Periodic solutions in unbounded domains for the Boussinesq system

Periodic solutions in unbounded domains for the Boussinesq system ∗ Elder Jes´us VILLAMIZAR-ROA Corresponding author, Escuela de Matem´aticas, Universidad Nacional de Colombia, Sede Medell´ın A.A. 3840, Medell´ın, Colombia, ejvillamizar[email protected] Mar´ıa ´ Angeles RODR´ IGUEZ-BELLIDO Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Aptdo. 1160 -41080 Sevilla, Spain, [email protected] & Marko Antonio ROJAS-MEDAR Dpto. de Ciencias B´asicas, Facultad de Ciencias, Universidad del B´ıo B´ıo, Campus Fernando May, Casilla 447, Chill´an, Chile, mark[email protected] Abstract Assuming that the external forces of the system are small enough, the reference temperature being a periodic function, we study the existence, the uniqueness and the regularity of time-periodic solutions for the Boussinesq equations in several classes of unbounded domains of Rn.Our analysis is based on the framework of weak-Lpspaces. Keywords: Boussinesq equations, Strong Periodic Solutions, Unbounded domains. MRSubClass: 35B35, 35B40, 35Q35, 76D03. 1 Introduction Let Ωbe as either the whole space Rn,n≥3,either the half space Rn +,n≥3,either a bounded domain in Rn,n≥3,or an exterior domain in Rn,n≥4,with boundary ∂Ωof class C2+µ(µ>0). ∗The second and third authors have been partially supported by M.E.C. (Spain), Project MTM2006-07932. The second author has been partially supported by by Junta de Andaluc´ıa, Project P06-FQM-02373. The third author has been partially supported by Fondecyt-Chile, No. 1080628. 1 We consider the following nonstationary Boussinesq equations in Ω: ∂u ∂t−ν∆u+(u·∇)u+1 ρ∇p=βθg+Ψ,x∈Ω,t∈R,(1.1) ∇·u=0,x∈Ω,t∈R,(1.2) ∂θ ∂t−χ∆θ+(u·∇)θ=f, x∈Ω,t∈R,(1.3) u=0on ∂Ω,(1.4) θ= 0 on ∂Ω,(1.5) where grepresents the gravitational field at x,fis the reference temperature, Ψis an external force and ρ,ν,β,χare positive physical constants which denote, respectively, the density, the kinematic viscosity, the coefficient of volume expansion and the thermal conductance. The unknowns are u(x,t)∈Rn,p(x,t)∈Rand θ(x,t)∈Rrepresenting respectively, the velocity field, the pressure and the temperature of the fluid at point (x, t)∈Ω×R.Boussinesq equations describe the evolution of the temperature and velocity field of a viscous incompressible Newtonian fluid. For an extensive discussion on the physical origin of the equations (1.1)-(1.3), see [4]. We are interested in the study of the time-periodic solutions for the system (1.1)-(1.5) when the reference temperature is a periodic function with the same period. Without loss of generality, we have taken the constants ρ,ν,β,χequal to one. To avoid some technical complexities in the study of (1.1)-(1.5), throughout this paper we assume Ψ=0.Several works have been made in the mathematical analysis of system (1.1)-(1.5); see, for instance, [3], [5], [7], [13], [6] and papers cited therein. The time-periodic solutions for the Boussinesq equations in bounded domains was considered in [13]. The analysis was made via the Galerkin’s method. Indeed, in [13] it was considered a class of nonlinear evolution equations in a separable Hilbert space generalizing several models of hydrodynamics. However, the study of periodic solutions for system (1.1)-(1.5) has not been investigated in unbounded domains. Hence, the purpose of the present paper is to prove the existence and uniqueness of strong periodic solutions for problem (1.1)-(1.5) in the framework of Semigroups Theory on the Lorentz spaces, more explicitly, on the theory of weak-Lpspaces. We construct the time-periodic solutions using Lp,q −Lr,s estimates for the semigroups generated by the Stokes and Laplace operators. If Ωis an exterior domain, we need to assume n≥4 in order to obtain the gradient bounds for the semigroups generated