem again, we obtain

∫ Ω t Φ 2 ⁢ ( u ) ⁢ ψ 2 ⁢ | D ⁢ U | 2 ⁢ p + 2 ⁢ 𝑑 x ≤ ∥ U ∥ BMO ⁢ ( Ω t ) ⁢ ∥ g ∥ ℋ 1

≤ ∥ U ∥ BMO ⁢ ( Ω t ) ( sup B t | D ψ | I 1 1 / 2 I * 1 / 2 + C Φ [ I ¯ 1 1 / 2 ( I 1 1 / 2 + I ^ 1 1 / 2 ) + I 1 1 / 2 I 2 1 / 2 ] ) .

A simple use Young’s inequality, the definition of I* given in (A.15) and then the fact that ψ=1 in Bs give (3.2). The proof is complete. ∎

Acknowledgements

The author is grateful to the anonymous referees for their careful reading of the first draft of this paper, corrections and suggestions.

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Received: 2017-02-22

Revised: 2017-03-03

Accepted: 2017-03-03

Published Online: 2017-04-21

Published in Print: 2018-02-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ans-2017-0013

Keywords for this article

Cross Diffusion Systems; Hölder Regularity; Global Existence

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