on the anisotropic orlicz spaces applied in the problems...

On the anisotropic Orlicz spacesapplied in the problems of

continuum mechanics

P. Gwiazda, P. Minakowski, A. Swierczewska-Gwiazda

Preprint no. 2011 - 019

Ph.D. Programme: Mathematical Methods in Natural Sciences (MMNS)e-mail: [email protected]://mmns.mimuw.edu.pl

Manuscript submitted to Website: http://AIMsciences.orgAIMS’ JournalsVolume X, Number 0X, XX 200X pp. X–XX

ON THE ANISOTROPIC ORLICZ SPACES APPLIED IN THE

PROBLEMS OF CONTINUUM MECHANICS

Piotr Gwiazda

Institute of Applied Mathematics and MechanicsUniversity of Warsaw

Banacha 2, 02-097 Warszawa, Poland

Piotr Minakowski and Agnieszka Swierczewska-Gwiazda

Institute of Applied Mathematics and MechanicsUniversity of Warsaw

Banacha 2, 02-097 Warszawa, Poland

(Communicated by the associate editor name)

Abstract. The paper concerns theory of anisotropic Orlicz spaces and its

applications in continuum mechanics. Our main motivations are e.g. flowof non-Newtonian fluid and response of inelastic materials with non-standard

growth conditions of the Cauchy stress tensor. The set of basic definitionsand theorems with proofs is presented. We prove the existence of a weak

solutions to the generalized Stokes system. Overview covering recent results in

the referred topic is given.

1. Introduction. Our interest is dedicated to the properties of the anisotropicOrlicz spaces and their applications in continuum mechanics. By anisotropy influid mechanics we mean dependence on all components of the strain tensor, notonly on the absolute value. The main motivation of the presented theory is tocover the response of the anisotropic fluids. Such fluids have one or more specificdirections in which one can observe anisotropic behaviour. This phenomena couldbe caused by internal structure of the material or external stimulus.

Liquid crystals are the state of matter which has the properties of liquids, aswell as the solid crystals. It means that they can flow like liquids, but its moleculesmay be oriented in a crystal-like way. On the other hand we consider materialswhich exhibit drastic changes in their rheological properties upon the applicationof an outer field (electric or magnetic). The electrorheological fluid consists ofdielectric particles suspended in non-conducting oil. Dielectric in the outer electricfield undergo polarization. Neighbouring dipolar particles are attracted to eachother and are aligned with the lines of external field, thus producing the fibrousstructures. The direction parallel to field lines is distinguished.

Anisotropic fluids are widely applicable in the common life. Most of the modernelectronic displays are liquid crystal based. We can mention also magnetorheolog-ical shock absorber of buildings or in the automotive industry, magnetorheologicaldamper and electroreheological clutch.

2000 Mathematics Subject Classification. 35Q35, 46E30.Key words and phrases. non-Newtonian fluids, Orlicz Spaces.

1

2 P. GWIAZDA, P. MINAKOWSKI AND A. SWIERCZEWSKA-GWIAZDA

One of the main aims of this paper is to collect basic definitions and theoremsconcerning theory of anisotropic Orlicz spaces. We present the complete proofs inorder to create the comprehensive reference to considered topic.

We focus on the steady flow of non-Newtonian incompressible fluids describedby the system

div (u⊗ u)− div Σ(x,D(u)) +∇p = f in Ω, (1)

divu = 0 in Ω, (2)

u(x) = 0 on ∂Ω, (3)

where Ω ⊂ Rd is an open, bounded set with a sufficiently smooth boundary ∂Ω,u : Ω → Rd is the velocity of a fluid, f given body force, p : Ω → R the pressure,Σ − Ip is the Cauchy stress tensor and D(u) symmetric part of velocity gradientD(u) = 1

2

(∇u+ (∇u)T

). We assume that Σ satisfies the following conditions

(S1) Σ is a Caratheodory function (i.e., measurable w.r.t. x and continuous w.r.t.the last variable).

(S2) There exists an N -function (def. 2.1) M : Ω × Rn×nsym → R+, an integrable

function k : Ω→ R+ and a constant c > 0 such that for all ξ ∈ Rn×nsym

Σ(x, ξ) : ξ ≥ c (M(x, ξ) +M∗(x,Σ(x, ξ)))− k(x). (4)

(S3) For all ξ,η ∈ Rd×dsym and for a.a. x ∈ Ω

(Σ(x, ξ)−Σ(x,η)) : (ξ − η) ≥ 0. (5)

By standard growth conditions we understand polynomial growth, see e.g. Ref.[8] , namely

|Σ(x, ξ)| ≤ c(1 + |ξ|)q−1

Σ(x, ξ) : ξ ≥ c|ξ|q.(6)

We will show in Sec. 1.1 why the anisotropic Orlicz spaces are proper spacesto cover flow of non-Newtonian fluids with non-standard growth conditions of theCauchy stress tensor.

