Existence of weak solutions for unsteady motions of generalized Newtonian fluids

Existence of weak solutions for unsteady motions of generalized Newtonian fluids

LARSDIENING, MICHAELR ˚UZIˇ CKA ANDˇ J ¨ORGWOLF

Abstract. We prove the existence of weak solutionsu:Q_T →Rⁿof the equa- tions of unsteady motion of an incompressible fluid with shear-dependent viscos- ity in a cylinderQ_T =×(0,T), where⊂Rⁿ denotes a bounded domain.

Under the assumption that the extra stress tensorSpossesses aq-structure with q > _n²ⁿ₊₂, we are able to construct a weak solutionu ∈ L^q(0,T;W₀^1,q())∩ C_w([0,T];L²())with divu=0. Our approach is based on the Lipschitz trun- cation method, which is new in this context.

Mathematics Subject Classification (2000):76D03(primary);35D05,46,34A34 (secondary).

1.

Introduction. Statement of the main result

Let ⊂

R

^{n, n ≥} 2, be a bounded domain. For 0 < T < ∞we set Q_T :=

×(0,T). The isothermal motion of a homogeneous, incompressible fluid through is governed by the balance equations for linear momentum and mass, which read¹

ρ ∂tu+ρdiv(u⊗u)−divS+ ∇p=ρf in Q_T, (1.1) divu=0 in Q_T, (1.2) where u = (u¹, . . . ,uⁿ)is the velocity, p the pressure,S = (S_{i j})ⁿ_i,j=1 the extra stress tensor,f =(f¹, . . . ,fⁿ)the external body force andρthe constant density.

In the following we divide the equation (1.1) by the constant densityρand relabel S/ρand p/ρagain asSandp, respectively. Moreover, for mathematical simplicity we assume thatf=divF, whereF=(F_{i j})ⁿ_i,j=1is a given tensor. The above system (1.1), (1.2) has to be completed by boundary and initial conditions and by consti- tutive assumptions for the extra stress tensor. Concerning the former we assume at

1Here, divv:=

i∂x_ivⁱ, where∂x_i = _∂^∂_xi(i=1, . . . ,n). Received June 3, 2008; accepted in revised form March 10, 2009.

the initial timet =0

u(0)=u₀ in , (1.3)

whereu₀denotes the initial velocity, which is a given vector field with divu₀ =0.

On the boundary∂we assume the following condition of adherence

u=0 on ∂×(0,T). (1.4)

If one assumes a linear dependence of the extra stress tensorSon the symmetric part of the velocity gradient D = D(u) := ¹₂(∇u+ ∇u ), then system (1.1)- (1.2) is just the classical Navier–Stokes equations. However, there exists many homogeneous, incompressible fluids that cannot be adequately described by such a simple constitutive relation. Such fluids are usually called non-Newtonian fluids.

There are many ways in which a non–Newtonian behaviour can manifest itself and we refer the reader to [4] and [26] for a general continuum mechanical background and to [3, 6, 23, 44, 50], and [35] for a detailed discussion of non-Newtonian fluids.

There is a large class of non-Newtonian fluids for which the dominant departure from a Newtonian behaviour is that in a simple shear flow the viscosity and the shear rate are not proportional. Such fluids are called fluids withshear-dependent viscosityorgeneralized Newtonian fluidsand are often modeled by the constitutive law

S=µ(D_{I I})D, (1.5)

whereD_{I I} =D:Dis the second invariant²ofDandµis the generalized viscosity function. The model (1.5) includes all power-law and Carreau-type models, which are quite popular among rheologists. Such models are used in many areas of en- gineering sciences such as chemical engineering, colloidal mechanics, glaciology, geology, and blood rheology (cf.[36] for a discussion of such models and further references). Typical examples for the constitutive relation (1.5) are

S=µ0

δ+ |D|q−2

D, S=µ0

δ²+ |D|^2q−2

2 D, (1.6)

with 1<q<∞,δ≥0 andµ0>0.

Motivated by the above discussion we impose the following conditions onS.

