Coupling the Stokes and Navier-Stokes equations with two scalar nonlinear parabolic equations

Mathematical Modelling and Numerical Analysis M2AN, Vol. 33, No1, 1999, p. 157–167 Mod´elisation Math´ematique et Analyse Num´erique COUPLING THE STOKES AND NAVIER–STOKES EQUATIONS WITH TWO SCALAR NONLINEAR PARABOLIC EQUATIONS∗ Macarena G´ omez M´ armol1and Francisco Orteg´ on Gallego2 Abstract. This work deals with a system of nonlinear parabolic equations arising in turbulence modelling. The unknowns are the Ncomponents of the velocity field ucoupled with two scalar quantities θand ϕ. The system presents nonlinear turbulent viscosity A(θ, ϕ) and nonlinear source terms of the form θ2|∇u|2and θϕ|∇u|2lying in L1. Some existence results are shown in this paper, including L∞-estimates and positivity for both θand ϕ. R´esum´e.Nous´etudions un syst`eme non-lin´eaire d’´equations du type parabolique provenant de la mod´elisation de la turbulence. Les inconnues sont les Ncomposantes du champ des vitesses ucoupl´ees avec deux grandeurs scalaires θet ϕ.Cesyst`eme pr´esente un terme de diffusion non-lin´eaire sous forme matricielle A(θ,ϕ)etlestermessourcesnon-lin´eaires θ2|∇u|2et θϕ|∇u|2appartenant `a L1. On d´emontre alors quelques r´esultats d’existence de solutions, ainsi que des estimations dans L∞et positivit´epourθet ϕ. AMS Subject Classification. 35K45, 35K60, 35Q35. Received: December 3, 1996. Revised: November 18, 1997. 1. Introduction In this paper, we state some existence results of a weak solution to the nonlinear parabolic system (S)                              ∂u ∂t −∇·(A(θ,ϕ)∇u)+∇p=f, ∇·u=0,in Q ∂θ ∂t −∇·(A(θ, ϕ)∇θ)=1−θ 2 |∇u|2,in Q ∂ϕ ∂t −∇·(A(θ, ϕ)∇ϕ)=−ϕ θ|∇u|2+1 θ+r!,in Q u(x, 0) = u0(x),θ(x, 0) = θ0(x),ϕ(x, 0) = ϕ0(x),in Ω u(x, t)=0,θ(x, t)=a, ϕ(x, t)=b, on ∂Ω×(0,T) where Q=Ω×(0,T), Ω ⊂RNis a bounded domain with Lipschitz boundary ∂Ω, N≥2, T>0, aand bare non-negative constants and r>0 is a small parameter. Positivity of both θand ϕis also shown in this work. System (S) derives from the so-called k-turbulence model (see [8]). Here, u=(u 1 ,...,u N) 0stands for the mean velocity field (the symbol 0here means vector transposition), pis the mean pressure and f=(f 1 ,...,f N) 0 Key words and phrases. Nonlinear parabolic systems, Stokes and Navier-Stokes equations. ∗Research partially supported by DGICYT grant PB93–1196. 1Departamento de Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla, Spain. e-mail: [email protected] 2Departamento de Matem´aticas, Universidad de C´adiz, Spain. e-mail: [email protected]s c EDP Sciences, SMAI 1999 158 M. G´ OMEZ M´ ARMOL AND F. ORTEG ´ ON GALLEGO is a given function describing a distributed force field over Q. The magnitudes θand ϕare obtained from both, the mean turbulent kinetic energy, k, and the mean turbulent viscous dissipation, . Indeed, θ=k−1, ϕ=2k−3and the θand ϕequations are both deduced from those of kand (see [6–8] for details). The resulting viscosity is of the form A(θ, ϕ)=(ν 0+1 θϕ)I,ν0>0 being a constant value. In order to avoid a zero denominator we