Another field where the problem formulation in Orlicz spaces is appropriate is aninelastic deformation theory. The anisotropy of the material arises from differentplastic properties in different directions. As an example we present a quasistaticsystem capturing the viscoplastic deformation behaviour of solids at small strain.

div xT (x, t) = −F (x, t) in Ω× (0, T ), (7)

T (x, t) = D(ε(u(x, t))− εp(x, t)) in Ω× (0, T ), (8)

ε(u(x, t)) =1

2(∇xu(x, t) +∇Txu(x, t)) in Ω× (0, T ), (9)

εpt (x, t) = G (PT (x, t)) in Ω× (0, T ), (10)

where Ω ⊂ R3 is a bounded domain with a smooth boundary ∂Ω, u : Ω×(0, T )→ R3

is the displacement field, T : Ω × (0, T ) → R3×3sym is the Cauchy stress tensor,

εp : Ω× (0, T )→ R3×3sym is the inelastic deformation tensor. Moreover, the function

F : Ω×(0, T )→ R3 describes the external forces acting on the material, D : R3×3sym →

R3×3sym is the elasticity tensor which is assumed to be constant in time and space,

symmetric and positive definite. Moreover, G : R3×3sym → PR3×3

sym is the inelastic

constitutive function and the map P is defined by PT = T − 13 trT ·I. We consider

ON THE ANISOTROPIC ORLICZ SPACES 3

system (7)–(10) with the following boundary condition of mixed type: the Dirichletboundary condition on Γ1 ⊂ ∂Ω

u(x, t) = gD(x, t) for x ∈ Γ1 and t ≥ 0 (11)

and the Neumann boundary condition on Γ2 ⊂ ∂Ω

T (x, t) · n(x) = gN (x, t) for x ∈ Γ2 and t ≥ 0 (12)

where n(x) is the exterior unit normal vector to the boundary ∂Ω at the point x, Γ1

and Γ2 are open in ∂Ω, disjoint, “smooth enough” sets satisfying ∂Ω = Γ1 ∪Γ2 andH2(Γ1) > 0, where H2 denotes the 2-dimensional Hausdorff measure. Moreover,the functions gD, gN are given boundary data. Finally, the initial condition for theinelastic strain tensor is in the form

εp(x, 0) = εp,0(x) (13)

with a given initial data εp,0 : Ω→ PR3×3sym.

We assume that G satisfies the following conditions

(G1) G is continuous and G(0) = 0,(G2) There exist positive constants c1, c2 and an N−function M such that for all

ξ ∈ R3×3sym it holds

G(ξ) : ξ ≥ c1 (M(ξ) +M∗(G(ξ))− c2(G3) G is strictly monotone

∀ σ1,σ2 ∈ R3×3sym σ1 6= σ2 ⇒ (G(σ1)− G(σ2)) : (σ1 − σ2) > 0.

1.1. Constitutive theory. To complete system (1)-(3) it is necessary to add theconstitutive relation which describes the behaviour of the fluid namely the Cauchystress tensor. In this section we will consider fluids with outer field dependencein particular electrorheological fluids. General form of the constitutive relation forelectrorheological fluids, can be assumed as, cf. [12]

Σ = α1I + α2E ⊗E + α3D(u) + α4D(u)2

+ α5(D(u)E ⊗E +E ⊗ D(u)E) + α6(D(u)2E ⊗E +E ⊗ D(u)

2E),

(14)

where E(x) : Ω → Rd is a continuous function represents outer field dependenceand αi i = 1, . . . , 6, are scalar functions of the invariants

|E|2, trD(u), trD(u)2, trD(u)

3, tr (D(u)E ⊗E), tr (D(u)

2E ⊗E).

We consider simplified form of (14) by taking into account only two terms, namely

Σ = α3D(u) + α5(D(u)E ⊗E +E ⊗ D(u)E), (15)

where α3 and α5 are scalar functions of non-negative invariants tr(D(u)

2)

and

tr(D(u)

2E ⊗E

). Respectively

α3(D(u)) = 2β(

trD(u)2)β−1

, (16)

α5(D(u),E) = γ(

tr (D(u)2E ⊗E)

)γ−1

, (17)

where β, γ ∈ R and β, γ > 1.Now we show that the above example of the constitutive relation satisfies con-

ditions (S1)-(S3) and does not fulfil the standard growth conditions (6). Moreoverstress tensor given by (15) is thermodynamically admissible.


The condition (S1) is obviously fulfilled. We consider the convex, N -function

(def. 2.1) M which takes as an argument symmetric part of velocity gradient anddepends on x by electric field

M(x,D(u)) = (trD(u)2)β + (tr (D(u)

2E(x)⊗E(x)))γ .

The Cauchy stress Σ is a gradient of function M

Σ = ∇D(u)

((trD(u)

2)β + (tr (D(u)

2E(x)⊗E(x)))γ

), (18)

which implies that Fenchel-Young inequality (rem. 1) becomes the equality, i.e.

Σ(x,D(u)) : D(u) = M(x,D(u)) + M∗(x,Σ(x,D(u))).

Monotonicity condition (S3) also follows form convexity of M and (18)

(Σ(D(u)1,E)−Σ(D(u)2,E)) : (D(u)1 − D(u)2) ≥ 0 (19)

for all D(u)1,D(u)2 ∈ Rd×dsym and a.a. x ∈ Ω. More details concerning theory ofconvex functions can be found in [11].

Moreover we justify that given example is thermodynamically admissible, namely

Σ(x,D(u)) : D(u) ≥ 0,

(α3(D(u))D(u) + α5(D(u),E)(E ⊗ D(u)E + D(u)E ⊗E)) : D(u) =

= α3(D(u))|D(u)|2 + 2α5(D(u),E) tr (D(u)2E ⊗E) ≥ 0.

2. Generalized Orlicz spaces.

Definition 2.1. Let Ω be a bounded open domain in Rd, a functionM : Ω×Rd×dsym →R+ is said to be an N-function if it satisfies the following conditions:

1. M is Caratheodory function such that M(x, ξ) = 0 if and only if ξ = 0,M(x, ξ) = M(x,−ξ) a.e. in Ω,

2. M(x, ξ) is a convex function w.r.t ξ,

3. lim|ξ|→0 supx∈ΩM(x,ξ)|ξ| = 0,

4. lim|ξ|→∞ infx∈ΩM(x,ξ)|ξ| =∞.