by the Stokes and the Laplace operators in L(p,∞)(see Lemma 3.2). This work is motivated by the existence results of periodic solutions for the Navier-Stokes equations. In unbounded domains, this subject has been investigated in [11], [14], [15], [18] and [19]. In particular, in [14] was proved the existence of a unique time-periodic solution on the whole space R3for small external force. The problem in the half-space R3 +was considered in [15]. In [11], making use of Lp−Lrestimates for the semigroup generated by the Stokes operator, time-periodic solutions were constructed for small time-periodic forces. The stability 2 of these solutions was considered in [18]. However, the existence of strong time-periodic solutions in general unbounded domains is still an open problem. More complete references, including results for bounded domains, can be found in [11], [14], [15]. This paper is organized as follows. In Section §2, we give some preliminaries about Lorentz spaces and state our main results. Section §3 is devoted to prove the existence and the uniqueness of strong periodic solutions. 2 Preliminaries and Results Before stating our results we introduce some functional spaces. C∞ 0,σ(Ω) denotes the set of all C∞−real functions ϕ=(ϕ1, ..., ϕ n) with compact support in Ω,such that div ϕ=0.The closure of C∞ 0,σwith respect to norm Lr,1<r<∞,is denoted by Lr σ(Ω). Let us recall the Helmholtz decomposition: Lr(Ω) = Lr σ(Ω)⊕Gr(Ω),1<r<∞,where Gr(Ω) = {∇p∈Lr(Ω) : p∈Lr loc(Ω)}(c.f. [8]). Prdenotes the projection operator from Lr(Ω) onto Lr σ(Ω).The Stokes operator Ar=−Pr∆with domain D(Ar)={u∈H2,r(Ω) : u|∂Ω=0}∩Lr σ.Itiswellknown that −Argenerates a uniformly bounded analytic semigroup {e−tAr}t≥0of class C0in Lr σ(c.f. [9]). We denote by Bqthe Laplace operator in Lq(Ω),1<q<∞,with homogeneous Dirichlet boundary conditions: Bq=−∆with domain D(Bq)=W2,q(Ω)�W1,q 0(Ω).We recall that the operator −Bqgenerates a uniformly bounded analytic semigroup {e−tBq}t≥0in Lq(Ω) of class C0. Now we introduce some preliminaries about the Lorentz spaces. For details see [1]. Let 1<p≤∞and 1 ≤q≤∞.A Lebesgue measurable function fdefined on a domain Ω⊂Rn belongs to Lorentz space L(p,q)(Ω) if the quantity �f�(p,q)=  �q p�∞ 0�t1 pf∗∗(t)�qdt t�1 q,if 1<p<∞,1≤q<∞ supt>0t1 pf∗∗(t),if1<p≤∞,q=∞ , is finite, where f∗∗(t)=1 t�t 0 f∗(s)ds, f∗(t)=inf{s>0:m{x∈Ω:|f(x)|>s}≤t},t>0, with mdenoting the Lebesgue measure on Rn.The spaces L(p,q)with the norm �f�(p,q)are Banach spaces. Note that Lp(Ω) = L(p,p)(Ω). When q=∞,L (p,∞)(Ω) are called the Marcinkiewicz spaces or weak-Lpspaces. Moreover, L(p,q1)(Ω)⊂Lp(Ω)⊂L(p,q2)(Ω)⊂L(p,∞)(Ω) for 0 < q1≤p≤q2≤∞. We recall that the space C∞ 0(Ω) is not dense in L(p,∞)(Ω).Borchers and Miyakawa [2] established the following Helmholtz decomposition of the Lorentz spaces extending the operator Prto a bounded operator on L(r,d)(Ω),which we denote by Pr,d.Setting L(r,d) σ(Ω) = Range(Pr,d) and G(r,d)(Ω) = Kernel(Pr,d), then L(r,d)(Ω) = L(r,d) σ(Ω)⊕G(r,d)(Ω), 3 with L(r,d) σ(Ω) = {u∈L(r,d)(Ω) : ∇·u=0,u·n|∂Ω=0}and G(r,d)(Ω) = {∇v∈L(r,d)(Ω) : v∈L(r,d) loc (¯ Ω)}.For simplicity, we shall abbreviate the projection operator and the Stokes and Laplace operators on Lorentz spaces by P,A,B, respectively. In view of [2], the operators −A,−Bgenerate bounded analytic semigroups on L(p,q) σ(Ω) and L(p,q)(Ω),respectively. However, we recall that if q=∞,this semigroups are not strongly continuous at t=0. Applying the operator Pon the equations (1.1)-(1.2), from (1.1)-(1.5) we obtain the following problem of parabolic type: ut+Au +P[(u·∇)u]=P(θg),t∈R(2.6) θt+Bθ+(u·∇)θ=f, t ∈R.