must change this expression in some way; for example, we may take a perturbation of the form A(θ,ϕ)=(ν 0+1 |θϕ|+r)I. Here we shall consider a general matrix expression for A(θ,ϕ). Also, we point outthattheterm− ϕ θappearing in the original modelling of the equation for ϕhas been changed to −ϕ |θ|+rfor the same reason as above. System (S) lacks transport terms; we have first considered these equations (without transport terms) in order to establish some existence results, independently on the space dimension. The resolution of the full model (C) is discussed in Section 6 below. As we are just considering system (S) from the mathematical standpoint, we may assume that the involved physical quantities are dimensionless and that the physical constants which actually appear are taken to be equal to one. The interest in introducing the θ-ϕapproach in turbulence modelling is that (S) may be considered as a stabilization of the k-model in the sense that we can state some results concerning the existence, regularity, positivity and L∞-estimates of certain solutions of (S). As one can readily see, system (S) presents some mathematical difficulties, namely 1. nonlinear source terms and nonlinear viscosity; 2. the whole system is coupled through these nonlinear terms; 3. the regularity result for the Stokes (or even the Navier-Stokes equations) yields a velocity field u∈ L2([0,T]; H1 0(Ω)N) if, for example, f∈L2([0,T]; H−1(Ω)N). So both the θand ϕequations contain nonlinear source terms, namely, θ2|∇u|2and θϕ|∇u|2, which lie in L1(Q)ifθand ϕbelong to, say, L∞(Q). Some partial results concerning the existence and positivity of solutions (θ,ϕ) may be found in [6,7]. In these papers, it is assumed that u∈L∞([0,T]; W1,∞(Ω)) is a given data and verifies the rather restrictive condition ess infQ|∇u|>0. The goal of this work is to state the existence of a weak solution to the whole system (S), such that θ,ϕ ∈L∞(Q) when the initial data lie in L∞(Ω), together with θ≥0andϕ≥0 almost everywhere in Q (Theorems 1 and 2). The proofs are based in some different standard techniques, including truncation ( [2]) and a priori estimates. But from the physical (and numerical) standpoint, the results about the L∞(Q) regularity and positivity of both, θand ϕ, are very important since, in general, there is an enormous lack of this kind of results in turbulence modelling (e.g. in the k-model). The resolution of a system like (S), or (C) in section 6, may be regarded, from the mathematical point of view, as a pioneering work in turbulence modelling, since it is the first time that an existence result is shown for a two equations turbulence model. 2. Functional spaces and weak formulation The following notation and functional spaces will be adopted throughout this paper: Ω⊂RNis a bounded domain with Lispchitz boundary ∂Ω; N≥2 is the space dimension, and Qis the cylinder Ω×(0,T), with T>0 the final time. D(Ω) def =spaceofC ∞functions with compact support in Ω. For an integer m≥0and1≤p≤+∞we define Wm,p(Ω) def =v∈Lp(Ω) /∂|α|v ∂xα1 1···∂xαN N ∈Lp(Ω), COUPLING THE STOCKES AND NAVIER–STOKES EQUATIONS... 