Definition 2.2. We say that an N-function M satisfies ∆2−condition (or equiv-alently is ∆2−regular) if for some non-negative, integrable in Ω function h and aconstant k > 0

M(x, 2ξ) ≤ kM(x, ξ) + h(x) for all ξ ∈ Rd×dsym and a.a x ∈ Ω.

Definition 2.3. The complementary function M∗ to a function M is defined by

M∗(x,η) = supξ∈Rd×dsym

(ξ : η −M(x, ξ))

for η ∈ Rd×dsym, and a.a. x ∈ Ω.

The complementary function M∗ is also an N-function.

Remark 1. (Fenchel-Young inequality).Let M be an N-function and M∗ a complementary to M . Then the followinginequality is satisfied

|ξ : η| ≤M(x, ξ) +M∗(x,η) (20)

for all ξ,η ∈ Rd×dsym and a.a x ∈ Ω.


Proof. The proof follows from definition 2.3 and is analogous to the isotropic case.

Definition 2.4. The generalized Orlicz class LM (Ω) is a set of all measurablefunctions ξ : Ω→ Rd×dsym such that∫

Ω

M(x, ξ)dx <∞.

Definition 2.5. The generalized Orlicz space LM (Ω) is defined as a set of allmeasurable functions ξ : Ω→ Rd×dsym which satisfy∫

Ω

M(x, λξ)dx→ 0 as λ→ 0.

Definition 2.6. Orlicz norm, for ξ ∈ LM (Ω) we have

‖ξ‖M = sup

∫Ω

|η : ξ|dx : η ∈ LM∗(Ω),

∫Ω

M∗(x,η)dx ≤ 1

. (21)

Definition 2.7. Luxemburg norm, for ξ ∈ LM (Ω) we have

‖ξ‖LM = inf

λ > 0 :

∫Ω

M

(x,ξ

λ

)dx ≤ 1

. (22)

Orlicz and Luxemburg norms are equivalent. The proof in less general case (i.e.M(x, |ξ|)) can be found in [9].

Definition 2.8. By EM (Ω) we mean the closure in LM (Ω) of all measurable andbounded functions.

Definition 2.9. We say that a sequence ξi∞i=1 converges modularly to ξ in LM (Ω)if there exists λ > 0 such that∫

Ω

M

(x,ξi − ξλ

)dx→ 0. (23)

We will use the notation ξiM−→ ξ for the modular convergence in LM (Ω).

3. Basic theorems.

Theorem 3.1. The generalized Orlicz space is a Banach space with respect to theOrlicz norm (21) or the equivalent Luxemburg norm (22).

Proof. We will prove the completeness w.r.t Orlicz norm. Let ξn∞n=1 be a Cauchysequence in LM (Ω) such that ∀ε>0∃Nε>0 holds

sup

∫Ω

η : (ξm − ξn)dx ; η ∈ LM∗(Ω),

∫Ω

M∗(x,η)dx ≤ 1

< ε ∀ n,m > Nε.

(24)Let λ > 0 be such that∫

Ω

M∗(x,η)dx ≤ 1 for all η ∈ L∞(Ω;Rd×d), ‖η‖∞ ≤ λ.

By putting

η =

λ ξm−ξn|ξm−ξn|

if ξm 6= ξn0 otherwise

(25)


to (24) we obtain ∫Ω

|ξm − ξn|dx ≤ε

λfor all m,n ≥ Nε.

Therefore ξn∞n=1 is a Cauchy sequence in L1(Ω). Hence, by Fatou’s lemma∫Ω

|(ξ − ξn) : η|dx =

∫Ω

limm→∞

|(ξm − ξn) : η|dx ≤ lim infm→∞

∫Ω

|(ξm − ξn) : η|dx < ε.

Thus ξ ∈ LM (Ω) and ‖ξ − ξn‖M → 0 with n→∞. This completes the proof.

Lemma 3.2. Generalized Holder inequalityLet M be an N -function and M∗ its complementary, then∣∣∣∣∫

Ω

ξ : η dx

∣∣∣∣ ≤ 2‖ξ‖M‖η‖M∗ , (26)

where ξ ∈ LM (Ω) and η ∈ LM∗(Ω).

Proof. From (20) by putting ξ = ξ(x)‖ξ‖M , η = η(x)

‖η‖M∗we obtain∫

Ω

∣∣∣∣ ξ(x)

‖ξ|Mη(x)

‖η‖M∗

∣∣∣∣ dx ≤ ∫Ω

M

(x,ξ(x)

‖ξ‖M

)dx+

∫Ω

M∗(x,

η(x)

‖η‖M∗

)dx ≤ 2.

We finish the proof of (26) by multiplying above inequality by ‖ξ‖M‖η‖M∗ .

Theorem 3.3. The space LM∗(Ω) is a dual space of EM (Ω), i.e. (EM (Ω))∗ =LM∗(Ω).

Before we prove Theorem 3.3, we will show the following lemma.

Lemma 3.4. Let η ∈ LM∗(Ω). The linear functional Fη defined by

Fη(ξ) =

∫Ω

ξ : η dx (27)

belongs to the space (EM (Ω))∗ and its norm in that space fulfils

‖Fη‖ ≤ 2‖η‖M∗ . (28)

Proof. It follows by Holder inequality (26) that

|Fη(ξ)| ≤ 2‖ξ‖M‖η‖M∗holds for all ξ ∈ LM (Ω) confirming the inequality (28).

