On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications

DOI:10.1051/cocv:2007049 www.esaim-cocv.org

ON LIPSCHITZ TRUNCATIONS OF SOBOLEV FUNCTIONS

(WITH VARIABLE EXPONENT) AND THEIR SELECTED APPLICATIONS

Lars Diening

, Josef M´ alek

and Mark Steinhauer

Abstract. We study properties of Lipschitz truncations of Sobolev functions with constant and vari- able exponent. As non-trivial applications we use the Lipschitz truncations to provide a simplified proof of an existence result for incompressible power-law like fluids presented in [Frehseet al., SIAM J.

Math. Anal. 34(2003) 1064–1083]. We also establish new existence results to a class of incompressible electro-rheological fluids.

Mathematics Subject Classification.35J55, 35J65, 35J70, 35Q35, 76D99 Received April 14, 2006. Revised August 25, 2006.

Published online September 21, 2007.

1. Introduction

Letλbe a large positive number,p≥1. Sobolev-functions fromW₀^1,p can be approximated by λ-Lipschitz functions that coincide with the originals up to sets of small Lebesgue measure. The Lebesgue measure of these non-coincidence sets is bounded by the Lebesgue measure of the sets where the Hardy-Littlewood maximal function of the gradients are aboveλ. See for example [3,19,23,26,29,38].

Lipschitz truncations of Sobolev functions are used in various areas of analysis in different aspects. To name a few, we refer to the articles with applications in the calculus of variations [1,19,20,27,35,36], in the existence theory of partial differential equations [13,18,23,34,37] and in the regularity theory [2,14].

The purpose of this article is four-fold. First of all, in Section2 we recall, survey, and strengthen properties ofW₀^1,∞-truncations ofW₀^1,p-functions that are useful from the point of view of the existence theory concerning nonlinear PDE’s. We illustrate the potential of this tool by establishing the weak stability for the system of p-Laplace equations with very general right-hand sides.

Then, in Section3 we exploit Lipschitz truncations in the analysis of steady flows of generalized power-law fluids. In this case we reprove in a simplified way the existence results established in [18].

Keywords and phrases.Lipschitz truncation ofW₀^1,p/W₀^1,p(·)-functions, existence, weak solution, incompressible fluid, power-law fluid, electro-rheological fluid.

1 Abteilung f¨ur Angewandte Mathematik, Universit¨at Freiburg, Eckerstr. 1, 79104 Freiburg i. Br., Germany;

diening@mathematik.uni-freiburg.de

2 Mathematical Institute, Charles University, Sokolovsk´a 83, 18675 Prague 8, Czech Republic;malek@karlin.mff.cuni.cz Supported by the Czech Science Foundation, the project GA ˇCR 201/03/0934, and by MSMT, the project MSM 0021620839.

3 Mathematical Seminar, University of Bonn, Nussallee 15, 53115 Bonn, Germany;MSteinh@uni-bonn.de Supported by SFB 611 at the University of Bonn.

Article published by EDP Sciences c EDP Sciences, SMAI 2007

Next, in order to apply this method to a class of electro-rheological fluids which are characterized by power- law index varying with the spatial variables we extend the Lipschitz truncation method to Sobolev functions of variable exponents W^1,p(·). The properties of Lipschitz truncations are presented in Section4.

Finally, we establish new existence results to an electro-rheological fluid model in Section5.

We wish to mention that our main interest in investigating properties of Lipschitz truncations of Sobolev functions comes from studies of equations describing flows of certain incompressible fluids. In order to explain how the properties of Lipschitz truncations can be used in the analysis of nonlinear partial differential equations to those readers who are not familiar with (or not interested in) analysis of generalized incompressible Navier- Stokes equations we decided to consider first the following problem: for a given vector field F= (F₁, . . . , Fd), to findv= (v₁, . . . , vd) solving¹

−div

|Dv|^p−2Dv

=F in Ω⊂R^d,

v=0 on∂Ω. (1.1)

Here Ω is a bounded domain with Lipschitz boundary, p >1 and Dv denotes either the gradient of v or its symmetric part.

Ifp= 2, (1.1) represents a non-linear problem. A key issue in the proof of the existence of a weak solution to (1.1) is the stability of weak solutions with respect to weak convergence. This property, calledweak stability of (1.1), can be made more precise in the following way: assume that we havevⁿ enjoying the properties

Ω

|Dvⁿ|^p−2Dvⁿ·Dϕdx=Fⁿ, ϕ for all suitableϕ, (1.2) and

Ω

|Dvⁿ|^pdx≤K <∞ for alln∈N, Fⁿ, ϕ → F, ϕ for all suitableϕ.