159 ∀α=(α 1 ,...,α N)∈Z N +,|α|=α 1+···+α N≤m . Also, for s>0, and 1 ≤p≤+∞one can introduce the space Ws,p(Ω) by interpolation (see [1]). Ws,p 0(Ω) def = closure of D(Ω) with the standard norm of Ws,p(Ω), W−s,p0(Ω) def = dual space of Ws,p 0(Ω),1 p+1 p0=1,1≤p<∞ H 1 (Ω) def =W1,2(Ω),H 1 0 (Ω) def =W1,2 0(Ω),H −1 (Ω) def =W−1,2(Ω), ∇·vdef =∂v1 ∂x1 +···+∂vN ∂xN ,divergence of v=(v 1 ,...,v N) 0, Vdef =v∈H1 0(Ω)N/∇·v=0 ,V 0 def = dual space of V. Let n=n(x) be the outward unitary normal vector to ∂Ωinx∈∂Ω, then we define Hdef =v∈L2(Ω)N/∇·v=0inΩ,v·n=0on∂Ω , For a Banach space Xand 1 ≤p≤+∞,wedenotebyL p (X)thespaceL p ([0,T]; X), that is, the set of (equivalence class of) functions f:[0,T]7→ Xmeasurables and such that t∈[0,T]7→ kf(t)kXis in Lp(0,T). Forafunctionf∈L p (X) we put kfkLp(X) def = ZT 0 kf(t)kp Xdt!1/p ,1≤p<+∞,kfk L ∞ (X) def = ess sup t∈[0,T] kf(t)kX. It is well-known that Lp(X),k·k L p(X) is a Banach space. Notice that by Fubini’s theorem we can identify the space Lp(Lp(Ω)) with Lp(Q) (the reader is refer to [4] for more properties about these spaces). W1 def =v∈L2(V)/dv dt ∈L2(V0), W(q) 2 def =v∈L2(H1 0(Ω)) /dv dt ∈L2(W−1,q(Ω)),1≤q<+∞. W (q) 3 def =v∈Lq(W1,q 0(Ω)) /dv dt ∈L1(W−1,q(Ω)),1≤q<+∞. In these definitions, all derivatives are assumed to be taken in the sens of distributions. It is well known that all these spaces are Banach spaces provided with their standard norms. Moreover, V,Hand W(2) 2are in fact 160 M. G´ OMEZ M´ ARMOL AND F. ORTEG ´ ON GALLEGO Hilbert spaces. Also, the following imbeddings hold algebraically and topologically    W1,→C(H); W(q) 2,→C(L2(Ω)),∀q≤2N N−2<∞; W(q) 3,→C(W−1,q(Ω)),∀q<∞; (1) where we are denoting C(X)=C([0,T]; X), Xa Banach space, the space of continuous functions v:[0,T]7→ X, provided with the norm kvkC(X)=max [0,T]kv(t)kX. We will make use of the following compactness lemma (see [9]): Lemma 1. Let X,Band Ybe three Banach spaces such that X,→B,→Y, every imbedding being continuous and the inclusion X,→Bcompact. For 1≤p, q < +∞, let Wbe the Banach space defined as W= v∈Lp(X)/dv dt ∈Lq(Y). Then, the inclusion W,→Lp(B)holds and is compact. Finally, we will use the abbreviation ‘a.e.’ meaning ‘almost everywhere’. Now, we may introduce the weak formulation of system (S). Let the initial data u0,θ0and ϕ0(in Ω), the boundary constants aand b, and the forcing term f(in Q) be given. Then, we search for u=(u 1 ,...,u N) 0,θ and ϕ, in certain suitable spaces, such that: ZT 0du dt ,v+ZQ A(θ, ϕ)∇u∇v=ZT 0 hf,vi,∀v∈L2(V),(2) ZT 0dθ dt ,Ψ+ZQ A(θ,ϕ)∇θ∇Ψ=ZQ1−θ 2 |∇u|2Ψ,∀Ψ∈D(Q),(3) ZT 0dϕ dt ,Ψ+ZQ A(θ,ϕ)∇ϕ∇Ψ=−ZQ ϕθ|∇u|2+1 |θ|+rΨ,∀Ψ∈D(Q),(4) u(x, 0) = u0,θ(x, 0) = θ0(x),ϕ(x, 0) = ϕ0(x),in Ω,(5) θ(x, t)=a, ϕ(x, t)=b, on ∂Ω×(0,T). (6) Remarks 1. In (2)–(4), the symbol h·,·i stands for certain duality products specified below. 2. As usual, the pressure pdoes not appear in the weak formulation (2)–(6) and it can be retrieved by the classical de Rham’s argument (see [10]). 