Proof. of the Theorem 3.3Lemma 3.4 has already shown that any element η ∈ LM∗(Ω) defines a boundedlinear functional Fη on EM (Ω) which is given by (27). It remains to show thatevery bounded linear functional on EM (Ω) is of the form Fη for any η ∈ LM∗(Ω).

Let F ∈ (EM (Ω))∗, we define a measure λ on the measurable subsets S of Ω

λ(S) = F (τ IS)

where IS denotes the characteristic function of S, τ ∈ Rd×dsym, |τ | = 1. Let

A(r) = supx∈Ω,|ξ|=r

M(x, ξ)

be an auxiliary function and r ∈ [0,∞). This function is needed to generalise theapproach presented in [1]. Since∫

Ω

M

(x,A−1

(1

|S|

)ISτ

)dx ≤

∫S

supx∈S

M

(x,A−1

(1

|S|

)τ

)dx ≤

∫S

1

|S|≤ 1,


we have

|λ(S)| = |F (τ IS)| ≤ ‖F‖‖τ IS‖M ≤c‖F‖

A−1(1/|S|). (29)

Since the right-hand side of (29) converges to zero when |S| converges to zero, themeasure λ is absolutely continuous w.r.t Lebesgue measure. By Radon-Nikodymand Riesz theorems, cf. [15], λ can be expressed in the form

λ(S) =

∫S

η(x)dx

for some η integrable on Ω. Therefore

F (ξ) =

∫Ω

ξ : η dx

holds for measurable bounded functions ξ.If ξ ∈ EM (Ω) we can find a sequence of measurable functions ξi which converges

a.e. to ξ and satisfies |ξi| ≤ |ξ| on Ω. Since |ξi : η| converges a.e to |ξ : η|, Fatou’slemma yields ∣∣∣∣∫

Ω

ξ : η dx

∣∣∣∣ ≤ ∫Ω

|ξ : η|dx ≤ lim infi→∞

∫Ω

|ξi : η|dx

≤ lim infi→∞

2‖ξi‖M‖η‖M∗ ≤ 2‖ξ‖M‖η‖M∗ .

Hence the linear functional

Fη(ξ) =

∫Ω

ξ : η dx

is bounded on EM (Ω) when η ∈ LM∗(Ω). Since Fη and F achieve the same valueson the measurable, simple functions, a set which is dense in EM (Ω), they agree onEM (Ω) and the proof is completed.

Theorem 3.5. The space EM (Ω) is separable.

Proof. The theorem can be proved in two steps. First we approximate u ∈ EM (Ω)by simple functions. Then a dominated convergence argument shows that the simplefunctions converge in norm to u in EM (Ω).

Theorem 3.6. The space LM (Ω) is separable if and only if M is ∆2−regular.

Proof. We begin the proof with the fact that if M is ∆2−regular then EM (Ω) =LM (Ω). Let τ ∈ EM (Ω) and τ b be a bounded function on Ω such that ‖τ−τ b‖M <12 . Then from ∆2−condition we obtain M(x, 2τ−2τ b) ≤ 2M(x, τ−τ b)+h(x) whatimplies that 2τ − 2τ b ∈ LM (Ω). As 2τ b ∈ LM (Ω), convexity of LM (Ω) providesthat ξ = 1

2 [2(τ − τ b) + 2τ b] ∈ LM (Ω) ⊂ LM (Ω).The second part of the proof can be shown by contradiction using Luzin’s theo-

rem, cf. [7]. And by definition of EM (Ω) we obtain separability of LM (Ω).

Theorem 3.7. LM (Ω) is reflexive if and only if both M and M∗ are ∆2-regular.

Proof. Form the proof of the theorem 3.6 we know that if M is ∆2-regular thenEM (Ω) = LM (Ω). By theorem 3.3 (EM (Ω))∗ = LM∗(Ω), so we obtain reflexivity

((LM (Ω))∗)∗

= ((EM (Ω))∗)∗

= (LM∗(Ω))∗

= (EM∗(Ω))∗

= LM (Ω).


Lemma 3.8. Let ξi : Ω → Rd×dsym be a measurable sequence. Then ξiM−→ ξ in

LM (Ω) modularly if and only if ξi → ξ in measure and there exists some λ > 0such that the sequence M(x, λξi)∞i=1 is uniformly integrable, i.e.,

limR→∞

(supi∈N

∫x:|M(x,λξi)|≥R

M(x, λξi)dx

)= 0.

For the proof see [3].

Proposition 1. Let M be an N -function and M∗ its complementary function.Suppose that the sequences ψj : Ω → Rd×d and φj : Ω → Rd×d are uniformly

bounded in LM (Ω) and LM∗(Ω) respectively. Moreover ψjM−→ ψ modularly in

LM (Ω) and φjM∗−−→ φ modularly in LM∗(Ω). Then ψj : φj → ψ : φ strongly in

L1(Ω;Rd×dsym).

For the proof see [3].In the statement of the next theorem, an N−function has the same properties

as before, but it is defined on R+. To avoid confusion, we denote it with a smallletter m.

Theorem 3.9. Let Ω be a bounded domain with a Lipschitz boundary. Let m bean N -function satisfying ∆2-condition and such that mγ is quasiconvex for someγ ∈ (0, 1). Then, for any f ∈ Lm(Ω;R) such that∫

Ω

fdx = 0,

the problem of finding a vector field v : Ω→ Rn such that

div v = f in Ω

v = 0 on ∂Ω(30)

has at least one solution v ∈ Lm(Ω;Rn) and ∇v ∈ Lm(Ω;Rn×n). Moreover, forsome positive constant c ∫

Ω

m(|∇v|)dx ≤ c∫

Ω

m(|f |)dx.

