Reproductive solution for grade-two fluid model in two dimensions

HRevista Integración Escuela de Matemáticas Universidad Industrial de Santander Vol. 27, No. 1, 2009, pág. 15–24 Reproductive solution for grade-two fluid model in two dimensions L. Friz∗ F. Guillén-González∗∗ M. A. Rojas-Medar∗∗∗ Abstract. We treat the existence of reproductive solution (weak periodic solution) of a second-grade fluid system in two dimensions, by using the Galerkin approximation method and compactness arguments. 1. Introduction For a general incompressible fluid of grade 2, the Cauchy stress tensor is given by T=−pI+µA1+α1A2+α2A2 1, where µ≥0is the viscosity, α1,α2are material coefficients, namely normal stress moduli, pis the pressure and A1,A2are the first two Rivlin-Ericksen (see [8] or [9]) tensors defined by A1=∇u+ (∇u)T, A2=d dtA1+A1∇u+ (∇v)TA1. 0Keywords: Reproductive solution, two-grade fluid, Galerkin method. 0MSC2000: 35Q35, 76D03. 0∗Grupo de Matemáticas Aplicadas, Departamento de Ciencias Básicas, Universidad del Bío-Bío, Chillán, Chile. Casilla 447. e-mail:[email protected] 0∗∗ Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Sevilla, 41012, España. e-mail:[email protected] 0∗∗∗Grupo de Matemáticas Aplicadas, Departamento de Ciencias Básicas, Universidad del Bío-Bío, Chillán,Casilla 447. e-mail:[email protected] 15 16 L. Friz, F. Guillén-González, & M. A. Rojas-Medar From the thermodynamical principles we have that α1+α2, and the requirement that the free energy be a minimum in equilibrium implies that α1≥0. With all these conditions the equations of motion for an incompressible fluid of grade two are given by (∂ ∂t(u−α∆u)−ν∆u+curl(u−α∆u)×u+∇p=fin Ω×]0, T [, divu= 0 in Ω×]0, T [, (1) with homogeneous Dirichlet boundary conditions u= 0,on ∂Ω, and initial condition u(0) = u0,in Ω. Here, ν > 0represents the Kinematic viscosity and fthe external forces. The study of this kind of fluids was initiated by Dunn and Fosdick in [4] and by Fosdick and Rajapogal in [5]. The first successful mathematical analysis of (1) was done by Cioranescu and El Hacène in [1]. Another interesting work is due to Galdi and Sequeira [6], where the authors obtain some existence results. Later Cioranescu and Girault in [2] establish existence, uniqueness and regularity of a global weak solution of (1) with small data fand u(0) and the same result on some interval for arbitrary data. The existence is obtained by applying Galerkin’s method with a special basis. In this paper we seek reproductive solutions of the two-grade fluid system, i.e. solutions of the following system:          ∂ ∂t(u−α∆u)−ν∆u+curl(u−α∆u)×u+∇q=fin Ω×]0, T [, divu= 0 in Ω×]0, T [, u=0on ∂Ω×]0, T [, u(0) = u(T), (2) by supposing that fdepends on the time t(notice that if fdoes not depend on t, the solution of the associated steady-state system of the secondgrade fluid is actually a reproductive solution). As the reader can see, the usual initial condition has been changed by a time periodic condition. The next theorem is the main result of this paper. Theorem 1.1. For any f∈L2(0, T ;H(curl; Ω)) ∩L∞(0, T ;L2(Ω)), there exists a weak solution of the two-grade fluid system (2). [Revista Integración