(1.3)

The uniform estimate (1.3)₁ implies (modulo a suitably taken subsequence) that

vⁿ v weakly inW₀^1,p(Ω)^d. (1.4)

Ifvis also a weak solution to (1.1) then we say that system (1.1) posseses the weak stability property.

SettingT(B) :=|B|^p−2B(p=_p−1^p ), we can reformulate our task differently. Noticing that forp=_p−1^p the uniform bound (1.3)₁ implies that

Ω

|T(Dvⁿ)|^pdx≤c(K), (1.5)

we conclude thatT(Dvⁿ) χweakly inL^p(Ω)^d×d (at least for a subsequence). The weak stability of (1.1) is thus tantamount to show thatT(Dv) =χ.

To provide an affirmative answer to the issue of stability of weak solutions, it is enough to show that for a not relabeled subsequence

lim sup

k→∞

Ω

T(Dvⁿ)−T(Dv)

·D(vⁿ−v) dx= 0. (1.6)

1In (1.1) we could replace thep-Laplace operator by anyp-coercive, strictly monotone operator of (p−1)-growth.

Indeed, knowing thatTis strictly monotone,i.e.,

(T(ζ)−T(z))·(ζ−z)>0 for allζ,z∈R^d×d (ζ=z), one concludes from (1.6) that

Dvⁿ →Dv almost everywhere in Ω, (1.7)

at least for a not relabeled subsequence. Vitali’s theorem then completes the proof allowing to pass to the limit in the nonlinear term.

Note that (1.6) can be weakened, still giving (1.7), see also [4]. More precisely, to obtain (1.7) it is enough to show for some 0< θ≤1 that there is a not relabelled subsequence of {vⁿ} such that

lim sup

n→∞

Ω

T(Dvⁿ)−T(Dv)

·D(vⁿ−v)θ

dx= 0. (1.8)

We distinguish two cases how to achieve (1.6), or (1.8) respectively.

Simple case. The problem is simply solvable if we assume thatFⁿ,F∈(W₀^1,p(Ω)^d)^∗andFⁿ→Fstrongly in (W₀^1,p(Ω)^d)^∗. In fact, to obtain (1.6), it is natural to take ϕ = vⁿ −v in (1.2), which is a suitable test function (all terms are meaningful). Then we obtain, after subtracting the term

ΩT(Dv)·D(vⁿ−v) dxfrom both sides of the equation

Ω

T(Dvⁿ)−T(Dv)

·D(vⁿ−v) dx=Fⁿ,vⁿ−v −

Ω

T(Dv)·D(vⁿ−v) dx.

For n→ ∞, the right-hand side vanishes due to weak convergence of {vⁿ} and strong convergence of{Fⁿ}, and (1.6) follows.

Difficult case. More difficult and also more interesting is the case when

Fⁿ= divGⁿ withGⁿ→G strongly ∈L¹(Ω)^d×d. (1.9) Thenuⁿ:=vⁿ−vis not anymore a suitable test function sincedivG,uⁿor−G,∇uⁿdo not have a clear meaning. However, we can replaceuⁿ by its Lipschitz truncation and conjecture that uniform smallness of the integrand on the sets where the Lipschitz truncation differs fromuⁿ can lead to (1.8). Note thatFⁿ = divGⁿ with {Gⁿ} bounded in L¹(Ω)^d×d is not sufficient for the estimate (1.3)₁. However, in our applications in Theorems3.1and5.1the right hand side will have additional structure (due to the incompressibility constraint involved in the problem) to ensure the validity of (1.3)₁.

To proceed further, we need to study carefully the properties of Lipschitz truncations of Sobolev functions.

This is the subject of the next section, where we also complete the proof of the weak stability of (1.1) in the difficult case.

2. Lipschitz truncations of standard Sobolev functions

LetZ⊂R^d. ThenZ denotesR^d\Z and|Z|denotes thed-dimensional Lebesgue measure ofZ.

Assumption 2.1. We assume that Ω⊂R^d is an open bounded set with the property: there exists a constant A₁≥1 such that for allx∈Ω

|B_{2 dist(}x,Ω)(x)| ≤A₁|B_{2 dist(}x,Ω)(x)∩Ω|. (2.1) Remark 2.2. If Ω⊂R^d is an open bounded set with Lipschitz boundary then Ω satisfies Assumption2.1.

For anyp∈[1,∞), we use standard notation for the Lebesgue spaces (Lp(Ω), · p) and the Sobolev spaces (W₀^1,p(Ω),·1,p), being the completions of smooth, compactly supported functions w.r.t. the relevant norms. If X is a Banach space of scalar functions thenX^dandX^d×dstand for the spaces of vector-valued or tensor-valued functions whose components belong toX.