For the proof see e.g. [14].

4. Generalized Stokes problem. The present section is directed to the existenceof weak solutions of the steady generalized Stokes problem. Omitting the convec-tive term we can skip the assumption on the lower growth of the N−function. Weshall also provide the proof without an assumption that M∗ satisfies ∆2−condition.Contrary to most of the results mentioned in Section 5 where shear thickening fluidswere considered, here we direct our attention to shear thinning fluids. The frame-work is based on the closures of smooth functions with respect to various topologies.Because of the smoothing procedure we may consider the case of still anisotropic,but homogenous function M : Rd×dsym → R+. We will consider simplification ofsystem (1)-(3) namely the generalized Stokes system

−div Σ(x,D(u)) +∇p = f in Ω, (31)

divu = 0 in Ω, (32)

u(x) = 0 on ∂Ω, (33)


where Ω ⊂ Rd is an open, bounded set with a sufficiently smooth boundary ∂Ω,u : Ω→ Rd is the velocity of a fluid and p : Ω→ R the pressure, Σ−pI is the Cauchystress tensor. We assume that Σ satisfies conditions (S1)-(S3). The main result ofthis section concerns existence of the weak solutions to the system (31)-(33).

4.1. Definitions and notation. The following notation is used

V := u ∈ C∞c (Ω); divu = 0,L2

div (Ω) := closure of V in L2-norm,

where C∞c (Ω) is the set of compactly supported smooth functions.We define the closure of C∞c (Ω) with respect to two topologies, namely

1. modular topology of LM (Ω), which we denote by YM0 , namely

YM0 =u ; divu = 0, D(u) ∈ LM (Ω) ∃ uj∞j=1 ⊂ V :

D(uj)M−→ D(u) modularly in LM (Ω)

(34)

2. weak-star topology of LM (Ω), which we denote by ZM0 , namely

ZM0 =u ; divu = 0, D(u) ∈ LM (Ω) ∃ uj∞j=1 ⊂ V :

D(uj)∗ D(u) weakly star in LM (Ω).

(35)

Moreover, by BDM (Ω) we denote the space of functions with symmetric gradientin LM (Ω), namely

BDM (Ω) := u ∈ L1(Ω;Rd) ; D(u) ∈ LM (Ω).

Remark 2. The space BDM (Ω) is a Banach space with a norm

‖u‖BDM (Ω) := ‖u‖L1(Ω) + ‖D(u)‖Mand it is a subspace of the space of bounded deformations BD(Ω)

BD(Ω) := u ∈ L1(Ω;Rd) ; [D(u)]i,j ∈M(Ω), for i, j = 1, . . . , d,

here M(Ω) denotes the space of bounded measures on Ω.

We also define the subspace and the subset of BDM (Ω) as follows

BDM,0(Ω) := u ∈ BDM (Ω) | γ0(u) = 0,

BDM,0(Ω) := u ∈ BDM (Ω) | D(u) ∈ LM (Ω) and γ0(u) = 0.Where according to [13, Theorem 1.1.] γ0 is a unique continuous operator from

BD(Ω) onto L1(∂Ω;Rd) such that the generalized Green formula

2

∫Ω

φ[D(u)]i,jdx = −∫

Ω

(uj∂φ

∂xi+ ui

∂φ

∂xj

)dx+

∫∂Ω

φ (γ0(ui)nj + γ0(uj)ni) dHd−1

(36)holds for every φ ∈ C1(Ω), where n is the unit outward normal vector on ∂Ω andγ0(ui) is the i-th component of γ0(u) and Hd−1 is the (d− 1)−Hausdorff measure.

The operator γ0 is a generalization of the concept of trace in Sobolev spaces tothe case of BD(Ω) space. Moreover, if u ∈ C(Ω;Rd), then γ0(u) = u|∂Ω. Notice

that in case of u ∈ W 1,10 (Ω;Rd) it coincides with the classical trace operator in

Sobolev spaces.By [13, Proposition 1.1.] there exists an extension operator from BD(Ω) to

BD(Rd).


Let us define two auxiliary functions m, m : R+ → R+ as follows

m(r) := minξ∈Rd×dsym,|ξ|=r

M(ξ),

m(r) := maxξ∈Rd×dsym,|ξ|=r

M(ξ).(37)

The N−functions m and m are defined on R+.

4.2. Existence result.

Theorem 4.1. Let condition C1. or C2. be satisfied

(C1) Ω is a bounded star-shaped domain,(C2) Ω is a bounded non-star-shaped domain and

m(r) ≤ cm(

(m(r))dd−1 + 1

)(38)

for all r ∈ R+, and m satisfies ∆2-condition.

Let M be an N -function and Σ satisfy conditions (S1)-(S3). Then, for given |f | ∈Em∗(Ω) there exists u ∈ ZM0 such that∫

Ω

Σ(x,D(u)) : D(ϕ)dx =

∫Ω

f ϕdx (39)

for all ϕ ∈ V.

We will not present the complete proof of existence of the weak solutions to theboundary value problem (31)-(33). The following lemmas 4.3 and 4.4 provide thecrucial step in the proof. The omitted parts follow the same steps as presented in[4].

Lemma 4.2. Let m be an N -function and Ω be a bounded domain, Ω ⊂ [− 14 ,

14 ]d,

and u ∈ BDM,0(Ω). Then

‖m(|u|)‖L

dd−1 (Ω)

≤ Cd‖m(|D(u)|)‖L1(Ω). (40)

Proof. The above variant of the Korn-Sobolev inequality was prooved in [5].