We say that the extra stress tensor S possesses a q-structure if there exist q ∈(1,∞)andδ≥0, such that

(I) S: Q_T ×

M

ⁿsym→

M

ⁿsym is a Carath´eodory function³;

2For two matricesA,B∈Rⁿ²byA:Bwe denote the sum

i j A_{i j}B_{i j}.

3 Mⁿsymis the vector space of all symmetricn×nmatricesξ = (ξi j)ⁿ_i,j=1. We equipMⁿsym with the scalar productξ:ηand the norm|ξ| :=(ξ:ξ)^1/2.Bya·bwe denote the usual scalar product inRⁿand by|a|we denote the Euclidean norm.

Growth condition:

(II) |S(x,t,ξ)| ≤ c₀(δ+ |ξ|)^q−2|ξ| +κ1(x,t) for all ξ ∈

M

ⁿsym, almost all (x,t)∈Q_T (c₀>0, κ1 ∈L^q(Q_T), κ1≥0);

Coercivity:

(III) S(x,t,ξ):ξ ≥ν0(δ+ |ξ|)^q−2|ξ|²−κ2(x,t) for all ξ ∈

M

ⁿsym,almost all (x,t)∈Q_T (ν0>0, κ2∈L¹(Q_T), κ2≥0);

Strict monotonicity:

(IV)

(S(x,t,ξ)−S(x,t,η)):(ξ−η) >0

∀ξ,η∈

M

ⁿsym(ξ=η), almost all(x,t)∈ Q_T.

Before we introduce the notion of a weak solution of the system (1.1)-(1.4) let us provide some notation and function spaces which will be used throughout the paper. As usual letC₀^∞()denote the space of all smooth functions having compact support in . By W^k,q(), k ∈

N

^{,1 ≤ q ≤ ∞}, we denote the usual Sobolev spaces. We define W₀^k,q()as the closure of C₀^∞()in W^k,q(). We will not distinguish between scalar-valued, vector-valued or tension-valued versions of these function spaces. The Lebesgue measure in

R

ⁿ⁺¹will be denoted by

L

n+1(·). For A⊂

R

^{n+1with 0}<

L

n+1(A) <∞andg∈L¹_loc(

R

ⁿ⁺¹)we denote byg_Athe mean value ofgover the setA,i.e.

vA :=

A

vdX = 1

L

ⁿ+1(A)

A

vdX.

Let (X, · X)be a normed space. By L^q(0,T;X)we denote the space of all Bochner measurable functionsϕ:(0,T)→ X, such that

ϕL^q(0,T;X):= ^T

0

ϕ(t)^q_Xdt _q¹

<+∞ if 1≤q <+∞, ϕL^∞(0,T;X):=ess sup

t∈(0,T)ϕ(t)X <+∞ if q= +∞.

We set

H_q = H_q():= {ϕ∈C₀^∞()| divϕ=0}^·Lq(), V_q =V_q():= {ϕ∈C₀^∞()| divϕ=0}^·Vq, where

ϕVq := D(ϕ)L^q().

In the case thatq =2 we omit the subscriptqand denoteH :=H₂andV :=V₂. It immediately follows from our definition ofW₀^1,q()and [39, 18] that there exists a positive constantγ0, such that the following Korn’s inequality holds

∇vL^q()≤γ0D(v)L^q() ∀v∈W₀^1,q(). (1.7) Definition 1.1. Assume thatSsatisfies (I) and (II) withq ∈ [_n²ⁿ₊₂,∞). LetF ∈ L¹(Q_T)andu₀ ∈ H. A vector-valued functionu∈ L^q(0,T;V_q)∩L^∞(0,T;H) is called aweak solutionto (1.1)-(1.4) if the following identity

−

QT

u·∂tϕdxdt +

QT

S(x,t,D(u)):D(ϕ)dxdt

−

Q_T(u⊗u):D(ϕ)dxdt

=

QT

F: ∇ϕdxdt+

u₀·ϕ(0)dx

(1.8)

holds for allϕ∈C^∞(Q_T)with divϕ=0 and supp(ϕ)⊂⊂× [0,T)⁴.