Forf ∈L¹(R^d), we define the Hardy-Littlewood maximal function as usual through (M f)(x) := sup

r>0

1

|Br(x)|

B_r(x)

|f(y)|dy=: sup

r>0fB_r(x).

Similarly, foru∈W^1,1(R^d) we defineM(∇u) :=M(|∇u|) and foru∈W^1,1(R^d)^d we setM(Du) :=M(|Du|).

Theorem 2.3. Let Ω⊂ R^d satisfy Assumption 2.1. Let v ∈ W₀^1,1(Ω)^d. Then for every θ, λ > 0 there exist truncations vθ,λ∈W₀^1,∞(Ω)^d such that

||vθ,λ||∞≤θ, (2.2)

||∇vθ,λ||∞≤c₁A₁λ, (2.3)

wherec₁>0 does only depend on the dimension d. Moreover, up to a nullset (a set of Lebesgue measure zero) {vθ,λ=v} ⊂Ω∩

{Mv> θ} ∪ {M(∇v)> λ}

. (2.4)

Theorem 2.3 summarizes the facts established earlier in original papers [3] or [23], and presented in the monograph [26], among others. Since Theorem2.3serves as a basic stone in proving Theorem2.5(for standard Sobolev functions) and Theorem 4.4(for functions from the Sobolev space with variable exponent), we give a proof of Theorem2.3here for the sake of completness. Before doing so we recall the following extension theorem ([17], p. 201 or also [15], p. 80 and [26], p. 40 for the scalar case).

Lemma 2.4. Let v:E →R^m, defined on a nonempty setE ⊂R^d, be such that for certainλ >0 and θ >0 and for all x, y∈E

|v(y)−v(x)|R^m ≤λ|y−x|_R^d and |v(x)|R^m ≤θ . (2.5) Then there is an extension vθ,λ:R^d→R^mfulfilling (2.5)for allx, y ∈R^d, and vθ,λ=v onE.

Let us return to the proof of Theorem2.3.

Proof of Theorem 2.3. We first extendv by zero outside of Ω and obtainv∈W₀^1,1(R^d)^d.

The following facts are provede.g. in [26]: for a functionh∈W₀^1,1(R^d) letL(h) be the set of its Lebesgue points. Then|L(h)|= 0, and for all ballsBr(x₀)⊂R^d and for allξ, ζ ∈ L(h)∩Br(x₀) it holds

|h(ξ)− hB_r(x₀)| ≤c r M(∇h)(ξ),

|h(ζ)− hB_r(x₀)| ≤c r M(∇h)(ζ), (2.6) which implies that

|h(ξ)−h(ζ)| ≤c r

M(∇h)(ξ) +M(∇h)(ζ) .

Then for anyx, y ∈ L(h) we takex₀=x,r= 2|y−x|,ξ=xandζ=y in the above inequality and obtain

|h(x)−h(y)| ≤c|x−y|

M(∇h)(x) +M(∇h)(y)

. (2.7)

Forλ >0 we define

Hθ,λ:=L(v) ∩ {Mv≤θ} ∩ {M(∇v)≤λ}.

Then it follows from (2.7) that for allx, y∈Hθ,λ

|v(x)−v(y)| ≤c λ|x−y| and|v(x)| ≤θ. (2.8) If Ω =R^d, the statements of Theorem2.3follow from Lemma2.4applied to E=Hθ,λ.

If Ω =R^d, we need to proceed more carefully in order to arrange that the Lipschitz truncations vanish on the boundary. Letx∈Hθ,λ ∩ Ω andr:= 2 dist(x,Ω). Then by Assumption 2.1and sincevis zero on Ω we have

−

B_r(x)

|v(z)− vB_r(x)|dz≥ 1

|Br(x)|

B_r(x)∩Ω|v(z)− vB_r(x)|dz

=|Br(x)∩Ω|

|Br(x)| |vB_r(x)|

≥ 1

A₁|vB_r(x)|.

(2.9)

By a variant of the Poincar´e inequality,e.g.in [26],

−

B_r(x)

|h(z)− hB_r(x)|dz≤c r−

B_r(x)

|∇h(z)|dz

we observe from (2.9) that forx∈Hθ,λ ∩ Ω

|vB_r(x)| ≤c A₁r −

B_r(x)

|∇v(z)|dz≤c A₁r M(∇v)(x)≤c A₁r λ.