Lemma 4.3. Let M : Rd×dsym → R+ be an N -function, Ω be a bounded star-shaped

domain and YM0 , ZM0 be the function spaces defined by (34) and (35). Then YM0 =ZM0 .

Moreover, if χ ∈ LM∗(Ω), f ∈ Lm∗(Ω) and

− divχ = f in D′(Ω), (41)

then ∫Ω

χ : D(u)dx =

∫Ω

f · u dx. (42)

Proof. The proof follows the similar lines as the proof of unsteady case from [5].We focus only on the inclusion

ZM0 ⊂ YM0 . (43)

As the modular topology is stronger than weak-star, obviously we have YM0 ⊂ ZM0 .For this purpose we want to extend u by zero outside of Ω to the whole Rd andthen mollify it. To extend u we observe that ZM0 ⊂ BDM,0(Ω). Notice thatu ∈ ZM0 is an element of BD(Ω). We concentrate on showing that it vanishes onthe boundary. Take the sequence uk∞k=1 of compactly supported smooth functions


with the properties prescribed in the definition of the space ZM0 . By putting thissequence into (36) we get

2

∫Ω

φ[D(uk)]i,jdx = −

∫Ω

(ukj

∂φ

∂xi+ uki

∂φ

∂xj

)dx. (44)

Now we pass to the weak-star limit in (44), it is possible, due to linearity of allterms. It implies that the boundary term vanishes. Let x0 be a vantage point of Ωand λ ∈ (0, 1). We define auxiliary function vλ(x) as follows

vλ(x) := v(λ(x− x0) + x0). (45)

Let ελ = 12dist (∂Ω, λΩ) where λΩ := y = λ(x− x0) + x0 | x ∈ Ω. Define then

uλ,ε(x) := %ε ∗ uλ(x) (46)

where %ε = 1εd%(xε ) is a standard regularizing kernel on Rd (i.e. % ∈ C∞(Rd),

% has a compact support in B(0, 1) and∫Rd %(x)dx = 1,%(x) = %(−x)) and the

convolution is done w.r.t. space variable x, ε < ελ2 . Note that uλ,ε also vanishes on

the boundary.

Now we pass to the limit with ε→ 0 and hence D(uλ,ε

)ε→0−−−→ D(u

λ) in L1(Ω;Rd×d).

The function D(uλ,ε

) ∈ L1(Ω;Rd×d) and %ε ∗ D(uλ)ε→0−−−→ D(u

λ) in L1(Ω;Rd×d)

and hence %ε ∗ D(uλ)ε→0−−−→ D(u

λ) in measure on the set Ω.

By the analogous argumentation as in the proof of Lemma 4.2, we show the

uniform integrability of M(D(uλ,ε

))ε>0.The convexity of M implies that for all θ > 0 the following inequality holds∫

Ω

∣∣∣∣M(D(uλ))− 1√

θ

∣∣∣∣+

dx ≥∫

Ω

∣∣∣∣M(%ε ∗ D(uλ))− 1√

θ

∣∣∣∣+

dx. (47)

Since βD(uλ) ∈ LM (Ω) for some β > 0, then also

∫Ω|M(x, βD(u

λ)) − 1√

θ|+dx

is finite. Hence taking supremum over ε ∈ (0, ελ2 ) in (47) provides the uniform

integrability of the sequence M(βD(uλ,ε

))ε>0.

Modular convergence in LM (Ω) of D(uλ,ε

)ε→0−−−→ D(u

λ) is provided by lemma 3.8

We pass to the lmit with λ→ 1 and obtain that D(uλ)λ→1−−−→ D(u) in L1(Ω;Rd×d)

and D(uλ)

λ→1−−−→ D(u) modularly in LM (Ω). Consequently YM0 = ZM0 , whichcompletes the first part of the proof.

It remains to prove (41). We define

uλ,ε(x) := %ε ∗ uλ(x) (48)

where ε < ελ2 . We test equation (41) with sufficiently regular test function uλ,ε∫

Ω

χ : D(uλ,ε)dx =

∫Ω

f · uλ,εdx. (49)

To treat the left-hand side of (49) we follow the ideas presented in the first part ofthe proof. For proving the convergence of the term

∫Ωf ·uλ,εdx we apply Lemma 4.2

to m and obtain(∫Ω

(m(|uλ,ε|))dd−1 dx

) d−1d

≤ Cd∫

Ω

m(|D(uλ,ε

)|)dx.


Moreover using the Holder inequality and the definition of m we get∫Ω

(m(|uλ,ε|))dx ≤ CΩ,d

∫Ω

(m(|D(uλ,ε

)|)dx ≤ CΩ,d

∫Ω

M(D(uλ,ε

))dx. (50)

Hence (50) provides that modular convergences D(uλ,ε

)ε→0−−−→ D(u

λ), D(u

λ)λ→1−−−→

D(u) in LM (Ω) imply that uλ,εε→0−−−→ uλ, uλ

λ→1−−−→ u modularly in Lm(Ω). UsingProposition 1 for N -functions m∗ and m we obtain

limε→0,λ→1

∫Ω

f · uλ,εdx =

∫Ω

f · udx.

Analogously Proposition 1 for N -functions M and M∗ provides the convergence

limε→0,λ→1

∫Ω

χ : D(uλ,ε

)dx =

∫Ω

χ : D(u)dx.

By passing to the limit with ε, λ in (49) we obtain that∫Ω

χ : D(u)dx =

∫Ω

f · u dx. (51)

In the case of non-star-shaped the additional condition is assumed. We controlthe anisotropy in terms of the spread between auxiliary functions m(r) and m(r).