3. We will show below that, when θ0,ϕ 0∈L ∞(Ω) (resp. θ0,ϕ 0∈L 1(Ω)) then, θ−aand ϕ−bbelong to some space W(q) 2(resp. W(q) 3)forsomeq∈(1,2). In particular, thanks to the imbeddings given in (1), the initial conditions (5) will make sense at least in L2(Ω)Nfor u,andinL 2 (Ω) (resp. W−1,q(Ω)) for θ and ϕ. 3. The main results We will consider the following hypotheses: (H1) f∈L2(H−1(Ω)N), u0∈H; COUPLING THE STOCKES AND NAVIER–STOKES EQUATIONS... 161 (H2) a≥0, b≥0 are real constants; (H3) θ0≥0, ϕ0≥0; (H4) A:R×R7→ RN×Nis continuous and there exists a constant α>0 such that A(s1,s 2)ξξ ≥α|ξ|2, for all s1,s2∈R,ξ∈RN. The main results now follow. The difference between them is the assumed regularity for the initial data θ0 and ϕ0. Theorem 1. (L∞initial data) Under hypotheses (H1−H4),ifθ 0 ,ϕ 0∈L ∞(Ω), then there exists (u, θ, ϕ) solution to (2)-(6), such that u∈W1,θ−a, ϕ −b∈W(q) 2∩L∞(Q),∀q∈1,N N−1,(7) 0≤θ(x, t)≤max{kθ0kL∞(Ω),a}+t, a.e. in Q,(8) 0≤ϕ(x, t)≤max{kϕ0kL∞(Ω),b},a.e. in Q.(9) Theorem 2. (L1initial data) Let assume hypotheses (H1−H4) and also (H5) there exists β>0such that kA(s1,s 2)k≤βfor all s1,s2∈R,k·k being some matrix norm. If θ0,ϕ 0∈L 1(Ω), then there exists (u, θ, ϕ)solution to (2)-(6), such that u∈W1,θ−a, ϕ −b∈W(q) 3∩L∞(L1(Ω)),∀q∈1,N+2 N+1,(10) θ≥0,ϕ≥0,a.e. in Q,(11) θ2|∇u|2∈L1(Q),ϕθ|∇u|2∈L1(Q).(12) Remarks 1. In both theorems, due to (1), the initial conditions given in (5) make sens at least in L2(Ω)Nfor uand in W−1,q(Ω) for θand ϕ. 2. In Section 6 we give two existence results concerning the full system in 2D with transport terms. 3. We may assume a more general version of the diffusion matrix A. Indeed, Theorem 1 also holds if (H4) is changed to: A:Q×R×R7→ RN×Nis a Caratheodory matrix function and there exist a constant α>0 and a non-decreasing function d:R+7→ R+such that (H40)α|ξ|2≤A(x, t, s1,s 2)ξξ ≤d(|s1|+|s2|)|ξ|2,a.e. in Q, forall s1,s 2∈R. Also, instead of just A, we may think of three different viscosities Au,Aθand Aϕfor the respective equations of u,θand ϕ. If these three matrix functions verify (H40) then Theorem 1 still holds, and if d is constant, then Theorem 2 also holds. 162 M. G´ OMEZ M´ ARMOL AND F. ORTEG ´ ON GALLEGO The next sections are devoted to develop the proofs of Theorems 1 and 2. The basic ideas of the proof are summarized here: 1. To divide the two main difficulties, namely (i) the presence of the terms θ2|∇u|2and θϕ|∇u|2,whichonly belong to L1; and (ii) the coupling of the N+ 2 equations (2)–(4) through nonlinearities. 2. To avoid L1-terms, we use a regularization method based on truncations. This will lead to approximated systems. 3. Finally, the same arguments due to Boccardo and Gallou¨et [2] may be applied in our context, and this yields the necessary estimates for the approximated solutions to pass to the limit. 