Consequently, using also (2.6), we obtain

|v(x)| ≤c r M(∇v)(x) +|vB_r(x)| ≤c A₁r λ. (2.10)

It follows from (2.10) that for allx∈Hθ,λ ∩ Ω and ally∈Ω holds

|v(x)−v(y)|=|v(x)| ≤c A₁dist(x,Ω)λ≤c A₁|x−y|λ. (2.11)

Sincevis zero on Ωit follows from (2.8) and (2.11) that

|v(x)−v(y)| ≤c A₁|x−y|λ for allx, y∈Hθ,λ ∪ Ω. (2.12)

In other words, we have shown thatv is Lipschitz continuous onGθ,λ:= Hθ,λ ∪ Ω with Lipschitz constant bounded by c A₁λ. Since, Mv ≤ θ on Hθ,λ and v = 0 on Ω, we also have |v| ≤ θ on Gθ,λ. Therefore, applying Lemma 2.4toE=Gθ,λ there exists an extensionvθ,λ∈W^1,∞(R^d) ofv|G_θ,λ withv(x) =vθ,λ(x) for allx∈Gθ,λ,||∇vθ,λ||_∞≤c A₁λ, and||vθ,λ||_∞≤θ. This proves (2.2) and (2.3). Fromvθ,λ=0on Ω (since it is contained inGθ,λ) we conclude thatvθ,λ∈W₀^1,∞(Ω). Finally, (2.4) follows observing thatv=vθ,λonGθ,λ,

|L(v)|= 0, and

G_θ,λ= Ω∩H_θ,λ = Ω∩

L(v)∪ {Mv> θ} ∪ {M(∇v)> λ}

.

The proof of Theorem2.3is complete.

Theorem 2.5. Let 1 < p < ∞. Let Ω ⊂ R^d be a bounded domain which satisfies Assumption 2.1. Let uⁿ∈W₀^1,p(Ω)^d be such thatuⁿ0weakly in W₀^1,p(Ω)^d asn→ ∞. Set

K:= sup

n ||uⁿ||1,p<∞, (2.13)

γn:=||uⁿ||p→0 (n→ ∞). (2.14) Let θn>0 be such that (e.g. θn:=√

γn)

θn→0 and γn

θn

→0 (n→ ∞).

Let µj:= 2^2j. Then there exist a sequenceλn,j>0 with

µj≤λn,j≤µj+1, (2.15)

and a sequence u^n,j∈W₀^1,∞(Ω)^d such that for all j, n∈N

||u^n,j||∞≤θn →0 (n→ ∞), (2.16)

||∇u^n,j||∞≤c λn,j≤c µj+1. (2.17)

Moreover, up to a nullset

{u^n,j=uⁿ} ⊂Ω ∩

{Muⁿ> θn} ∪ {M(∇uⁿ)>2λn,j}

. (2.18)

For all j∈Nandn→ ∞

u^n,j →0 strongly inL^s(Ω)^d for all s∈[1,∞], (2.19) u^n,j 0 weakly inW₀^1,s(Ω)^d for alls∈[1,∞), (2.20)

∇u^{n,j ∗} 0 *-weakly inL^∞(Ω)^d. (2.21)

Furthermore, for alln, j∈N

||∇u^n,jχ_{un,j=uⁿ}||p≤c||λn,jχ_{un,j=uⁿ}||p≤cγn

θn

µj+1+c j, (2.22)

wherej :=K2^−j/p vanishes asj → ∞. The constantc depends onΩvia Assumption 2.1.

The assertions (2.16)–(2.21) summarize the properties of Lipschitz truncations established earlier in [3] and [23]. To our best knowledge, the estimate (2.22) seems to be new. More specifically, the Acerbi-Fusco approxi- mation lemma says, see [3], that|{u^n,λn,j =uⁿ}| ≤ ^C_λ^upnp1,p

n,j . Applying this estimate we obtain

∇u^n,λn,jχ_{{un,λn,j=un}}p≤λn,j|{u^n,λn,j =uⁿ}|^1/p≤Cuⁿ1,p≤K .

Thus one concludes just boundedness of the above term from the Acerbi-Fusco approximation lemma while (2.22) says that for suitable Lipschitz truncations this term can be so small as needed.

Proof of Theorem 2.5. First, observe that (2.13) and (2.14) are direct consequences ofuⁿ 0inW^1,p(Ω)^d and the compact embedding ofW₀^1,p(Ω) intoL^p(Ω).

Since 1< p <∞the Hardy-Littlewood maximal operatorM is continuous fromL^p(R^d) toL^p(R^d). This and (2.13) imply

sup

n

Ω

|Muⁿ|^pdx+ sup

n

Ω

|M(∇uⁿ)|^pdx≤c K^p. (2.23)

Next, we observe that forg∈L^p(R^d) with||g||p≤K we have K^p≥ ||g||^pp=

R^d|g(x)|^pdx=p

R^d

_∞

0

t^p−1χ_{|g|>t}dtdx

=p

R^d

m∈Z

₂^m+1

2^m

t^p−1χ_{|g|>t}dtdx

≥

R^d

m∈Z

2^mp

χ_{|g|>2^m+1}dx

≥

R^d

m∈N

2^mp

χ_{|g|>2^m+1}dx

=

j∈N 2^j+1−1

k=2^j

R^d

2^kp

χ_{|g|>2^k+1}dx.