Lemma 4.4. Let M be an N -function such that m(r) ≤ cm((m(r))dd−1 + 1) for r ∈

R+ and let m satisfy ∆2-condition. Let Ω be a bounded domain with a sufficientlysmooth boundary, YM0 , ZM0 be the function spaces defined by (34) and (35). ThenYM0 = ZM0 .

Moreover, let χ ∈ LM∗(Ω), |f | ∈ Lm∗(Ω) and

− divχ = f in D′(Ω). (52)

Then ∫Ω

χ : D(u)dx =

∫Ω

f · u dx (53)

holds.

Proof. If Ω is a Lipschitz domain, then there exists a finite family of star-shapeddomains Ωii∈J such that

Ω =⋃i∈J

Ωi

see e.g. [10]. We introduce the partition of unity θi with 0 ≤ θi ≤ 1, θi ∈C∞c (Ωi), supp θi = Ωi,

∑i∈J θi(x) = 1 for x ∈ Ω. By Lemma 4.2 applied to m

and function (uθi)λ,ε and the definition of m we obtain∫

Ω

(m(|(uθi)λ,ε|))dd−1 dx ≤ Cd

(∫Ω

m(|D((uθi)

λ,ε)|)dx

) dd−1

≤ Cd(∫

Ω

M(D((uθi)

λ,ε)dx) dd−1

).

Since m(r) ≤ cm((m(r))dd−1 + 1), ∇θ ∈ L∞(Ω;Rd) and

(D(uλ)θλi )ε +1

2(u⊗∇θi)λ,ε +

1

2(∇θi ⊗ u)λ,ε = D

((uθi)

λ,ε),


where Ωi = supp θi, we conclude that D((uθi)

λ,ε)∈ LM (Ωi).

We concentrate on the function∑i∈J %ε ∗ u θiλ, where ·λ is defined by

(45). To solve the problem that∑i∈J %ε ∗ u θiλ may be not divergence-free, we

introduce the function ϕλ,ε ∈ Lm

dd−1

(Ω) which is a solution to the problem

divϕλ,ε =∑i∈J

%ε ∗ u · ∇θiλ

in Ω,

ϕλ,ε = 0 on ∂Ω.

(54)

The existence of such ϕλ,ε is provided by Theorem 3.9 applied to the N -function

mdd−1 which satisfies ∆2-condition. The function mγ with γ = d−1

d is quasiconvex.Then we complete the proof in a similar way as in the case of star-shaped domains.Instead of the sequence defined by (46), we take

ψλ,ε(x) :=∑i∈J

%ε ∗ uθiλ −ϕλ,ε(x).

It remains to show that ϕλ,ε → 0 with λ → 1 and ε → 0. Indeed, Theorem 3.9implies that∫

Ω

mdd−1 (|D(ϕλ,ε)|)dx ≤

∫Ω

mdd−1 (|∇ϕλ,ε|)dx

≤ c∫

Ω

mdd−1

(∣∣∣∣∣∑i∈J

%ε ∗ u · ∇θiλ

∣∣∣∣∣).

(55)

Since for every i ∈ J the sequence

%ε ∗ u · ∇θiλ m

dd−1

−−−−→ u · ∇θi modularly in Lm

dd−1

(Ω) (56)

as ε→ 0 and λ→ 1 and∑i∈J u · ∇θi = 0, we conclude that

∑i∈J

%ε ∗ u · ∇θiλ m

dd−1

−−−−→ 0 modularly in Lm

dd−1

(Ω) (57)

as ε→ 0 and λ→ 1. Therefore

D(ϕλ,ε)m

dd−1

−−−−→ 0 modularly in Lm

dd−1

(Ω). (58)

Again, following the similar lines as the star-shape case, instead of the functiondefined by (48), we test (52) with

ζλ,ε(x) :=∑i∈J

%ε ∗ u θiλ −ϕλ,ε(x). (59)

Passing to the limit with λ→ 1 and ε→ 0 in (49) it again remains to show that allthe terms related with function ϕλ,ε vanish in the limit, namely

limε→0,λ→1

∫Ω

χ : D(ϕλ,ε) dx = 0,

limε→0,λ→1

∫Ω

f ·ϕλ,ε dx = 0.

(60)


Since D(ϕλ,ε)m

dd−1

−−−−→ 0 (see (58)), m(r) ≤ cm((m(r))dd−1 + 1), then M(αD(ϕλ,ε))

is uniformly integrable with some α > 0. Moreover, by Lemma 3.8 the modu-lar convergence in L

mdd−1

(Ω) to zero implies the convergence in measure to zero.

Consequently by Lemma 3.8 applied to a function M we obtain that D(ϕλ,ε) → 0modularly in LM (Ω) as ε→ 0 and λ→ 1. Therefore (60)1 holds.

The convergence passage in (60)2 is a simple consequence of (58). Finally weconclude that ∇ϕλ,ε → 0 modularly in Lm(Ω) and since ϕ = 0 on ∂Ω we obtain

ϕλ,ε → 0 modularly in Lm(Ω;Rn) as ε → 0 and λ → 1. To complete the proof wefollow the case of star-shaped domains.

5. Overview of recent results. We provide an overview of recent results con-cerning applications of anisotropic Orlicz-Musielak spaces.

The system (1)-(3) with assumptions of type (S1)–(S3) has been extensivelystudied both for steady and unsteady flows. The convective term div (u⊗u) imposesalways some condition for the lower growth of the function M . We formulate it asa consecutive assumption.

(S4) Let an N -function M satisfy for some c > 0 and q ≥ 3d+2d+2 the condition

M(x, ξ) ≥ c|ξ|q (61)

and M∗ satisfies ∆2−condition.