(2.7) The system (2.6)-(2.7), with periodic in time conditions, has associated the following system of integral equations u(t)=−�t −∞ e−(t−s)AP[(u·∇)u]ds +�t −∞ e−(t−s)AP(θg)ds (2.8) θ(t)=−�t −∞ e−(t−s)B(u·∇)θds+�t −∞ e−(t−s)Bfds. (2.9) Throughout this paper we assume the following assumptions on the external force fand the field g: Assumption 1. (CASE 1). If Ωis either the whole space Rn,a bounded domain in Rn,with boundary of class C2+µ(µ>0),or the half space Rn +,n≥3,we consider r, ˜r,q, ˜qverifying 2 <r,˜r<n, n 2< q, ˜q<n, 1 r−1 ˜r<min{2 n−1 q,2 n−1 ˜q}. (CASE 2). If Ωis an exterior domain in Rn,n≥4,with boundary of class C2+µ(µ>0),we consider r, ˜r,q, ˜qsuch that 2n (n−1) ≤r, ˜r<n, n 2<q,˜q<n, 1 r−1 ˜r<min{2 n−1 q,2 n−1 ˜q}. For each r, ˜rand q, ˜qwe assume that fsatisfies f∈BC(R;L(˜p,∞)(Ω)∩L(˜ l,∞)(Ω)),(2.10) for 1 <˜p,˜ l<∞with 1 ˜r+2 n<1 ˜p,1 ˜q<1 ˜ l<1 ˜q+1 nprovided n≥4 in both CASES (1,2). (Note that as n<2˜q,˜r<n,then the inequality 1/˜q<1/˜ l<1/˜q+1/n implies that 1/˜ l<2/n +1/˜r). If n=3,in the CASE 1, we assume that fsatisfies �f∈BC(R,L (˜ l,∞)(Ω)) such that f(s)=Bδ ˜p,∞h(s) for some h∈BC(R,D(Bδ ˜p,∞)),(2.11) 4 for 1 <˜p < min{˜r, ˜q},and δ>0 satisfying 3 2˜p+δ>1+max{1+ 3 2˜r,1 2+3 2˜q}and 1/˜q<1/˜ l< 1/˜q+1/3, where Bδ ˜p,∞denotes the power δof the operator Bon L˜p,∞.With respect to the field gwe make the following assumptions: g∈L(a,∞)(Ω)∩L(b,∞)(Ω), where aand bare such that: 1 a>2 n+1 r−1 ˜r,1 b<1 n+1 q−1 ˜r,(b>1,a>1). Remark 2.1 Condition (2.11) can be replaced by f(s)=∇·G(s),G(s)=(G1,...,G n)∈ BC(R;L(˜p,∞)(Ω)) with ∇G(t)∈BC(R;L(˜p,∞)(Ω))nfor 1<˜p<∞with 1/˜r+1/3<1/˜p. This implies that f(s)=∆h(s)for some h∈BC(R;D(B˜p,∞)). Our main results are stated as follows: Theorem 2.2 Let fbe a periodic function with period τ>0(i.e, for all t∈R,f(t)=f(t+τ)) satisfying Assumption 1. Then, if the quantities sup s∈R �f(s)�(˜p,∞)+sup s∈R �f(s)�(˜ l,∞),n≥4,in the CASES 1 and 2, sup s∈R �h(s)�(˜p,∞)+sup s∈R �f(s)�(˜ l,∞),n=3,in the CASE 1, �g�(b,∞)+�g�(a,∞),in the CASES 1 and 2, are small enough, then there exists a periodic solution (u,θ)of (2.8),(2.9), with the same period τof the external force, in the class u∈BC(R;L(r,∞) σ(Ω)),θ∈BC(R;L(˜r,∞)(Ω)),with ∇u∈ BC(R;L(q,∞)(Ω))n,∇θ∈BC(R;L(˜q,∞)(Ω))n.Moreover, within this class, if sup s∈R �u(s)�(r,∞)+sup s∈R �∇u(s)�(q,∞),sup s∈R �θ(s)�(˜r,∞)+sup s∈R �∇θ(s)�(˜q,∞) are small enough, then the solution is unique. Theorem 2.3 Under the assumptions of Theorem 2.2, if fis H¨older continuous with values in L(r∗,∞)(Ω)and g∈L(r∗,∞)(Ω),then, for all n<r ∗<q ∗=nq/(n−q),the periodic solution given by Theorem 2.2 satisfies 1. u∈BC(R;L(n,∞) σ(Ω)) ∩C1(R;Lr∗ σ(Ω)),θ∈BC(R;L(n,∞)(Ω)) ∩C1(R;Lr∗(Ω)). 2. Anu∈C(R;Lr∗ σ(Ω)),Bnθ∈C(R;Lr∗(Ω)). 3. For all t∈R,(2.6) and (2.7) are satisfied in Lr∗ σ(Ω)and Lr∗(Ω), respectively. 5 3 Existence, Uniqueness and Regularity of Periodic Solutions In this section we prove Theorem 2.2 and Theorem 2.3. Let us first recall the H¨older’s inequality and some L(r,∞)−L(p,∞)estimates for the semigroups {e−tA}t≥0,{e−tB}t≥0. Proposition 3.1 (Generalized H¨older’s inequality [16]) Let 1<p 1,p2,r<∞,f∈ L(p1,q1)(Ω)and g∈L(p2,q2)(Ω)where 1 p1+1 p2<1, then fg belongs to L(r,s)(Ω)where 1 r=1 p1+1 p2, and s≥1is any number such that 1 q1+1 q2≥1 s. Moreover, �fg�(r,s)≤C(r)�f�(p1,q1)�g�(p2,q2).(3.12) Lemma 3.2 ([2], [19]). 1. Let Ωbe either the whole space Rn,aboundeddomaininR3with boundary ∂Ωof class C2+µ(µ>0),or the half space Rn +,n≥3.Then �∇je−tAa�(r,1) ≤ct−n/2(1/p−1/r)−j/2�a�(p,1),1<p≤r<∞, for all a∈L(p,1) σ(Ω),j =0,1and all t>0,where c=c(n, p, r). 2. Let Ωbe an exterior domain in Rn,n≥4with boundary ∂Ω,of class C2+µ(µ>0). Then �e−tAa�(r,1) ≤ct−n/2(1/p−1/r)�a�(p,1),1<p≤r<∞, �∇e−tAa�(r,1) ≤ct−n/2(1/p−1/r)−1/2�a�(p,1),1<p≤r≤n, for all a∈L(p,1) σ(Ω)and all t>0,where c=c(n, p, r). Remark 3.3 Estimates in Lemma 3.2 for e−tAaand ∇e−tAain the norm L(r,∞)with respect to the data in L(p,∞)are true, because they are obtained by duality. Similar estimates hold for the semigroup {e−tB}t≥0.The estimates above hold in the particular case of Lpspaces (c.f. [11]). Lemma 3.4 ([2]). Let Ωbe as the CASE 1 and CASE 2 and suppose that 1<q<n,1≤d≤∞ and q∗=nq/(n−q).If φ∈L(p,∞)(Ω)for some p<∞and ∇φ∈L(q,d)(Ω)n,then φ∈L(q∗,d)(Ω) and the estimate �φ�(q∗,d)≤C�∇φ�(q,d)holds with C>0independent of φ. We denote by Xthe space of scalar functions {u∈BC(R;L(˜r,∞)):∇u∈BC(R;L(˜q,∞))n} with the norm �·� Xdefined as �u�X≡sup s∈R �u(s)�(˜r,∞)+sup s∈R �∇u(s)�(˜q,∞). 