(2.24)

The choiceg=M(∇uⁿ) implies

j∈N 2^j+1−1

k=2^j

R^d

2^kp

χ_{|M(∇uⁿ)|>2·2^k}dx≤K^p.

Especially, for allj, n∈N

2^j+1−1

k=2^j

R^d

2^kp

χ_{|M(∇uⁿ)|>2·2^k}dx≤K^p.

Since the sum contains 2^j summands, there is at least one indexkn,j such that

R^d

2^kn,jp

χ_{|M(∇un)|>2·2^kn,j}dx≤K^p2^−j. (2.25)

Defineλn,j := 2^kn,j andµj:= 2^2j. Then

µj= 2^2j ≤λn,j <2^2j+1=µj+1 (2.26) and we conclude from (2.25) that

R^d

λn,j

p

χ_{|M(∇uⁿ)|>2λ_n,j}dx≤K^p2^−j. (2.27)

Next, we notice that

(λn,j)^pχ_{Muⁿ>θ_n} ∪ {M(∇uⁿ)>2λ_n,j}dx

≤ λn,j

θn p

θ_n^pχ_{Muⁿ>θ_n}dx+

(λn,j)^pχ_{M(∇uⁿ)>2λ_n,j}dx

≤ λn,j

θn p

||Muⁿ||^pp+K^p2^−j.

≤c λn,j

θn p

||uⁿ||^pp+K^p2^−j.

=c

λn,jγn

θn p

+K^p2^−j.

(2.28)

For eachn, j∈Nwe apply Theorem2.3and set

u^n,j:= (uⁿ)θ_n,λ_n,j. Due to Theorem2.3(withθn and 2λn,j) we have for alln, j∈N

||u^n,j||_∞≤θn, (2.29)

||∇u^n,j||_∞≤2c₁A₁λn,j =:c λn,j ≤c µj+1 (2.30) and up to a nullset

{u^n,j=uⁿ} ⊂Ω ∩

{Muⁿ> θn} ∪ {M(∇uⁿ)>2λn,j}

. (2.31)

Using (2.28), (2.30), and (2.31) we observe

||∇u^n,jχ_{un,j=uⁿ}||^pp≤c||λn,jχ_{un,j=uⁿ}||^pp≤c

λn,jγn

θn p

+cK^p2^−j. (2.32) Taking thep-th root of (2.32) with the help of (2.26) we conclude (2.22).

SinceD(Ω) is dense inL^s(Ω) for alls ∈[1,∞) and (2.29) implies that

Ω

∇u^n,jϕdx=−

Ω

u^n,j∇ϕdx→0 asn→ ∞, for allϕ∈ D(Ω),

(2.20) and (2.21) follow fors∈(1,∞] using also (2.30). The cases= 1 then also follows.

We complete this section by proving the weak stability of (1.1) in the case when F = divG with G ∈ L¹(Ω)^d×d. It means that we have {vⁿ} such that (1.2), (1.3), (1.4), (1.5) and (1.9) hold and we want to prove (1.8). Recall that the choice ϕ = uⁿ, where uⁿ := vⁿ −v, is not admissible test function in (1.2).

Observing, however, that {uⁿ} fulfills the assumptions of Theorem2.5, its application leads to the sequence {u^n,j} possessing the properties (2.16)–(2.22); in particular, u^n,j ∈W₀^1,∞(Ω)^d is an admissible (suitable) test function. Insertingϕ=u^n,j into (1.2) we obtain

Ω

T(Dvⁿ)−T(Dv)

· Du^n,j

dx=−

Ω

(Gⁿ−G) +G+T(Dv)

·(Du^n,j) dx (2.33)

and the term at the right hand side vanishes asn→ ∞thanks to (2.21) and (1.9). Especially, we have

nlim→∞

Ω

T(Dvⁿ)−T(Dv),Du^n,j

dx= 0. (2.34)

We will show below in Lemma2.6that (2.34) or even the weaker condition (2.35) implies exactly condition (1.8) that remained to complete the weak stability of (1.1) in the case (1.9) (compare the discussion around (1.5)–(1.8) for details).