Reproductive solution for grade-two fluid model in two dimensions 17 2. Preliminaries Let Ωbe a bounded domain of R2of the class C2,1. To solve a grade 2 fluid system means to find a vector valued function u= (u1, u2)and a scalar function pdefined on Ω×]0, T [satisfying (2). Since we are in two dimensions (see [7]), the curl operator is defined by curl u=∂u2 ∂x1 −∂u1 ∂x2 , and if zis a scalar function, we define z×u= (−zu2, zu1). In what follows, the spaces in bold face represent spaces of bi-dimensional vector functions. We define the Hilbert spaces Hand Vin the following manner: H={Ψ∈L2(Ω) : div Ψ=0,Ψ·n= 0 on ∂Ω}, V={v∈H1(Ω) : div v= 0,v=0,on ∂Ω}, H(curl; Ω) = {v∈L2(Ω) : curl v∈L2(Ω)}. For α∈R+, we introduce the space (see [1] and [2]) V2=v∈V: curl (v−α∆v)∈L2(Ω),(3) equipped with the scalar product (u,v)V2= (u,v) + α(∇u,∇v) + (curl (u−α∆u),curl (v−α∆v)),(4) and associated norm and semi-norm kvkV2= (v,v)1/2 V2,|v|V2=kcurl (v−α∆v)kL2(Ω).(5) In the following lemma it is proved that the semi-norm | · |V2is a norm in H3. Lemma 2.1 ([1] p 182).Let Ωbe a bounded, simply-connected open set of R2of the class C2,1. Then every v∈V2belongs to H3(Ω). Moreover, there exists C > 0such that kvkH3(Ω) ≤Ckcurl (v−α∆v)kL2(Ω). An easy but tedious computation gives us the following equality: ZΩ curl (u−α∆u)×u·vdx =b(u;u,v)−αb(u; ∆u,v) + αb(v; ∆u,u); Vol. 27, No. 1, 2009] 18 L. Friz, F. Guillén-González, & M. A. Rojas-Medar here b(u;v,w) = 3 X i,j=1 ZΩ ui ∂vj ∂xi wjdx. From this, the variational formulation of the problem (1) is the following: Given f∈L2(0, T ;H(curl; Ω) ∩L∞(0, T ;L2(Ω)) and u0∈V2, find u∈L∞(0, T ;V2)such that (u′,v) + α(∇u,∇v) + ν(∇u,∇v) + b(u;u,v) −αb(u; ∆u,v) + αb(v,∆u,u) = (f,v),∀v∈V.(6) 3. A priori estimates of the Galerkin solutions By following the ideas given in [1] and [2] we consider the basis {wj}j∈N, the eigenfunctions of the problem: For j∈N,wj∈V2is the solution of (wj,v)V2=λj{(wj,v) + α(∇wj,∇v)},∀v∈V2,(7) where (·,·)V2is the scalar product in V2. Since the imbedding of V2into Vis compact, there exists a sequence of eigenvalues (λj)j≥1and a sequence of eigenfunctions (wj)j≥1 that constitutes a basis of V2. Lemma 3.1 ([2] p 326).Let Ωbe a bounded simply-connected open set of R3with a boundary Γof class C3,1. Then the eigenfunctions of the problem (7) belong to H4(Ω). For every m∈N, we define Vm 2the vector space spanned by the first meigenfunctions {w1,...,wm}, and by Pmthe orthogonal projection on Vm 2for the scalar product in V2. In order to construct a periodic solution of the problem (2) we will use Galerkin’s discretization. Indeed, for j∈ {1,2,...,m}we find um(t) = m X j=1 cm j(t)wj, solution of (u′ m(t),wj) + α(∇u′ m(t),∇wj) + ν(∇um(t),∇wj) + b(um(t); um(t),wj), −αb(um(t); ∆um(t),wj) + αb(wj,∆um(t),um(t)) = (f(t),wj),(8) um(0) = Pm(u0).