Remark 1.2. By virtue of Sobolev’s embedding theorem and H¨older’s inequality one easily verifies

L^q(0,T;V_q)∩L^∞(0,T;H) → L^qn+2n (Q_T). (1.9) The aim of the present paper is to prove the existence of a weak solution to the system (1.1)-(1.4) for suchqthat div(u⊗u)merely belongs to the spaceL¹(0,T; (W₀^1,∞())^∗). Our main result is the following:

Theorem 1.3 (Main theorem). Let ⊂

R

^{n, n ≥}2, be a bounded open set, and 0<T <∞. Assume thatSsatisfies(I), (II), (III), and(IV)for some q with

2n

n+2 <q <∞ (1.10)

andδ ≥0. Suppose thatu₀ ∈ H andF∈ L^q(Q_T)are given. Then there exists a weak solutionu∈L^q(0,T;V_q)∩C_w([0,T];H)to(1.1)-(1.4).

Remark 1.4.

1. The mathematical analysis of the the system (1.1)-(1.3) was initiated by La- dyzhenskaya (cf.[25, 27, 28]). She proved the existence of weak solutions in the case of Dirichlet boundary conditions (1.4) forq≥1+_n²ⁿ₊₂using the monotone operator theory combined with compactness arguments. This weak solution is unique ifq ≥ ⁿ⁺₂² andu₀∈ H. The same results have been proved in [30] for the extra stress tensorSdepending on the full velocity gradient∇urather than onD(u)only.

4HereA⊂⊂BmeansA,Bare open subsets ofRⁿ,Ais bounded andA¯ ⊂B.

2. The special caseq =2, for which (1.1)-(1.2) are just the classical Navier-Stokes equations, cannot be treated with the theory of monotone operators ifn ≥ 3.

However, the existence of weak solutions in the case of Dirichlet boundary con- ditions (1.4) is well known (cf.[29, 22]).

3. The existence of measure-valued solutions for the system(1.1)-(1.3) with Dirich- let boundary conditions (1.4) was proved forq > _n²ⁿ₊₂ in [41, 21] (cf.[33, 32]) using Ball’s theorem on Young measures. The existence of solutions satisfying a variational inequality instead of (1.8) was proved for the system (1.1)-(1.3) with Dirichlet boundary conditions (1.4) forq≥2,n≥2 in [8].

4. The lower bound for the existence of weak solutions of the system (1.1)-(1.3) with space periodic boundary conditions was extended toq> _n³ⁿ₊₂by Neˇcas and his collaborators (cf.[33, 5, 32]). The results have been obtained by using−u as a test function and deriving higher fractional differentiability in space. At the same time this technique gives the existence of a more regular solution, namely u ∈ L²(0,T;W^2,2())∩ L^∞(0,T;V_q)∩L^q(0,T;V^qn

n−2),∂tu ∈ L²(Q_T)for q ≥1+ _n²ⁿ₊₂. This solution is unique in the class of weak solutions, provided u₀ ∈ V₂. The question of the uniqueness of weak solutions for values of p below 1+_n²ⁿ₊₂is still an open problem, which is not adressed in the paper. Note that the extra stress tensorSmust satisfy slightly more restrictive assumptions than (I)-(IV), namely instead of (IV) one needs thatSis uniformly monotone, i.e.^∂S_∂^(ξ)_ξ :η⊗η≥c(1+|ξ|)^q−2|η|²holds for allξ,η∈

M

ⁿsym(cf.[45,46]). The method was extended in [34] to Dirichlet boundary condition proving for 2 ≤ q < 3,n = 3, the existence of weak solutions and some additional regularity properties for ⁹₄ ≤ q < 3,n = 3 (cf.[10, 11] for some improvements). The special caseq = 3,n =2,3, which corresponds to the Smagorinsky model in turbulence, was treated in [42], proving the existence of a unique weak solution which additionally satisfies∂tu∈ L²(0,T;V₂)∩L^∞(0,T;H), providedu₀ ∈ H∩W^2,2(). The existence of local in time strong solutions of the system (1.1)- (1.3) with space periodic boundary conditions was proved in [32] forq > ⁵₃, n =3. This result was improved in [14] toq > ⁷₅,n=3, also providing better regularity properties of the solution. The case of Dirichlet boundary conditions was considered in [7]. In that paper Sis assumed to be uniformly monotone withq ≥1. The case of small data and global in time strong solutions has been treated in [2].