Lemma 2.6. LetΩandpbe as in Theorem2.5. Letvⁿ,v∈W₀^1,p(Ω)withvⁿ vinW₀^1,p(Ω). Letuⁿ:=vⁿ−v and letu^n,j be the approximations of uⁿ as in Theorem2.5. Assume that for all j∈N we have

nlim→∞

Ω

T(Dvⁿ)−T(Dv),Du^n,j

dx≤δj, (2.35)

wherelimj→∞δj = 0. Then for any0< θ <1

lim sup

n→∞

Ω

T(Dvⁿ)−T(Dv)

·(Dvⁿ−Dv) θ

dx= 0.

Proof. For allj∈N, (2.35) implies that

lim sup

n→∞ In:= lim sup

n→∞

{u^n,j=uⁿ}

T(Dvⁿ)−T(Dv)

· Duⁿ

dx

≤lim sup

n→∞

{u^n,j=uⁿ}

T(Dvⁿ)−T(Dv)

· Du^n,j

dx +δj

= lim sup

n→∞

Ω

T(Dvⁿ)−T(Dv)

· Du^n,j

χ_{un,j=uⁿ}dx +δj.

Note that sincevⁿ vin W₀^1,p(Ω), alsov satisfies (1.3)₁ and (1.5). Applying H¨older’s inequality to the last integral, and using (1.5) and (2.22) valid for all j∈Nwith ^γ_θⁿ

n →0 asn→ ∞, we obtain lim sup

n→∞ In ≤c(K) lim sup

n→∞ ∇u^n,jχ_{un,j=uⁿ}p+δj

≤c(K) lim sup

n→∞

cγn

θn

µj+1+c j+δj

≤c(K)j+δj, (2.36)

with µj, j as in Theorem2.5. Since the last estimate holds for all j ∈Nand limj→∞j = limj→∞δj = 0, we finally conclude from (2.36) that

lim sup

n→∞ In = 0. (2.37)

Then with H¨older’s inequality

Ω

T(Dvⁿ)−T(Dv)

·(Duⁿ) θ

dx

=

{uⁿ=u^n,j}

T(Dvⁿ)−T(Dv)

·(Duⁿ)dx

θ

|Ω|^1−θ

= +

{uⁿ=u^n,j}

T(Dvⁿ)−T(Dv)

·(Duⁿ)dx

θ {uⁿ=u^n,j}^1−θ

=:Yn,j,1+Yn,j,2,

wherej ∈Nis arbitrary. Since (T(Dvⁿ)−T(Dv))·(Duⁿ)≥0, we have Yn,j,1≤(In)^θ|Ω|^1−θ. And therefore with (2.37)

lim sup

n→∞ Yn,j,1= 0. (2.38)

On the other hand from (2.22),L^p(Ω)→L¹(Ω), andλn,j ≥1 we deduce lim sup

n→∞ |{uⁿ=u^n,j}|= lim sup

n→∞ ||χ_{uⁿ_=u^n,j_}||1

≤lim sup

n→∞ c λ⁻¹_n,j||λn,jχ_{un=u^n,j}||p

≤lim sup

n→∞ c ||λn,jχ_{un=u^n,j}||p

≤c j.

(2.39)

Now, H¨older’s inequality, (1.3)₁, (1.5), and (2.39) prove Yn,j,2≤c(K)

lim sup

n→∞ |{uⁿ=u^n,j}|₁₋θ

≤c(K) (j)^1−θ.

(2.40)

Sincej∈Nis arbitrary and limj→∞j= 0, we get from (2.40) and (2.38) lim sup

n→∞

Ω

T(Dvⁿ)−T(Dv)

·(Dvⁿ−Dv)θ

dx= 0.

This proves Lemma2.6.

3. An application: existence result for power-law fluids

We consider the following problem of nonlinear fluid mechanics. For Ω⊂R^dwith Lipschitz boundary∂Ω we look for (v,p) : Ω→R^d×R, representing the velocity and the pressure, satisfying

div(v⊗v)−div

T(Dv)

=−∇p +f, div v= 0 in Ω (3.1) and

v=0on∂Ω, (3.2)

where f : Ω → R^d is given, Dv denotes the symmetric part of the gradient of v, and T : R^d_sym^×d → R^d_sym^×d is a known continuous function having the following properties: for fixed p ∈ (1,∞) there are certain positive constantsC₁ andC₂ such that for allη∈R^d_sym^×d

T(η)·η≥C₁(|η|^p−1), (3.3)

|T(η)| ≤C₂(|η|+ 1)^p−1 (3.4) and for allη₁, η₂∈R^d_sym^×d

(T(η1)−T(η2))·(η₁−η₂)>0 if η₁=η₂. (3.5) System (3.1)–(3.2) describes steady flows of incompressible fluids exhibiting no-slip on the boundary. The fluid is non-Newtonian as its viscosity is not constant and depends on |Dv|, the quantity that reduces in a simple shear flow to the shear rate. A special class of such fluids with shear rate dependent viscosity are the power-law fluids for whichT, the Cauchy stress, takes the formT(η) =ν₀|η|^p−2η.