First results in this field are related with the strict monotonicity condition in theplace of monotonicity condition (S3), namely

(S3’) For all ξ,η ∈ Rd×dsym, ξ 6= η and for a.a. x ∈ Ω

(Σ(x, ξ)−Σ(x,η)) : (ξ − η) > 0. (62)

In this case the following facts on existence of weak solutions has been proved, cf. [4,Thm. 1.1]

Theorem 5.1. Let f be in the form f = divF with F ∈ Rd×dsym and F ∈ LM∗(Ω).Moreover, let T satisfy (S1), (S2), (S3’), (S4). Then there exists a weak solutionto (1)-(3), namely v ∈ L2

div (Ω), D(v) ∈ LM (Ω) and the following is satisfied for allϕ ∈ V ∫

Ω

(v · ∇v ·ϕ+ S(x,D(v)) : D(ϕ)) dx = −〈F ,D(ϕ)〉M . (63)

An analogue for the unsteady case was proved in [3, Thm. 1.1]. We formulatethe problem and state the existence theorem below

ut + div (u⊗ u)− div Σ(x,D(u)) +∇p = f in (0, T )× Ω, (64)

divu = 0 in (0, T )× Ω, (65)

u(t, x) = 0 on (0, T )× ∂Ω, (66)

u(0, x) = u0 in Ω. (67)

Theorem 5.2. Let (S1), (S2), (S3’) and (S4) be satisfied. Given f ∈ W−1,q′(Q)and v0 ∈ L2

div (Ω) there exists a weak solution to (64) -(67), that is to say v ∈L∞(0, T ;L2

div (Ω))∩Lq(0, T ;W 1,q0 (Ω)), D(v) ∈ LM (Q) and the following is satisfied

for all ϕ ∈ D(−∞, T ;V(Ω))∫Q

(−vϕt + v · ∇v ·ϕ+ S(x,D(v)) : D(ϕ)) dxdt+

∫Ω

v0ϕdx =

∫Q

fϕdxdt. (68)


In the unsteady case the Orlicz spaces are understood as the spaces defined ontime-space cylinder Q := (0, T ) × Ω with x−dependent function M , namely by anOrlicz class we mean the set of all measurable functions ξ : (0, T )×Ω→ Rd×dsym suchthat ∫

(0,T )×Ω

M(x, ξ)dxdt <∞.

The remaining definitions of Orlicz spaces, norms, modular convergence, etc. areformulated analogously.

The existence proof in both of the mentioned papers used the Young measurestools what required strict monotonicity for showing the reduction of the Youngmeasure to the Dirac measure. This restriction was abandoned in [6] for unsteadyand in [16] for steady case, where the authors used generalization of Minty trick fornon-reflexive spaces.

An analogous formulation was considered for Stokes problem in unsteady case.We recall here from [5]

Theorem 5.3. Let condition C1. or C2. be satisfied

(C1) Ω is a bounded star-shaped domain,(C2) Ω is a bounded non-star-shaped domain and

m(r) ≤ cm((m(r))dd−1 + |r|2 + 1) (69)

for all r ∈ R+, and m satisfies ∆2-condition.

Let M be an N -function and S satisfy conditions (S1)-(S3). Then, for given u0 ∈L2

div(Ω;Rd) and f ∈ Em∗(Q;Rd) there exists u ∈ ZM0 such that∫Q

−u · ∂tϕ+ S(t, x,D(u)) : D(ϕ)dxdt =

∫Q

f ·ϕdxdt−∫

Ω

u0ϕ(0)dx (70)

for all ϕ ∈ C∞c (−∞, T ;V).

In the above unsteady setting the meaning of the space ZM0 differs from (35),namely

ZM0 =u ∈ L∞(0, T ;L2div (Ω;Rd)), D(u) ∈ LM (Q;Rd×dsym) |

∃ uj∞j=1 ⊂ C∞c ((−∞, T );V) : uj∗ u in L∞(0, T ;L2

div (Ω;Rd))

and D(uj)∗ D(u) weakly star in LM (Q;Rd×dsym).

(71)

The studies of the non-homogeneous case with the stress tensor depending on thedensity can be found in [17], where the existence of weak solutions to the followingproblem is proved

ρt + div (ρu) = 0 in (0, T )× Ω, (72)

(ρu)t + div (ρu⊗ u)− div Σ(t, x, ρ,D(u)) +∇p = ρf in (0, T )× Ω, (73)

divu = 0 in (0, T )× Ω, (74)

u(t, x) = 0 on (0, T )× ∂Ω, (75)

ρ(0, x) = ρ0, u(0, x) = u0 in Ω. (76)

We complete this section by recalling the result on existence of weak solutionsto the quasistatic system describing viscoplastic deformation behaviour of solids atsmall strain, namely the system (7)–(8). For simplicity we provide the statement ofthe theorem in the case of homogenous boundary conditions. Moreover, let F = 0,


which immediately provides that the so-called save load conditions are satisfied ([2,Def. 3.2]).

Theorem 5.4. Let (G1)-(G3) be satisfied and let M∗ satisfy ∆2−condtion. Givenεp,0 ∈ L2(Ω;PR3×3

sym) there exists a weak solution to (7)–(8).

Acknowledgments. P.G. is the coordinator and P.M. is a Ph.D student in theInternational Ph.D. Projects Programme of Foundation for Polish Science operatedwithin the Innovative Economy Operational Programme 2007-2013 (Ph.D. Pro-

gramme: Mathematical Methods in Natural Sciences). A. S-G. is supported by thegrant of National Science Centre No. 6085/B/H03/2011/40.

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