5. The lower bound for the existence of weak solutions of the system (1.1)-(1.3) with space periodic boundary conditions was further extended to q > ²⁽_nⁿ₊⁺₂¹⁾ in [16] usingL^∞-test functions. The same result was obtained in [52] in the case of Dirichlet boundary conditions usingL^∞-test functions and the local pressure method.

6. The result in Theorem 1.3 is optimal in the following sense: Forq < _n²ⁿ₊₂ the

term

Q_T(u⊗u):D(ϕ)dxdt, ϕ∈L^∞(0,T;W₀^1,∞())

is not well defined using the parabolic embedding (1.9) sinceqⁿ⁺_n² < 2. Nev- ertheless it is still possible to give sense to the above integral using the informa- tionu∈L^∞(0,T;L²())coming from the time derivative only. Moreover, for q < _n²ⁿ₊₂ the space W₀^1,q()does not embed intoL²(). Thus we have that L^q(0,T;W₀^1,q()) → L^q(0,T;(W₀^1,q()_∗

). Consequently, there are prob- lems to give sense to the distributional time derivative.

The paper is organized as follows. In Section 2 we use the local pressure method to introduce the pressure for the problem (1.1), (1.2) and decompose it similar to the procedure in [52]. However, here we use only standard results for the Stokes system and the divergence equation. In Section 3 we establish the Lipschitz trunca- tion method for unsteady problems. For that we use a Whitney type decomposition to extend a function in which irregular regions have been cut off before. We also prove that this extension belongs to the spaceL^∞(0,T;W^1,∞())and that a cer- tain partial integration formula with respect to time holds true. Similar results have been proved in [24]. Here it is to the knowledge of the authors the first time that the Lipschitz truncation method is used to prove an existence result for unsteady prob- lems. For steady problems the Lipschitz truncation method was introduced in [1].

In Section 4 we approximate the problem (1.1), (1.2) appropriately. The approxi- mate problem possesses weak solutions (cf.[52]) for which we justify the limiting process using essentially the properties of the Lipschitz truncation established be- fore. This gives a proof of the main Theorem 1.3. The steady version of the problem (1.1), (1.2) was treated in [17, 13] with the help of the Lipschitz truncation method.

2.

Pressure representation

The aim of this section is to introduce the pressure for the problem (1.1)-(1.2).

We decompose the pressure corresponding to the respective terms appearing in the equation. This forms the basis for studying the local behaviour of the velocity and the pressure.

Throughout this section letG ⊂

R

ⁿbe a bounded domain with∂G∈C². Lemma 2.1. Let 1 < s < ∞. Letv^∗ ∈ (W₀^1,s(G))^∗ with v^∗,v = 0 for all v∈V_s(G). Then there exists a unique p∈L^s(G)with p_G =0, such that

v^∗,v =

G

p divvdx ∀v∈W₀^1,s(G).

Furthermore there holds

pL^s(G)≤cv^∗_(W^1,s

0 (G))^∗. (2.1)

Proof. [19, Theorems III.3.1 and III.5.2].

Arguing similar as in [52] we will introduce the pressure.