4. Proof of Theorem 1 4.1. Setting of the approximated problem (PM) For every M>0, we define the truncation function at height M,TM,as T M (s)=sif |s|≤M, Msignsif |s|>M,sign s=s/|s|if s6=0, 0ifs=0. (13) Then, we consider the approximated problem (PM) consisting in finding uM,θMand ϕMsuch that (PM)                                                  uM∈W1,θM−a,ϕM−b∈W(2) 2∩L∞(Q) and, ZT 0duM dt ,v+ZQ A(θ M,ϕ M)∇u M∇v=ZT 0 hf,vi,∀v∈L2(V), ZT 0dθM dt ,Ψ+ZQ A(θM,ϕ M)∇θ M∇Ψ =ZQ1−θ M|θ M|T M|∇uM|2Ψ,∀Ψ∈L2(H1 0(Ω)), ZT 0dϕM dt ,Ψ+ZQ A(θM,ϕ M)∇ϕ M∇Ψ =−ZQ ϕ M|θ M|T M|∇uM|2+1 |θM|+rΨ,∀Ψ∈L2(H1 0(Ω)), uM(0) = u0,θ M (0) = θ0,ϕ M (0) = ϕ0. The existence of solution to (PM) is guaranteed by the next Lemma 2. Under hypotheses (H1−H4), there exists a solution (uM,θ M,ϕ M)of problem (PM)such that 0≤θM(x, t)≤max{a, kθ0kL∞(Ω)}+t, a.e. in Ω,∀t∈[0,T] (14) 0≤ϕM(x, t)≤max{b, kϕ0kL∞(Ω)},a.e. in Ω,∀t∈[0,T].(15) The existence of (uM,θ M,ϕ M)solutionto(P M) and verifying the L∞estimates (14)–(15) can be found in [5]. Basically, it is obtained by an application of Schauder’s fix point theorem. 4.2. Estimates for (uM,θM,ϕ M ) Estimates for (uM) The classical estimates for the Stokes (or Navier-Stokes) equations follows inmediately by taking v= uMas a test function in the equation verified by uMin (PM). This leads to the existence of a constant COUPLING THE STOCKES AND NAVIER–STOKES EQUATIONS... 163 C1=C1(α, kfkL2(H−1(Ω),T,ku 0k L 2(Ω)) such that kuMkL∞(L2(Ω)N)≤C1,kuMkL2(V)≤C1.(16) Estimates for θMand ϕM First of all, Lemma 2 gives uniform L∞bounds for (θM)and(ϕ M). Now, taking φ=θM−aand φ=ϕM−b in the respective equations for θMand ϕM, yield kθMkL2(H1(Ω)) ≤C2,kϕMkL2(H1(Ω)) ≤C3(17) where C2=C2(C1,a,kθ 0k L ∞ (Ω))andC 3=C 3 (C 2 ,b,kϕ 0k L ∞ (Ω)). Time derivatives estimates From the estimates derived in the preceeding paragraphs, we may deduce that duM dt is bounded in L2(H−1(Ω)N), whereas dθM dt and dϕM dt are bounded in L2(H−1(Ω)) + L2(L1(Ω)). On the other hand, we know by Sobolev’s imbedding (see [3]) that L1(Ω) ,→W−1,q(Ω) whenever q< N N−1 , which implies the inclusion L2(L1(Ω)) ,→L2(W−1,q(Ω)). Finally, L2(H−1(Ω)) + L2(L1(Ω)) ,→L2(W−1,q(Ω)),∀q< N N−1 and we may conclude that dθM dt ,dϕM dt are bounded in L2(W−1,q(Ω)), ∀q< N N−1 · Now, the inclusions X=H1(Ω) ,→B=L2(Ω) ,→Y=W−1,q(Ω) verify the hypotheses of Lemma 1, and consequently the imbedding W=W(q) 2,→L2(Q) is compact. In particular, this implies that (θM)and(ϕ M ) are relative compact in L2(Q). 4.3. Passing to the limit in (PM) From the estimates obtained in the last section, we deduce that from (uM), (θM)and(ϕ M ) we may extract subsequences, denoted in the same way, such that uM→u∈W1           in L2(H1 0(Ω)N)-weakly, in L2(Q)N-strongly, a.e. in Q, in L∞(L2(Ω)N)-weak-∗, θM→θ, ϕM→ϕ, θ−a, ϕ −b∈W(q) 2,∀q< N N−1 ,            in L2(H1(Ω))-weakly, in L2(Q)-strongly, a.e. in Q, in L∞(Q)-weak-∗, duM dt →du dt ,in L2(H−1(Ω))-weakly 164 M. G´ OMEZ M´ ARMOL AND F. ORTEG ´ ON GALLEGO dθM dt →dθ dt ,dϕM dt →dϕ dt ,in L2(W−1,q(Ω))-weakly,∀q< N N−1 · These convergences are enough to pass to the limit in the velocity equation of (PM), obtaining then (2). But passing to the limit in the θMand ϕMequations cannot be done directly unless the convergence of (uM)to uis in L2(H1 0(Ω)N)-strongly. Fortunately, this is the case as it can be shown by taking v=uMin (PM)and passing to the limit. Consequently, |∇uM|2→|∇u| 2in L1(Q)-strongly and, without loss of generality, we may assume that the convergence also holds almost eveywhere in Q. Now, we may pass to the limit in the θMand ϕMequations and deduce (3) and (4), respectively. On the other hand, using (1), we readily obtain (5). This ends the proof of Theorem 1.  