6 We also defined by Ythe space of vector functions {u∈BC(R;L(r,∞) σ(Ω)) : ∇u∈BC(R;L(q,∞)(Ω))n} with the norm �·� Ydefined as �u�Y≡sup s∈R �u(s)�(r,∞)+sup s∈R �∇u(s)�(q,∞). Xand Yare Banach spaces. We define the following operators F1and Gon Y×Yand Y×X, respectively, by F1(u,v)(t)=−�t −∞ e−(t−s)AP[(u·∇)v](s)ds, (3.13) G(u,θ)(t)=−�t −∞ e−(t−s)B(u·∇)θ(s)ds. (3.14) 3.1 Proof of Theorem 2.2 We construct a periodic solution of Problem (2.8)-(2.9) according to the following scheme: um+1(t)=F(um,θ m)(t),θ m+1(t)=θ0(t)+G(um,θ m)(t),(3.15) where u0(t)=�t −∞ e−(t−s)AP(θ0g)ds, θ0(t)=�t −∞ e−(t−s)Bfds, (3.16) F(um,θ m)(t)=F1(um,um)(t)+ �t −∞ e−(t−s)A{P(gθm)}ds, (3.17) G(um,θ m)(t)=−�t −∞ e−(t−s)B(um·∇)θm(s)ds. (3.18) Remark 3.5 In (1.1), when Ψis not zero, in the scheme above we consider u0(t)=�t −∞ e−(t−s)AP(Ψ)(s)ds and um+1 =u0(t)+F(um,θ m)(t). Let us first obtain some estimates for approximations above. We shall need the following lemmas. Lemma 3.6 Let r, ˜r,q and ˜qbe as Theorem 2.2. Then, we have sup s∈R �F1(u,v)�(r,∞)≤c1sups∈R�u(s)�(r,∞)�sup s∈R �v(s)�(r,∞)+sup s∈R �∇v(s)�(q,∞)�, sup s∈R �∇F1(u,v)�(q,∞)≤c1sup s∈R �∇v(s)�(q,∞)�sup s∈R �u(s)�(r,∞)+sup s∈R �∇u(s)�(q,∞)�, sup s∈R �G(u,θ)�(˜r,∞)≤c2sup s∈R �u(s)�(r,∞)�sup s∈R �θ(s)�(˜r,∞)+sup s∈R �∇θ(s)�(˜q,∞)�, sup s∈R �∇G(u,θ)�(˜q,∞)≤c2sup s∈R �∇θ(s)�(˜q,∞)�sup s∈R �u(s)�(r,∞)+sup s∈R �∇u(s)�(q,∞)�, for every u,v∈Y,θ∈X, where c1=c1(n, r, q),c 2=c2(n, r, q, ˜r, ˜q). 7 Proof. The proof is an application of Lemma 3.2. In fact, G(u,θ)(t)=−�t−1 −∞ e−(t−s)B(u·∇)θ(s)ds −�t t−1 e−(t−s)B(u·∇)θ(s)ds =G1(t)+G2(t). Then for all ψ∈C∞ 0and for all t∈R,we have |(G1(t),ψ)|≤�t−1 −∞ �∇e−(t−s)Bψ�((r˜r/(r+˜r))�,1)�θu�(r˜r/(r+˜r),∞)ds ≤csup s∈R �θ(s)�(˜r,∞)sup s∈R �u(s)�(r,∞)�t−1 −∞ (t−s)−n/2r−1/2�ψ�(˜r�,1). Hence, by duality, for all t∈R,�G1(t)�(˜r,∞)≤csup s∈R �θ(s)�(˜r,∞)sup s∈R �u(s)�(r,∞). �G2(t)�(˜r,∞)≤�t t−1 (t−s)−n/2(1/r+1/˜q−1/˜r)�u(s)�(r,∞)�∇θ(s)�(˜q,∞)ds ≤csup s∈R �u(s)�(r,∞)sup s∈R �∇θ(s)�(˜q,∞). Now, using Lemma 3.2 and Lemma 3.4 (for d=∞), we get �∇G(u,θ)�(˜q,∞)≤�t−1 −∞ �∇e−(t−s)B(u·∇)θ(s)�(˜q,∞)ds +�t t−1 �∇e−(t−s)B[(u·∇)θ](s)�(˜q,∞)ds ≤csup s∈R �u(s)�(r,∞)sup s∈R �∇θ(s)�(˜q,∞)�t−1 −∞ (t−s)−n/2r−1/2ds +csup s∈R �u(s)�(q∗,∞)sup s∈R �∇θ(s)�(˜q,∞)�t t−1 (t−s)−n/2qds ≤c�sups∈R�u(s)�(r,∞)+sup s∈R�∇u(s)�(q,∞)�sups∈R�∇θ(s)�(˜q,∞), for all t∈Rand c=c(n, r, q).This complete the proof of the last two estimates of lemma. The first two estimates are obtained similarly. Lemma 3.7 Let θ0be defined as in (3.16). Then θ0∈X. Proof. If fsatisfies (2.10), then using Lemma 3.2 we obtain �θ0(t)�(˜r,∞)≤csup s∈R �f(s)�(˜p,∞)�t−1 −∞ (t−s)−n/2(1/˜p−1/˜r)ds +csup s∈R �f(s)�(˜ l,∞)�t t−1 (t−s)−n/2(1/˜ l−1/˜r)ds. This is valid for all t∈R.The constant c=c(n, ˜r, ˜p, ˜ l).From (2.10), that is, 1/˜r+2/n < 1/˜pand 1/˜ l<2/n +1/˜r, we conclude that each integral above is finite and consequently, �θ0(t)�(˜r,∞)≤ 8 c1sup s∈R �f(s)�(˜p,∞)+c2sup s∈R �f(s)�(˜ l,∞),where c1=c(n, ˜p, ˜r) and c2=c(n, ˜ l, ˜r). A similar analysis proves that �∇θ0(t)�(˜q,∞)≤�c1sup s∈R �f(s)�(˜p,∞)�t−1 −∞ (t−s)−n/2(1/˜p−1/˜q)−1/2ds +�c2sup s∈R �f(s)�(˜ l,∞)�t t−1 (t−s)−n/2(1/˜ l−1/˜q)−1/2ds, for all t∈Rand �c1=c(n, �p, �q) and �c2=c(n,�l, �q). As 1/˜p>1/˜r+2/n > 1/n +1/˜qand 1/˜ l<1/˜q+1/n, the two integrals above converge. Now, if n=3,the previous analysis is wrong because it will be necessary 3/2(1/˜p−1/˜r)>1, with ˜p>1 and this is not possible. Consequently, we