Theorem 2.2. Let u ∈ C_w([0,T];H(G)) and let H_i ∈ L^si(0,T;L^si(G)), 1<s_i <∞, i =1,2. Suppose that

− _T

0

G

u·∂tϕdxdt = _T

0

G(H₁+H₂): ∇ϕdxdt (2.2) holds for allϕ∈C₀^∞(G×(0,T))withdivϕ=0. Then there exist unique functions

p_i ∈L^si(0,T;L^si(G)),i =1,2, and p_h∈C_w([0,T];W^1,2(G)), such that

− _T

0

G

u·∂tϕdxdt= _T

0

G

(H₁+H₂): ∇ϕdxdt +

_T

0

G

(p₁+p₂)divϕ+

_T

0

G

∇p_h·∂tϕdxdt +

G

u(0)·ϕ(0)dx

(2.3)

for allϕ ∈ C^∞(G×(0,T))withsupp(ϕ) ⊂⊂ G × [0,T). In addition, we have

−p_h=0, p_h(0)=0and the apriori estimates

p_iL^si(G×(0,T))≤c_iH_iL^si(G×(0,T)), i =1,2, (2.4) p_h(t)_W1,2(G) ≤c_hu(t)−u(0)_L²_(G), t ∈ [0,T], (2.5) with constants c_i depending only on n,G, and s_i,i = 1,2, and a constant c_h depending only on n and G.

Proof. Arguing as in the proof of [52, Theorem 2.6] one gets the existence of a pressure p˜ ∈C_w([0,T];L¹(G)), such that

G

(u(t)−u(0))ψdx =

G

(H₁(t)+H₂(t)): ∇ψdx +

Gp˜(t)divψdx, ∀ψ ∈W₀^1,2(G),

(2.6)

for all 0≤t ≤T, where H_i(t)=

_t

0

H_i(τ)dτ, 0<t <T, i =1,2.

Without loss of generality we may assume that p˜(t)G =0 for all 0≤t ≤T. Note that in this case the pressure p˜is unique.

Next, letv_i(t)∈V_si(G),0≤t ≤T, denote the unique solution to the Stokes problem

−v_i(t)+ ∇πi(t)= −divH_i(t) in G, divv_i(t)=0 in G, v_i(t)=0 on ∂G,

(2.7)

fori =1,2. According to Lemma 2.1 we haveπi(t)∈ L^si(G)with(πi(t))G =0.

Replacing in (2.7) t byt +h,0 < h ≤ T −h, and taking the difference of both equations we get in view of Lemma 2.1 the estimate

πi(t+h)−πi(t)L^si(G)≤c

∇(v_i(t+h)−v_i(t))L^si(G)+H_i(t+h)−H_i(t)L^si(G) . From [20, Theorem 2.1], and [19, Theorem IV.6.1] we obtain

∇(v_i(t +h)−v_i(t))L^si(G) ≤cH_i(t +h)−H_i(t)L^si(G), which leads to

πi(t +h)−πi(t)^s_Lⁱsi(G)≤c _t+h

t

H_i(τ)dτ ^si

L^si(G).

Dividing both sides byh^si, integrating over the interval(0,T−h), applying H¨older’s inequality and Fubini’s theorem yields

_T−h

0

πi(t+h)−πi(t)^s_Lⁱsi(G)

h^si dt ≤cH_i^s_Lⁱsi(G×(0,T)).

This shows thatπi ∈ W^1,si(0,T;L^si(G)). Set p_i := ∂tπi. In the estimate above passing to the limit ash→0 implies (2.4)

Next, letv_h(t)∈V₂(G)∩W^2,2(G),0≤t ≤T, denote the unique solution to the Stokes problem

−v_h(t)+ ∇p_h(t)= −u(t)+u(0) in G, divv_h(t)=0, in G, v_h(t)=0 on ∂G.

(2.8)

Appealing to [19, Theorem IV.6.1] one finds p_h(t)∈W^1,2(G)with(p_h(t))G =0.

Moreover, we have

p_h(t)W^1,2(G) ≤cu(t)−u(0)L²(G). Whence, (2.5).

Definev(t):=v₁(t)+v₂(t)+v_h(t). From (2.6) follows

−divH₂(t)−divH₂(t)−u(t)+u(0)= ∇ ˜p(t).