5. Proof of Theorem 2 Now, we assume hypothesis (H5) and that θ0,ϕ 0∈L 1(Ω). For M>0, we consider the new approximated problem (QM) (QM)                                                  uM∈W1,θM−a,ϕM−b∈W(2) 2∩L∞(Q)andsuchthat ZT 0duM dt ,v+ZQ A(θ M,ϕ M)∇u M∇v=ZT 0 hf,vi,∀v∈L2(V), ZT 0dθM dt ,Ψ+ZQ A(θM,ϕ M)∇θ M∇Ψ =ZQ1−θ2 MT M(|∇uM|2)Ψ,∀Ψ∈L2(H1 0(Ω)), ZT 0dϕM dt ,Ψ+ZQ A(θM,ϕ M)∇ϕ M∇Ψ =−ZQ ϕ Mθ MT M(|∇uM|2)+ 1 θ M+rΨ,∀Ψ∈L 2 (H1 0(Ω)), uM(0) = u0,θ M (0) = TM(θ0),ϕ M (0) = TM(ϕ0). Since TM(θ0),T M(ϕ 0)∈L ∞(Ω) we can apply Lemma 2 and deduce that (QM) admits a solution (uM,θ M,ϕ M). It is straightforward that the estimates for (uM) given in (16) still hold. For (θM)and(ϕ M )wehavethe Lemma 3. Let ˜ θM=θM−aand ˜ϕM=ϕM−b. Then, there exists a constant C>0(depending on (kfkL2(H−1),α,a,b,T,kθ 0k L 1(Ω),kϕ0kL1(Ω))) such that for all M>0 ZΩ | ˜ θ M (t)|≤C, ∀t∈[0,T],ZQ θ2 MT M(|∇uM|2)≤C; (18) ZΩ |˜ϕM(t)|≤C, ∀t∈[0,T],ZQ ϕ Mθ MT M(|∇uM|2)≤C; (19) k∇Tj(˜ θM)kL2(H1 0(Ω)) ≤C, k∇Tj(˜ϕ M)kL 2(H1 0(Ω)) ≤C, ∀j>0; (20) COUPLING THE STOCKES AND NAVIER–STOKES EQUATIONS... 165 ZBj M A(θM,ϕ M)∇˜ θ M∇˜ θ M≤C, ZCj M A(θM,ϕ M)∇˜ϕ M∇˜ϕ M≤C, ∀j>0,(21) where Bj M={(x, t)∈Q/j≤| ˜ θ M|<j+1}and Cj M={(x, t)∈Q/j≤|˜ϕ M|<j+1}; lim j→∞ Z{θM>j} θ2 MTM(|∇uM|2) = lim j→∞ Z{ϕM>j} ϕMθMTM(|∇uM|2)=0.(22) Proof. All the estimates are obtained by using in (QM) suitable test functions. Indeed, (18) and (19) are deduced taking φ=1 εTε(˜ θM)andφ=1 ε T ε (˜ϕ M)intheθ Mand ϕMequations respectively, and then passing to the limit in ε↓0. Estimates in (20) are straightforward by putting φ=Tj(˜ θM)(resp.φ=T j (˜ϕ M)). Finally, to obtain (21) and (22) we just take φ=gj(˜ θM)(resp.φ=g j (˜ϕ M)) where gjis given by gj(s)=       0if|s|<j signsif |s|≥j+1 s−jsignsif j≤|s|<j+1.  Now, we may apply a result due to Boccardo and Gallou¨et [2] and deduce, from (20, 21) that (θM)and(ϕ M) are bounded in Lq(W1,q(Ω)), for all q<N+2 N+1 . Going back to (QM), we see that duM dt is bounded in L2(H−1(Ω)N), whereas dθM dt and dϕM dt are bounded in L1(Q)+L q (W−1,q(Ω)), for all q<N+2 N+1 (here, we have explicitly used hypothesis (H5)). But now, we have L1(Q),→L1(W−1,r(Ω)) whenever r< N N−1, and since N+2 N+1 <N N−1, N≥2, we also have L1(Q)+L q(W−1,q(Ω)) ,→L1(W−1,q(Ω)),∀q<N+2 N+1· This means that (˜ θM)and(˜ϕ M) lie in a bounded set of the space W(q) 3,∀q<N+2 N+1 , which is compactly imbedded in L1(Lq(Ω)) thanks to Lemma 1. 5.1. Passing to the limit in (QM) From the estimates obtained above, we deduce that from (uM), (θM)and(ϕ M) we may extract subsequences, denoted in the same way, such that uM→u∈W1           in L2(H1 0(Ω)N)-weakly, in L2(Q)N-strongly, a.e. in Q, in L∞(L2(Ω)N)-weak-∗, θM→θ, ϕM→ϕ, θ−a, ϕ −b∈W(q) 3,∀q<N+2 N+1 ,     in Lq(W1,q(Ω))-weakly, in L1(Lq(Ω))-strongly, a.e. in Q,