Our aim here is to reprove, in a simpler way, the result established in [18]. In [18] and in [25], the reader can find details related to mechanical and mathematical aspects of the considered system and related results dealing with an analysis of (3.1)–(3.2) as well.

Theorem 3.1. Let p > _d²₊₂^d ,d≥2. LetΩ⊂R^d be an open, bounded, connected set with Lipschitz boundary.

Assume thatf ∈(W₀^1,p(Ω)^d)^∗and (3.3)–(3.5)hold. Sets:= min{p, dp/(2(d−p)}ifp < dands:=p otherwise.

Then there exists a weak solution (v,p) to (3.1)–(3.2)such that

v∈W₀^1,p(Ω)^d and p∈L^s(Ω), (3.6)

div v= 0 a.e. inΩ and

Ω

p dx= 0, (3.7)

(T(Dv),Dϕ) = (v⊗v,Dϕ) + (p,divϕ) +f, ϕ for all ϕ∈W₀^1,∞(Ω)^d. (3.8) Proof. Let us for a fixedp∈(_d²₊₂^d , d) and q= _p²₋₁^p = 2p consider vⁿ ∈W₀^1,p(Ω)∩L^q(Ω) satisfying divvⁿ = 0 a.e. in Ω and

(T(Dvⁿ),Dϕ) + 1

n(|vⁿ|^q−2vⁿ, ϕ) =f, ϕ+ (vⁿ⊗vⁿ,Dϕ)

for allϕ∈W₀^1,p(Ω)^d∩L^q(Ω)^d, divϕ= 0. (3.9) Moreover, allvⁿ satisfy the uniform estimate²

Dv^npp+∇v^npp+1

nv^nqq ≤K (3.10)

and consequently, due to the growth condition (3.4) and Sobolev’s embedding theorem

T(Dvⁿ)p ≤c(K), (3.11)

v^{n dp}

d−p ≤c(K), (3.12)

vⁿ⊗v^{n dp}

2(d−p) ≤c(K). (3.13)

2To verify it, takeϕ=vⁿ in (3.9) and apply basic inequalities including the Korn one.

The existence of vⁿ solving (3.9) forn∈Nis standard and can be proved, for example,via Galerkin approx- imations combined with the monotone operator theory and the compactness for the velocity. An important feature and the advantage of this approximation consists in the fact that the space of test functions coincides with the space where the solution is constructed. The choice of the value for q is due to the quadratic term since forn∈N

(vⁿ⊗vⁿ,Dϕ)≤ vⁿ²₂pDϕp=vⁿ²qDϕp≤C(n).

Obviously, the estimate (3.10) implies the existence of v∈W₀^1,p(Ω), and a (not relabeled) subsequence{vⁿ} such that

vⁿ v weakly inW₀^1,p(Ω)^d, (3.14)

1

n(|vⁿ|^q−2vⁿ, ϕ)→0 for allϕ∈L^∞(Ω)^d, (3.15) and due to the compact embedding theorem

vⁿ→v strongly inL^σ(Ω)^d for allσ∈[1, dp

d−p). (3.16)

In particular,

vⁿ →v strongly inL²(Ω)^d provided thatp > 2d

d+ 2, (3.17)

which implies that

(vⁿ⊗vⁿ,Dϕ)→(v⊗v,Dϕ) for allϕ∈W₀^1,∞(Ω)^d. (3.18) Next goal is to prove that also

(T(Dvⁿ),Dϕ)→(T(Dv),Dϕ) for allϕ∈W₀^1,∞(Ω)^d. (3.19) It suffices, by virtue of (3.10), (3.11) and Vitali’s theorem, to show at least for a subsequence that

Dvⁿ →Dv a.e. in Ω. (3.20)

This follows, see for example [8] for details, from (3.5) provided that for a certainθ∈(0,1]

lim sup

n→∞

Ω

(T(Dvⁿ)−T(Dv))·(Dvⁿ−Dv)θ

dx= 0. (3.21)

To verify (3.21) (even withθ= 1) we takeϕ=vⁿ−v in (3.9) and letn→ ∞. It is then easy to observe that (3.21) is a consequence of

lim sup

n→∞ |(vⁿ⊗vⁿ,D(vⁿ−v))|= 0. (3.22)

Since (vⁿ⊗vⁿ,D(vⁿ−v)) = (vⁿ⊗vⁿ,∇(vⁿ−v)) =−(vⁿ⊗(vⁿ−v),∇v)) =−(vⁿ⊗(vⁿ−v),Dv)), (3.22) follows from (3.10), (3.12), (3.13) and H¨older’s inequality, provided that

p > 3d

d+ 2· (3.23)

In order to establish the existence result also for p∈ 2d

d+ 2, 3d d+ 2

, (3.24)

we first notice that owing to (3.10) and (3.14) the functions uⁿ:=vⁿ−v

fulfill the assumptions of Theorem 2.5 and we conclude the existence of a sequence {u^n,j} possessing the properties (2.16)–(2.22).