Thus, taking the sum of (2.7) and (2.8) leads to

−v(t)+ ∇(π1(t)+π2(t)+ p_h(t)− ˜p(t))=0 in G, divv(t)=0, in G, v(t)=0 on ∂G,

(2.9) which impliesv=0and p˜(t)=π1(t)+π2(t)+ p_h(t). Thus, insertingψ(·,t)=

∂tϕ(·,t),t ∈(0,T), into (2.6), integrating this identity over(0,T), and applying integration by parts yields (2.3).

3.

Parabolic Lipschitz truncation

In this section we will establish a Lipschitz truncation method for unsteady prob- lems. The idea is to regularize a given function by cutting off regions of irregularity and then to extend this restricted function by a Whitney type extension to the whole domain again. In a different context this method has been used in [24]. The ba- sis of this approach is a covering lemma of Whitney type, which is well-known (cf.[48, Chapter 6], [12, Chapter1], [9, Theorem 3.2]). Here we use these general results for the Euclidean space

R

ⁿ⁺¹with an anisotropic parabolic metric. The typ- ical points(x,t)and(y,s)of

R

ⁿ⁺¹are shortly denoted by X andY, respectively.

Givenα >0 we equip

R

ⁿ⁺¹with the following anisotropic (scaled parabolic) met- ric

d_α(X,Y):=max{|x−y|,|α⁻¹(s−t)|^1/2}, X,Y ∈

R

^n+1.

ForX₀∈

R

^{n+1andr >}0 we define ballsQ^α_r(X₀):= {X∈

R

^n+1|dα(X,X₀) <r}, which are equal to the cylindrical sets B_r(x₀)×(t₀−αr²,t₀+αr²). One easily checks that for allX ∈

R

^{n+1, 0<r <∞andα >0 holds}

L

ⁿ+1(Q^α_r(X))≤2ⁿ⁺²

L

ⁿ+1(Q^αr 2(X)).

This implies that(

R

^n+1,dα)satisfies the following homogeneity property:

There exists a number Nˆ ∈

N

, depending only on n, such that, each cylinder Q_r(X₀)contains no more thanN pointsˆ {X_i}with d_α(X_i,X_j)≥ ^r₄ for i = j . Thus we have (cf.[48, Chapter 6], [12, Chapter 1], [9, 47, Theorem 3.2]) the fol- lowing result:

Lemma 3.1 (Whitney type covering). Let E be a non-empty, open, bounded set of

R

ⁿ⁺¹equipped with the metric d_α,α >0. Then there exists a covering{Q^α_i} = {Q^α_rα

i(X_i^α)}of E, such that⁵

(W1)

i1

2Q^α_i = E,

(W2) for all i ∈

N

^{we have8Qα}i ⊂ E and16Q_i^α∩(

R

^{n+1\E)= ∅,} (W3) if Q^α_i ∩Q^α_j = ∅then ¹₂r^α_j ≤r_i^α<2r^α_j,

(W4) each X ∈ E belongs to at most120ⁿ⁺²of the sets4Q^α_i. We have denoted byγQ_i^α(γ > 0)the ball Q^α_γrα

i(X_i^α). Note that this scaling de- pends onα.

For the readers convenience an elementary proof can be found in the Appendix C of this paper.

5 ByγQ^α_i (γ >0)we denote the ballQ^α_γrα

i (X^α_i).

Remark 3.2.

1. The covering{Q^α_i}ofEhas the following additional properties:

(W5)

j

L

n+1(4Q^α_j)≤120ⁿ⁺²

L

n+1(E).

Property (W4) of the above lemma can be written as

iχQ^α_i ≤120ⁿ⁺², which implies

j

L

ⁿ+1(4Q^α_j)=

j

E

χ4Q^α_jdX ≤ 120ⁿ⁺²

L

ⁿ+1(E).

2. Define A_i :=

j ∈

N

²₃^Qαj ∩ ²₃Q^α_i = ∅

(i = 1,2, . . .). Note that from Lemma C.1 follows that #A_i ≤120ⁿ⁺². Moreover, we have

(W6) Q^α_j ⊂4Q^α_i ⊂ E for all j ∈ A_i.