assume a new condition; in fact, if f satisfies (2.11), using the following estimate (which is a consequence of the analytic properties of the semigroup): �Bδe−tBa�(˜p,∞)≤Ct −δ�a�(˜p,∞),∀a∈L(˜p,∞),t>0,c=c(˜p, δ),δ≥0, and using Lemma 3.2, we obtain �θ0(t)�(˜r,∞)≤�t−1 −∞ �e−(t−s)BBδh(s)�(˜r,∞)ds +�t t−1 �e−(t−s)Bf(s)�(˜r,∞)ds ≤c�t−1 −∞ (t−s)−3/2(1/˜p−1/˜r)�Bδe−(t−s)B/2h(s)�(˜p,∞)ds +c�t t−1 (t−s)−3/2(1/˜ l−1/˜r)�f(s)�(˜ l,∞)ds ≤c�t−1 −∞ (t−s)−3/2(1/˜p−1/˜r)−δ�h(s)�(˜p,∞)ds +sup s∈R �f(s)�(˜ l,∞)�t t−1 (t−s)−3/2(1/˜ l−1/˜r)ds ≤c�sups∈R�h(s)�(˜p,∞)+sup s∈R�f(s)�(˜ l,∞)�, for all t∈Rwith c=c(n, ˜r, ˜p, ˜ l,δ).A similar estimate can be obtained for �∇θ0�(˜q,∞),(n= 3). This proves the lemma. Now we will estimate the terms F(um,θ m) and G(um,θ m).We start with the following lemma Lemma 3.8 The terms �F(um,θ m)�Y,�G(um,θ m)�Xgiven by (3.17),(3.18) satisfy �F(um,θ m)�Y≤2c1�um�2 Y+c3�θm�X,(3.19) �G(um,θ m)�X≤2c2�um�Y�θm�X,(3.20) where c1,c 2are as in Lemma 3.6 and c3depends on gbut is independent of m. Proof. We will prove that ����t −∞ e−(t−s)AP(gθm)(s)ds���Y≤c3�θm�X.(3.21) 9 By duality, for t0<t<t 0+T, we get ����t t0 e−(t−s)B(vm·∇wm)(s)ds���(n/α,∞)≤C1,2Km,2Km,1(t−t0)−(1−α)/2. Now, using Lemma 3.4 we obtain �t t0 �e−(t−s)Bf(s)�(n/α,∞)≤c�t t0 �∇e−(t−s)Bf(s)�(q,∞) ≤c�f�BC(R;L(l,∞))(t−t0)−(1−α)/2+3/2−n/2l, for all t0<t<t 0+Twith c=c(n, q, l).Since 1/l < 1/q +1/n, we have (1 −α)/2<3/2−n/2l and hence the above estimate yields (t−t0)(1−α)/2����t t0 e−(t−s)Bf(s)���(n/α,∞)≤c�f�BC(R,L(l,∞))T(1−α)/2. Consequently,    sup t0<t<t0+T (t−t0)(1−α)/2�wm+1(t)�(n/α,∞) ≤K0,2+c�f�BC(R;L(l,∞))T(1−α)/2+C1,2Km,1Km,2. Then, we can take Km+1,1,K m+1,2being respectively, Km+1,1=K0,1+C1,1K2 m,1+C2,1Km,2,(3.45) Km+1,2=K0,2+c�f�BC(R;L(l,∞))T(1−α)/2+C1,2Km,1Km,2.(3.46) Setting Km= max(Km,1,K m,2),m=1,2,..., from (3.43), (3.44), (3.45) and (3.46) we have Km+1 ≤K0+� CK2 m+C2,1Km, K0=c1T(1−α)/2max{�a�(n/α,∞),�b�(n/α,∞)+�f�BC(R;L(l,∞))}(3.47) and � C= max{C1,1,C 1,2}.If we consider C2,1<1,K 0<(1 −C2,1)2 4� C,(3.48) we have that Km<(1 −C2,1)−�(1 −C2,1)2−4� CK0 2� C≡k< 1 2� C,∀m=0,1,2,... (3.49) Assuming (3.48) and working as Subsection 3.2, due to the uniform estimate with respect to m, we can conclude the existence of a couple (v,w) such that (3.40) holds and satisfying (3.41) −(3.42). Thus, we finish the proof of Lemma 3.12. 16 Lemma 3.13 If K0defined by (3.47) is small enough, then the limit (v,w)given by Lemma 3.12 satisfies the following estimate (t−t0)1/2∇(v(·),w(·)) ∈BC((t0,t 0+T); L(n,∞)(Ω)×L(n,∞)(Ω))n,(3.50) with lim m→∞ sup t0<t<t0+T (t−t0)1/2�∇vm(t)−∇v(t)�(n,∞)=0, lim m→∞ sup t0<t<t0+T (t−t0)1/2�∇wm(t)−∇w(t)�(n,∞)=0. Proof. The proof is done by induction. In fact, we will prove that sup t0<t<t0+T (t−t0)1/2�∇vm(t)�(n,∞)≤Jm,1,(3.51) sup t0<t<t0+T (t−t0)1/2�∇wm(t)�(n,∞)≤Jm,2,(3.52) for some constants Jm,1,J m,2which are independent of t0,m=0,1,... Note that by Lemma 3.2 �∇v0(t)�(n,∞)≤C(t−t0)−1/2�a�(n,∞),�∇w0(t)�(n,∞)≤C(t−t0)−1/2�b�(n,∞), where C=C(n) is independent of t0.Hence we can take J0,1and J0,2, being respectively, C�a�(n,∞),C�b�(n,∞). Supposed inequalities (3.51)-(3.52) are true. Then ���∇�t t0 e−(t−s)AP[(vm·∇)vm]���(n,∞)≤�t t0 (t−s)−n/2(α/n)−1/2�vm�(n/α,∞)�∇vm�(n,∞) ≤cKm,1Jm,1�t t0 (t−s)−α/2−1/2(s−t0)α/2−1ds ≤cKm,1Jm,1(t−t0)−1/2B((1 −α)/2,α/2) ≤C3,1kJm,1(t−t0)−1/2, for all t0<t<t 0+T, where C3,1=C3,1(n, q) is independent of t0.Moreover ���∇�t t0 e−(t−s)AP(gwm)ds���(n,∞)≤�t t0 (t−s)−n/2(α/n)−1/2�g�(n,∞)�wm�(n/α,∞)ds ≤c�g�(n,∞)Km,2�t t0 (t−s)−(α+1)/2(s−t0)−(1−α)/2ds ≤cB((1 −α)/2,(1 + α)/2)k�g�(n,∞)≤C4,1k�g�(n,∞). Therefore, sup t0<t<t0+T (t−t0)1/2�∇vm+1�(n,∞)≤J0,1+C3,1kJm,1+C4,1k�g�(n,∞)T1/2. Now, for any t,t0<t<t 0+T, ���∇�t t0 e−(t−s)B(vm·∇wm)ds���(n,∞)≤�t t0 (t−s)−n/2(α/n)−1/2�vm�(n/α,∞)�∇wm�(n,∞) ≤cKm,1Jm,2�t t0 (t−s)−α/2−1/2(s−t0)α/2−1ds ≤C2,2kJ m,2(t−t0)−1/2, 17 where