Note that the functionsu^n,j are in general not divergence free on the set{uⁿ=u^n,j}and we have to correct them in order to use them as a test function in (3.9). For 1< σ <∞define

L^σ₀(Ω) :=

h∈L^σ(Ω) :

Ω

hdx= 0

.

Since ∂Ω is Lipschitz, according to [5], there exists an linear operator B such that for all σ ∈ (1,∞) we haveB : L^σ₀(Ω)→W₀^1,σ(Ω)^d continuously and div(Bh) =h. In particular for allσ∈(1,∞) and allh∈L^σ₀(Ω) we have

div(Bh) =h,

||Bh||1,σ≤c||h||σ, (3.25)

where the constant depends only on Ω andσ. We define

ψ^n,j:=B(divu^n,j) =B(χ_{uⁿ_=u^n,j_}divu^n,j) Then

||ψ^n,j||1,p≤c||divu^n,jχ_{un=u^n,j}||p. Consequently, (3.14) and (2.16)–(2.22) yield forj∈N,n→ ∞,

ψ^n,j0 weakly inW^1,σ(Ω)^d for allσ∈(1,∞), (3.26) ψ^n,j→0 strongly inL^σ(Ω)^d for allσ∈(1,∞), (3.27) and

lim sup

n→∞ ψ^n,j1,p≤clim sup

n→∞

divu^n,jχ_{un=u^n,jp

≤clim sup

n→∞

∇u^n,jχ_{un=u^n,j}p

≤c j

(3.28)

with j :=K2^−pj. Note that we have used in (3.26) that a continuous linear operator preserves weak conver- gence.

Next, we take in (3.9)ϕof the form

ϕ^n,j =u^n,j−ψ^n,j. (3.29)

Note thatϕ^n,j ∈W₀^1,s(Ω)^d∩L^q(Ω)^d and by (3.25)

divϕ^n,j = 0. (3.30)

Note that due to (3.26) and (3.27) we have forj∈N,n→ ∞

ϕ^n,j0 weakly inW^1,σ(Ω)^d for allσ∈(1,∞), (3.31) ϕ^n,j→0 strongly in L^σ(Ω)^d for allσ∈(1,∞). (3.32) The weak formulation of the approximative problem (3.9) withϕ^n,j as a test function can be rewritten as

(T(Dvⁿ)−T(Dv),Du^n,j) = (T(Dvⁿ),Dψ^n,j)

−(T(Dv),Du^n,j)

− 1

n(|vⁿ|^q−2vⁿ, ϕ^n,j) +f, ϕ^n,j + (vⁿ⊗vⁿ,Dϕ^n,j)

:=J_n,j¹ +J_n,j² +J_n,j³ +J_n,j⁴ .

(3.33)

FromW₀^1,p(Ω)→→L²(Ω) (sincep > _d²₊₂^d ) and (3.14) we deduce vⁿ⊗vⁿ →v⊗v in L²(Ω).

Letting n→ ∞, we observe from (3.10) and (3.18) that

nlim→∞(J_n,j² +J_n,j³ +J_n,j⁴ ) = 0. (3.34) On the other hand with H¨older’s inequality, (3.11), and (3.28)

lim sup

n→∞ J_n,j¹ ≤c(K)j. (3.35)

Overall, (3.33), (3.34), and (3.35) imply for allj∈N lim sup

n→∞ (T(Dvⁿ)−T(Dv),Du^n,j)≤c(K)j. (3.36) Now, (3.21) follows immediately from (3.36) and Lemma2.6. This proves the validity of (3.9). This and (3.16), (3.18), as well as (3.19) prove that

(T(Dv),Dϕ) =f, ϕ+ (v⊗v,Dϕ) for allϕ∈W₀^1,∞(Ω)^d, divϕ= 0. (3.37) Next, we apply deRham’s theorem and the Neˇcas theorem on Sobolev spaces with negative exponents to reconstruct the pressure. Especially, there is p∈L^s₀(Ω) fulfilling

(T(Dv),Dϕ) =f, ϕ+ (v⊗v,Dϕ) + (p,divϕ)

for allϕ∈W₀^1,∞(Ω)^d. (3.38)

The proof of Theorem3.1is complete.