Indeed, if ²₃Q^α_j ∩ ²₃Q^α_i = ∅, then we haver^α_j ≤2r_i^α due to (W3). ForY ∈ Q^α_j we estimate

d_α(X_i^α,Y)≤d_α(X^α_i,X)+d_α(X,X^α_j)+d_α(X^α_j,Y)

< 2 3r_i^α+4

3r_i^α+2r_i^α=4r_i^α. Consequently,Y ∈4Q^α_i.

Next, let us introduce the notion of Lipschitz continuous functions with respect to the metricd_α.

Definition 3.3. For a given Lebesgue measurable set A ⊂

R

^{n+1 byC}d⁰_α^,1(A)we denote the space of all essentially bounded functionsu: A→

R

, satisfying

Lip_dα;A(u) <∞, where

Lip_dα;A(u):= sup

X,Y∈A X=Y

|u(X)−u(Y)|

d_α(X,Y) .

Partition of unity. LetEbe a non-empty, bounded, open subset of

R

^n+1equipped with the metricd_α. Let{Q^α_i}denote a Whitney type covering ofEfrom Lemma 3.1.

Letηi ∈C₀^∞(

R

ⁿ⁺¹⁾denote cut-off functions with 0 ≤ηi ≤1 in

R

^{n+1,ηi = 0 in}

R

^n+1\23Q^α_i,ηi =1 on ¹₂Q^α_i , such that

Lip_dα(ηi)+ |α ∂tηi|¹² ≤ c_n(r_i^α)⁻¹ in

R

^{n+1 (i ∈}

N

^{). (3.1)}

Recalling the definition of A_i (cf.Remark 3.2) we have

j

ηj =

j∈A_i

ηj in Q^α_i, and

jηj ∈C^∞(

R

ⁿ⁺¹⁾. Next, we set ψi(X):=ηi(X)

j

ηj(X), X ∈E (i ∈

N

^), which is well defined, since by (W1) there holds

jηj ≥1 inE. Clearly, we have thatψi ∈C^∞(

R

^{n+1)satisfyψi ≡0 in}

R

^n+1\23Q^α_i . Moreover, we have

j

ψj =

j∈A_i

ψj =1 inQ^α_i. (3.2)

Properties (W1), (W2), (W3), (W4), and (3.1) yield

|ψi(X)−ψi(Y)| ≤ |ηi(X)−ηi(Y)| +

j∈A_i

|ηj(X)−ηj(Y)|

≤c_n

(r_i^α)⁻¹+

j∈Ai

(r^α_j)⁻¹

d_α(X,Y) (3.3)

≤c_n(1+2·120ⁿ⁺²)(r_i^α)⁻¹d_α(X,Y).

Similarly, one proves that

|∂tψi| ≤ |∂tηi| +

j∈A_i

|∂tηj| ≤c²_n(1+4·120ⁿ⁺²) α⁻¹(r_i^α)⁻² in

R

^n+1. Thus,

Lip_dα(ψi)+ |α ∂tψi|¹² ≤ c_n(1+4·120ⁿ⁺²) (r_i^α)⁻¹ in

R

^{n+1. (3.4)} Definition of the truncation operator. LetG ⊂

R

ⁿbe a non-empty, open, bounded set and 0 < T <∞. For a non-empty, open set E ⊂ G×(0,T)let{Q^α_i}be the corresponding Whitney covering from Lemma 3.1 for the metric d_α, and{ψi}the associated partition of unity. Foru∈ L¹_loc(G×(0,T))we define

(

T

E^αu)(X) :=









u(X) if X ∈(G×(0,T))\E, ∞

i=1

ψi(X)u_Qα

i if X ∈E.

From the construction of

T

_E^αu it is clear that the restriction of

T

_E^{αu upon E is} smooth.

Next, we prove several properties of the operator

T

E^αu, which will be used in what follows.

Lemma 3.4. Let u∈L¹_loc(G×(0,T)). Then there holds

|

T

E^αu| ≤10ⁿ⁺²

M

^∗(|u|) a.e. in G×(0,T), (3.5) where c=const>0.