C2,2is independent of t0.As ���∇�t t0 e−(t−s)Bf(s)ds���(n,∞)≤c(t−t0)−1/2�f�BC(R;L(n,∞)), we conclude that sup t0<t<t0+T (t−t0)1/2�wm+1(t)�(n,∞)≤J0,2+C2,2kJm,2+c�f�BC(R;L(n,∞)). Then we can take Jm+1,1and Jm+1,2being respectively, Jm+1,1=J0,1+C3,1kJm,1+C4,1k�g�(n,∞)T1/2, Jm+1,2=J0,2+C2,2kJm,2+c�f�BC(R;L(n,∞)) . Let Jm= max{Jm,1,J m,2},m=1,2,... and J0= Max{J0,1+C4,1k�g�(n,∞)T1/2,J 0,2+c�f�BC(R;L(n,∞))}, then Jm+1 ≤J0+k� CJm, where � C= max{C3,1,C 2,2}.Consequently, if k<1/� C(3.53) we have a uniform estimate for the sequence {Jm}given by Jm≤J0 1− � Ck ≡J, m =0,1,... Assuming (3.53), we can see that the limits v,w satisfy (3.50) and the proof of Lemma 3.13 is finished. Lemma 3.14 The limit (v,w)given by Lemma 3.12 and Lemma 3.13 verifies (t−t0)1/4(v(·),w(·)∈BC((t0,t 0+T); L2n σ(Ω)×L2n(Ω)), with lim m→∞ sup t0<t<t0+T (t−t0)1/4�vm(t)−v(t)�2n=0, lim m→∞ sup t0<t<t0+T (t−t0)1/4�wm(t)−w(t)�2n=0. Proof. As the previous lemmas, the proof is done by induction. In fact, we will prove that there exist some constants Nm,1,N m,2,which are independent of t0,such that �vm(t)�2n≤Nm,1(t−t0)−1/4,�wm(t)�2n≤Nm,2(t−t0)−1/4.(3.54) 18 Since L(p0,∞)∩L(p1,∞)⊂Lpand �f�p≤C(p0,p 1,λ)�f�1−λ (p0,∞)�f�λ (p1,∞)provided that p0�= p1,0<λ<1 and 1/p =(1−λ)/p0+λ/p1,we have �v0(t)�2n≤C(t−t0)−1/4�a�(n,∞),�w0(t)�2n≤C(t−t0)−1/4�b�(n,∞), where C=C(n) is independent of t0.Hence, we define N0,1and N0,2as C�a�(n,∞)and C�b�(n,∞), respectively. Assuming true (3.54) for a given m, we can prove that (3.54) holds for the case m+1.In fact, note that for all φ∈C∞ 0,σ,ϕ∈C∞ 0,we get ����−�t t0 e−(t−s)AP[(vm·∇)vm](s)ds, φ����≤�t t0 �vm⊗vm�n�∇e−(t−s)Aφ�n�ds ≤C�t t0 �vm�2 2n(t−s)−3/4�φ�(2n)�ds ≤c(t−t0)−1/4B(1/4,1/2) N2 m,1�φ�(2n)� ����−�t t0 e−(t−s)B(vm·∇wm)(s)ds, ϕ����≤�t t0 �wm·vm�n�∇e−(t−s)Bϕ�n�ds ≤C�t t0 �vm�2n�wm�2n(t−s)−3/4�ϕ�(2n)�ds ≤c(t−t0)−1/4B(1/4,1/2) Nm,1Nm,2�ϕ�(2n)�. Hence by duality ����t t0 e−(t−s)AP(vm·∇vm)(s)ds���2n≤C1,1N2 m,1(t−t0)−1/4, ����t t0 e−(t−s)B(vm·∇wm)(s)ds���2n≤C1,2Nm,1Nm,2(t−t0)−1/4. We also note that ����t t0 e−(t−s)AP(gwm)(s)ds���2n≤�t t0 (t−s)−1/2�g�(n,∞)�wm(s)�(2n)ds ≤c�g�(n,∞)B(1/2,3/4) (t−t0)1/4≤C2,1Nm,2(t−t0)1/4 and ����t t0 e−(t−s)Bf(s)ds���2n≤�t t0 (t−s)−1/4�f(s)�(n,∞)ds ≤c�f�BC(R;L(n,∞))(t−t0)3/4. The inequalities above imply that sup t0<t<t0+T (t−t0)1/4�vm+1�2n≤N0,1+C1,1N2 m,1+C2,1Nm,2T1/2 sup t0<t<t0+T (t−t0)1/4�wm+1�2n≤N0,2+C1,2Nm,1Nm,2+c�f�BC(R;L(n,∞))T. As before, setting Nm= max(Nm,1,N m,2),m=1,2,...and N0= max(N0,2+c�f�BC(R;L(n,∞))T,N0,1), we obtain Nm+1 ≤N0+� CN2 m+� C2,1Nm,with � C2,1=C2,1T1/2where � C= max(C1,1,C 1,2).If we consider C2,1<1,N 0<(1 −C2,1)2 4� C,(3.55) 19 we have that the sequence {Nm}m=∞ m=0 is bounded with Nm≤(1 −C2,1)−�(1 −C2,1)2−4N0� C 2� C,m=0,1,... Assuming (3.55) and working as Lemma 3.12 and Lemma 3.13, we conclude the proof. Lemma 3.15 The limit (v,w)given by Lemma 3.12 and Lemma 3.13 verifies (v,w)∈BC((t0,t 0+T); L(n,∞) σ(Ω)∩L(q∗,∞) σ(Ω)×L(n,∞)(Ω)∩L(q∗,∞)(Ω)),(3.56) with sup t0<t<t0+T �vm(t)�(n/s,∞)≤M1,s,m,s=α, s =1,(3.57) sup t0<t<t0+T �wm(t)�(n/s,∞)≤M2,s,m,s=α, s =1,(3.58) and M1,s,m,M 2,s,m independent of t0. Proof. Computations similar to Lemma 3.12, Lemma 3.13, yield to M1,α,0=C�a�(n/α,∞),M 1,1,0=C�a�(n,∞), M2,α,0=C�b�(n/α,∞),M 2,1,0=C�b�(n,∞), where C=C(n, s) is independent of t0.Suppose by induction that (3.57), (3.58) are true. Note that ����−�t t0 e−(t−s)AP(vm·∇vm)(s)ds, φ����≤����t t0 (vm⊗vm(s),∇e−(t−s)Aφ)ds��� ≤�t t0 �vm�(n/α,∞)�vm�(n/s,∞)�∇e−(t−s)Aφ�(n/(n−α−s),1) ds ≤CKm,1M1,s,m �t t0 (t−s)−α/2−1/2(s−t0)−(1−α)/2·�φ�(n/(n−s),1) ds ≤CkM1,s,mB((1 −α)/2,(1 + α)/2)�φ�(n/(n−s),1), for all φ∈C∞ 0,σand all t0<t<t 0+Tand C=C(n, q, s) independent of t0. Consequently, by duality, for s=1,α,we have sup t0<t<t0+T����t t0 e−(t−s)AP(vm·∇vm)(s)ds���(n/s,∞)≤C5,1kM 1,s,m,(3.59) where C5,1independent of t0.Note that ����t t0 