Proof. If X ∈(G ×(0,T))\E, then|T_E^αu(X)| = |u(X)| ≤

M

^{∗(|u|)(X)by the} properties of

M

^{∗ (cf.} Appendix A). Now, let X ∈ E. Then by (W1) we have

X∈Q^α_i for somei. By the definition of

T

E^αuand of the setA_i it follows that (

T

E^αu)(X)=

j∈A_i

u_Q^α

jψj(X).

Taking into account (W6) and (W3) implies

|

T

E^αu(X)| ≤

j∈A_i

|u_Qα

j|ψj(X)≤8ⁿ⁺²

4Q^α_i

|u|dY. (3.6)

Since 4Q^α_i ⊂Q^α_5rα

i(X)we obtain

|

T

E^αu(X)| ≤10ⁿ⁺²

Q^α_5rα

i(X)|u|dY ≤10ⁿ⁺²

M

^∗(|u|)(X).

The proof of the lemma is completed.

Lemma 3.5. For every1≤ p≤ ∞there exists a constant c such that

T

E^αuL^p(G×(0,T))≤cuL^p(G×(0,T)) ∀u∈L^p(G×(0,T)), (3.7) where c depends only n.

Proof. By the definition of

T

E^α and the properties (W1), (W6), and (W5) of the Whitney covering we have

G×(0,T)|(

T

E^αu)(X)|dX =

G×(0,T)\E|u(X)|dX+

E|(

T

Eu)(X)|dX

≤ u_L¹_(G×(0,T))+

i

Q_i^α

|(

T

^Eu)(X)|dX

≤ uL¹(G×(0,T))+

i

Q_i^α

j∈A_i

Q_j|u(Y)|dY

dX

≤ u_L¹_(G×(0,T))+2ⁿ⁺²

i

j∈A_i

4Q^α_i

|u(X)|dX

≤

1+240ⁿ⁺²

uL¹(G×(0,T)).

From the definition of

T

E^αfollows

T

E^αuL^∞(G×(0,T))≤ uL^∞(G×(0,T)). The assertion thus follows from interpolation also for 1< p<∞.

Lemma 3.6. Let u∈L¹_loc(G×(0,T)). Then we have for all Y,Z ∈Q^α_i, i ∈

N

|

T

E^αu(Y)−

T

E^αu(Z)| ≤c(r_i^α)⁻¹d_α(Y,Z)

4Q^α_i

|u−u_4Q^α

i|dX, (3.8) where c depends only on n.

Proof. By the definition of

T

E^αone easily calculates for allY,Z ∈Q^α_i

T

E^αu(Y)−

T

E^αu(Z)=

j∈A_i

u_Q^α

j(ψj(Y)−ψj(Z))

=

j∈A_i

(u_Qα j −u_Qα

i)(ψj(Y)−ψj(Z)),

where we used (3.2). Using Jensen’s inequality along with property (W6) one gets

|u_Qα j −u_Qα

i| ≤4ⁿ⁺²·2

4Q^α_i

|u−u_4Qα

i|dX ∀j ∈ A_i. (3.9) The last two estimates together with the Lipschitz bound (3.4) and the properties (W3) and (W4) of the Whitney covering imply (3.8).

Lemma 3.7. Let u∈L¹_loc(G×(0,T)). Then for all Y ∈Q_i^α, i ∈

N

|

T

E^αu(Y)−u(Y)| ≤c

4Q^α_i

|u(X)−u(Y)|dX, (3.10) where c depends only on n.

Proof. LetY ∈Q^α_i,i ∈

N

. The definition of

T

_E^αand (W6) yield

|

T

E^αu(Y)−u(Y)| ≤

j∈A_i

ψj(Y)|u_Q^α

j −u(Y)| ≤c

4Q^α_i |u(X)−u(Y)|dX. Whence (3.10).

Note that from Lemma 3.7 one obtains the following estimate fori ∈

N

Q^α_i

|

T

E^αu−u|dY ≤c

4Q^α_i

|u−u_4Qα

i|dY, (3.11)

wherecdepends only onn.