e−(t−s)AP(gwm)(s)ds���(n/s,∞)≤c�g�(b,∞)�t t0 (t−s)−n/2b�wm(s)�(n/s,∞)ds ≤c�g�(b,∞)(t−t0)1−n/2b≤C6,1M2,s,m (t−t0)1−n/2b, 20 ����t t0 e−(t−s)Bf(s)ds���(n/α,∞)≤c�f�BC(R;L(l,∞))Ta, with a=α 2−n 2l+1>0.Moreover, for all ϕ∈C∞ 0and all t0<t<t 0+T, ����−�t t0 e−(t−s)B(vm·∇wm)(s)ds, ϕ����≤����t t0 (wm·vm(s),∇e−(t−s)B)ϕ)ds��� ≤�t t0 �vm�(n/α,∞)�wm�(n/s,∞)�∇e−(t−s)Bϕ�(n/(n−α−s),1) ds ≤cK m,1M2,s,m ��t t0(t−s)−α/2−1/2(s−t0)−(1−α)/2ds��ϕ�(n/(n−s),1) ≤ckM 2,s,m B((1 −α)/2,(1 + α)/2) �ϕ�(n/(n−s),1). Thus, for s=1,α, sup t0<t<t0+T����t t0 e−(t−s)B(vm·∇wm)(s)ds���(n/s,∞)≤C4,2kM 2,s,m,(3.60) where C4,2is independent of t0.Hence, from (3.59)-(3.60) we can take M1,s,m+1 =M1,s,0+C5,1kM 1,s,m +C6,1M2,s,m,(3.61) M2,s,m+1 =M2,s,0+c�f�BC(R;L(l,∞))Ta+C4,2kM 2,s,m.(3.62) Setting Ms,m = max{M1,s,m,M 2,s,m}, Ms,0= max{M1,s,0,M 2,s,0+c�f�BC(R;L(l,∞))Ta}, ˘ C= max{C5,1,C 4,2}, from (3.61),(3.62) we obtain Ms,m+1 ≤Ms,0+k˘ CM s,m +C6,1Ms,m,for m=0,1,...,s =1,α. Then, if k˘ C+C6,1<1,(3.63) we have Ms,m ≤Ms,0 1−k˘ C−C6,1,m=0,1,...,s=1,α,which yields to (3.56) with (3.57)-(3.58). Now, we continue the proof of Proposition 3.11. We can see that under conditions (3.48), (3.53) and (3.63), the limit (v,w) belongs to the class required in Proposition 3.11. Moreover, then following convergences hold in L(n,∞) σ(Ω), L(n,∞) σ(Ω) and L(n,∞)(Ω), respectively �t t0 e−(t−s)AP[(vm·∇)vm](s)ds −→ �t t0 e−(t−s)AP[(v·∇)v](s)ds, �t t0 e−(t−s)AP(wmg)(s)ds −→ �t t0 e−(t−s)AP(wg)(s)ds, �t t0 e−(t−s)BP[(vm·∇)wm](s)ds −→ �t t0 e−(t−s)AP[(v·∇)w](s)ds, (3.64) 21 uniformly in t∈(t0,t 0+t) as m→∞.In fact, note that by Lemma 3.2, Lemma 3.12 and Lemma 3.13, we have ����t t0 e−(t−s)AP[(vm·∇)vm](s)ds −�t t0 e−(t−s)AP[(v·∇)v](s)ds���(n,∞) ≤�t t0 (t−s)−α/2�vm(s)−v(s)�(n/α,∞)�∇vm(s)�(n,∞)ds +�t t0 (t−s)−α/2�v(s)�(n/α,∞)�∇(vm(s)−v(s))�(n,∞)ds ≤cB(1 −α/2,1/2) Jsup t0<s<t0+T (s−t0)(1−α)/2�vm(s)−v(s)�(n/α,∞) +B(1 −α/2,(α+ 1)/2) ksup t0<s<t0+T (s−t0)1/2�∇vm(s)−∇v(s)�(n,∞)), which converges to 0. On the other hand ����t t0 e−(t−s)AP(wmg)(s)ds −�t t0 e−(t−s)AP(wg)(s)ds���(n,∞) ≤�t t0 �e−(t−s)AP((wm−w)g)(s)�(n,∞)ds ≤�t t0 (t−s)−n/2b+(1−α)/2�g�(b,∞)�wm(s)−w(s)�(n/α,∞) ≤cT1−n/2bsup t0<s<t0+T (s−t0)(1−α)/2�wm(s)−w(s)�(n/α,∞)−→ 0. Analogously we obtain (3.64). Now we will prove the weak continuity on the initial dada. Firstly, we note that for any φ∈L(n�,1) σ(Ω) and ϕ∈L(n�,1)(Ω) we have |(e−(t−t0)Aa−a,φ)|=|(a,e −(t−t0)Aφ−φ)| ≤�a�(n,∞)�e−(t−t0)Aφ−φ�(n�,1) →0,t→t+ 0. |(e−(t−t0)Bb−b, ϕ)|=|(b, e−(t−t0)Aϕ−ϕ)| ≤�b�(n,∞)�e−(t−t0)Bϕ−ϕ�(n�,1) →0,t→t+ 0. As �v�(q∗,∞),�w�(q∗,∞)≤cand (t−t0)1/4�v�2n,(t−t0)1/4�w�2n≤c, we have lim t→t0��t t0 e−(t−s)AP(wg)(s)ds −�t t0 e−(t−s)AP[(v·∇)v](s)ds, φ�=0, lim t→t0��t t0 e−(t−s)Bf(s)ds −�t t0 e−(t−s)B(v·∇w)(s)ds, ϕ�=0. Indeed, note that when t→t+ 0, �t t0�e−(t−s)B[(v·∇)w],ϕ �≤�t t0 �v�(q∗,∞)�w�2n�∇e−(t−s)Bϕ�(1−1/q∗−1/2n,1) ≤c�t t0 (s−t0)−1/4(t−s)−n/2q+1/4�ϕ�(n�,1) ds ≤c(t−t0)1−n/2qB(3/4,−n/2q+5/4) �ϕ�(n�,1) →0, 22 �t t0�e−(t−s)AP[(v·∇)v],φ�≤�t t0 �v�(q∗,∞)�v�2n�∇e−(t−s)Aφ�(1−1/q∗−1/2n,1) ≤c�t t0 (s−t0)−1/4(t−s)−n/2q+1/4�φ�(n�,1) ds ≤c(t−t0)1−n/2qB(3/4,−n/2q+5/4) �φ�(n�,1) →0, �t t0�e−(t−s)AP(wg),φ�ds ≤�t t0 �e−(t−s)AP(wg)�(n,∞)�φ�(n�,1) ds ≤c�w�q∗�g�(n,∞)�φ�(n�,1) �t t0 (s−t0)−n/2q∗ds ≤c(t−t0)1−n/2q∗�w�q∗�g�(n,∞)�φ�(n�,1) →0, and �t t0�e−(t−s)Bf(s),ϕ �ds ≤�t t0 �e−(t−s)Bf(s)�(n,∞)�ϕ�(n�,1) ds ≤c�t t0 �f�(n,∞)ds �ϕ�(n�,1) ≤c(t−t0)→0. Collecting all the previous convergences and letting m→∞in (3.36)-(3.37), we see that (v,w) is a solution of (3.34)-(3.35). Finally, we will estimate the time-interval Tof existence in terms of the prescribed data. As kis determined by (3.49), there exists a constant ˜ kindependent of t0such that if K0≤˜ k, then conditions (3.48),(3.53), (3.63) are satisfied. Now, from (3.47) we see that Tmay be chosen as (3.33). Remark 3.16 The solution (v,w)of integral equations (3.34)-(3.35) satisfies (v,w)∈BC(t0,t 0+ T;Lp σ(Ω)×Lp(Ω)),for all p∈(n, q∗),with �v�p≤C�v�1−λ (n,∞)�u�λ (q∗,∞),�w�p≤C�w�1−λ (n,∞)�w�λ (q∗,∞), where λis such that 1/p =(1−λ)/n +λ/q∗. Note that being aand belements of L(n,∞) σ(Ω)∩L(q∗,∞) σ(Ω) and L(n,∞)(Ω)∩L(q∗,∞)(Ω), respectively, we have that aand bbelong to space Lr∗ σ(Ω) and Lr∗(Ω),respectively. Consequently, the norms �e−(t−t0)Aa�r∗≤C�a�r∗and �e−(t−t0)Bb�r∗≤C�b�r∗are finite. Moreover, it is not difficult to see that �����t t0 e−(t−s)AP[(v·∇)v],φ����≤C��t t0 �v�2 r∗(t−s)−n 2(1 (r∗)�−1 (r∗/2)�)−1 2��φ�(r∗)� ≤C�v�2 r∗��t t0 (t−s)−n 2r∗−1 2ds��φ�(r∗)�≤C�φ�(r∗)�, for all φ∈C∞ 0,σ.By duality, we have that �t t0e−(t−s)AP[(v·∇)v](s)ds ∈Lr∗ σ(Ω). Analogously, we can see that �t t0 e−(t−s)B(v·∇w)(s)ds +�t t0 e−(t−s)AP(wg)(s)ds ∈Lr∗(Ω),�t t0 e−(t−s)Bf(s)ds ∈Lr∗(Ω). 23 Therefore we conclude that equalities (3.34)-(3.35) are satisfied in Lr∗ σ(Ω) and Lr∗(Ω),respectively. The proof of Proposition 3.11 is finished. � Proof of Theorem 3.10. By the hypothesis of Theorem 3.10, we can apply Proposition 3.11 and hence, there exist T∈(0,1] and functions vand wsatisfying:              v∈BCw([t0,t 0+T); L(n,∞) σ(Ω)) ∩BC([t0,t 0+T); L(q∗,∞) σ(Ω)), w∈BCw([t0,t 0+T); L(n,∞)(Ω)) ∩BC([t0,t 0+T); L(q∗,∞)(Ω)), (t−t0)1/2∇v∈BC((t0,t 0+T); L(n,∞)(Ω))n, (t−t0)1/2∇w∈BC((t0,t 0+T); L(n,∞)(Ω))n, in such a way that for all t0∈Rthe integral system (3.34)-(3.35) is satisfied in the Lr∗-norm. Firstly, we study the uniqueness of that solution. Let (v1,w 1) another solution of (3.34)- (3.35), in the class given by Proposition 3.11, with the same initial condition. Let (V,W)= (v−v1,w−w1). Note that (V(t),W(t)) ∈Ln σ(Ω)×Ln(Ω) for all 0 <t<T,with sup t0<t<T �V(t)�n<∞,sup t0<t<T �W(t)�n<∞.(3.65) In fact, taking n<q<min{2n, r∗}we have for all φ∈C∞ 0,σand all t0<t<T, |(V(t),φ)|≤����t t0 (v−v1⊗v1,∇e−(t−s)Aφ)ds���+����t t0 (e−(t−s)AP(Wg),φ)ds��� ≤C(n, q) ��sup t0<s<T sn/2(1/n−1/q)�v�q�2 +�sup t0<s<T sn/2(1/n−1/q)�v1�q�2��φ�n� +C(n, q)��sup t0<s<T sn/2(1/n−1/q)�w(s)�q+sup t0<s<T sn/2(1/n−1/q)�w1(s)�q� ×�sup t0<s<T s1/2�g�(n,∞)���φ�n�. Analogously, for all ϕ∈C∞ 0and all t0<t<T |(W(t),ϕ)|≤C(n, q)�sup t0<s<T sn/2(1/n−1/q)�v(s)�qsup t0<s<T sn/2(1/n−1/q)�w(s)�q +sup t0<s<T sn/2(1/n−1/q)�v1(s)�qsup t0<s<T sn/2(1/n−1/q)�w1(s)�q��ϕ�n�. 24 By duality we can conclude (3.65). We define KV(t)≡sup t0<t<T �V(t)�(n,∞)and KW(t)≡ sup t0<t<T �W(t)�(n,∞).Let psatisfying 1/p =1−1/n −1/r∗.Then |(V(t),φ)|≤����t t0 (V⊗v(s)−v1,∇e−(t−s)Aφ)ds���+����t t0 (e−(t−s)AP(Wg)(s),φ)ds��� ≤�t t0 (�v(s)�r∗+�v1(s)�r∗)�V(s)�n�∇e−(t−s)Lφ�pds +�t t0 (t−s)−1/2�W(s)�n�g�(n,∞)�φ�n�ds ≤CK V(t)�sup t0<s<t0+t (s−t0)n/2(1/n−1/r∗)�v(s)�r∗ +sup t0<s<t0+t (s−t0)n/2(1/n−1/r∗)�v1(s)�r∗� ×��t t0 (t−s)−n/2(1/n�−1/p)−1/2(s−t0)−n/2(1/n−1/r∗)ds��φ�n� +KWsup t0<s<t0+t (s−t0)1/2�g�(n,∞)��t t0 (t−s)−1/2(s−t0)−1/2ds��φ�n�, for all φ∈C∞ 0,σand for all t0<t<T.Then, using duality we obtain �V(t)�n≤KV(t)�sup t0<s<t0+t (s−t0)n/2(1/n−1/r∗)�v(s)�r∗+ +sup t0<s<t0+t (s−t0)n/2(1/n−1/r∗)�v1(s)�∗ r�+C2KW(t)sup t0<s<t0+t (s−t0)1/2�g�(n,∞).(3.66) Analogously, |(W(t),ϕ)|≤CK V(t)sup t0<s<t0+t (s−t0)n/2(1/n−1/r∗)�w1(s)�r∗�ϕ�n� +CK W(t)sup t0<s<t0+t (s−t0)n/2(1/n−1/r∗)�v(s)�r∗�ϕ�n�, for all ϕ∈C∞ 0and all t0<t<T.By duality, �W(t)�n≤C3KV(t)sup t0<s<t0+t (s−t0)n/2(1/n−1/r∗)�w1(s)�r∗ +C4KW(t)sup t0<s<t0+t (s−t0)n/2(1/n−1/r∗)�v(s)�r∗.(3.67) Let K(t) = max{KV(t),K W(t)}.Then we have that max {�V(t)�n,�W(t)�n}≤C∗K(t)�sup t0<s<t0+t (s−t0)n/2(1/n−1/r∗)�v(s)�r∗ +sup t0<s<t0+t (s−t0)n/2(1/n−1/r∗)�v1(s)�r∗+sup t0<s<t0+t (s−t0)n/2(1/n−1/r∗)�w(s)�r∗ +sup t0<s<t0+t (s−t0)n/2(1/n−1/r∗)�w1(s)�r∗�+C∗∗ K(t)sup t0<s<t0+t (s−t0)1/2�g�(n,∞). We define the constant � C=C∗+C∗∗.From hypothesis of this theorem, there exists t0<t 1≤T, such that sup t0<s<t0+t (s−t0)n/2(1/n−1/r∗)(�w(s)�r∗+�w1(s)�r∗+�v(s)�r∗+�v1(s)�r∗) +sup t0<s<t0+t (s−t0)1/2�g�(n,∞)<1 2� C,for t0<t<t 1. 25