(9) By multiplying both sides of (8) by cm j(t)and summing with respect to j, from the anti-symmetry of bwe obtain the equality 1 2 d dt kum(t)k2 L2(Ω) +α|um|2 H1(Ω)+ν|um|2 H1(Ω) = (f(t),um(t)). [Revista Integración Reproductive solution for grade-two fluid model in two dimensions 19 Therefore, by integrating in time for t∈[0, T ]the above equality we obtain the following lemma. Lemma 3.2 ([2] p 327).The solution um(t)of the problem (8)-(9) satisfies the following differential inequality for each t∈[0, T ]: kum(t)k2 L2(Ω) +αk∇um(t)k2 L2(Ω) ≤e−νKt kum(0)k2 L2(Ω) +αk∇um(0)k2 L2(Ω)+P2 νZt 0 e−νK(t−s)kf(s)k2 L2(Ω)ds, where P>0is the Poincaré constant and K= (P2+α)−1. In order to obtain an estimation for the norm kumkV2, we adapt the proof of Theorem 4.4 in [2] and the proof of the differential inequality given in [1] p 189. At first, we define the vector-valued function F(um,um)by (F(um(t),um(t)),v) = ν(∇um(t),∇v) + b(um(t); um(t),v) −αb(um(t); ∆um(t),v) + αb(v; ∆um(t),um(t)),(10) for every v∈H1 0(Ω). For 1≤m≤m, by construction of F(um,um), (u′ m(t),wj) + α(∇u′ m(t),∇wj) + (F(um(t),um(t)),wj)−(f(t),wj) = 0.(11) From Lemma 3.1, F(um(t),um(t)) ∈H1(Ω). Next for each t, let vm(t)∈Vbe solution of the Stokes equation vm(t)−α∆vm(t) + ∇qm(t) = F(um(t),um(t)) −f(t).(12) By classical regularity results, vm(t)∈H3(Ω) and then, curl(vm(t)−α∆vm(t)) belongs to L2(Ω). Therefore, vm∈V2. By multiplying (12) by wj, we obtain (vm(t),wj) + α(∇vm(t),∇wj) = (F(um(t),um(t)) −f(t),wj), thus (11) can be written (u′ m(t),wj) + α(∇u′ m(t),∇wj) + (vm(t),wj) + α(∇vm(t),∇wj) = 0.(13) Multiplying equation (13) by λjcm j(t)and adding for j= 1,...,m, we get (u′ m,um)V2+ (vm,um)V2= 0, Vol. 27, No. 1, 2009] 20 L. Friz, F. Guillén-González, & M. A. Rojas-Medar in other words, (curl(u′ m−α∆u′ m),curl(um−α∆um)) + (curl(vm−α∆vm),curl(um−α∆um)) = 0. By taking curl in (12), curl(vm−α∆vm) = curl(F(um,um)−f), and thus (curl(u′ m−α∆u′ m),curl(um−α∆um)) + (curl(F(um,um)−f),curl(um−α∆um)) = 0. Using definition (10) we find 1 2 d dtkcurl(um−α∆um)k2 L2(Ω) + (curlF(um,um),curl(um−α∆um)) = (f,um) + (curlf,curl(um−α∆um)). (14) Now, we will estimate the term: T= (curlF(um,um),curl(um−α∆um)). Since div um= 0 and Ω⊆R2, it is not so difficult to prove that curl(curl(um−α∆um)×um) = um· ∇(um−α∆um), and (um· ∇(um−α∆um),curl(um−α∆um)) = 0, and thus T= (−ν∆curlum,curl(um−α∆um)) + (um· ∇(um−α∆um),curl(um−α∆um)) =ν αkcurl(um−α∆um)k2 L2(Ω) −ν α(curlum,curl(um−α∆um)). Therefore, the equation (14) can be written 1 2 d dtkcurl(um−α∆um)k2 L2+ν αkcurl(um−α∆um)k2 L2(Ω) = (curlf,curl(um−α∆um)) + ν α(curlum,curl(um−α∆um)). [Revista Integración Reproductive solution for grade-two fluid model in two dimensions 21 Then, we get the following inequality: 1 2 d dtkcurl(um−α∆um)k2 L2+ν αkcurl(um−α∆um)k2 L2(Ω) ≤ kcurlfkL2(Ω)kcurl(um−α∆um))kL2(Ω) +ν αkcurlumkL2(Ω)kcurl(um−α∆um)kL2(Ω) ≤1 2λkcurlfk2 L2(Ω) +1 λkcurl(um−α∆um))k2 L2(Ω) +ν 2αεkcurlumk2 L2(Ω) +1 εkcurl(um−α∆um))k2 L2(Ω). If we take ε= 2 and λ=2α νwe have that 1 2 d dtkcurl(um−α∆um)k2 L2+ν αkcurl(um−α∆um)k2 L2(Ω) ≤2ν αkcurlumk2 L2(Ω) +2α νkcurlfk2 L2(Ω). But kcurlumk2 L2(Ω) ≤2k∇umk2 L2(Ω), thus 1 2 d dtkcurl(um−α∆um)k2 L2+ν αkcurl(um−α∆um)k2 L2(Ω) ≤4ν αk∇umk2 L2(Ω) +2α νkcurlfk2 L2(Ω). From Lemma 3.2 1 2 d dtkcurl(um(t)−α∆um(t))k2 L2+ν αkcurl(um(t)−α∆um(t))k2 L2(Ω) ≤4ν α2(kum(0)k2 L2(Ω) +αk∇um(0)k2 L2(Ω)) + 4P2 ανK kfkL∞(0,T ;L2(Ω)) +2ν αkcurlf(t)k2 L2(Ω). From all this considerations, we have proved the following lemma. Lemma 3.3. The solution umof the problems (8) and (9) satisfies the a priori estimate for all t∈[0, T ]: kcurl(um(t)−α∆um(t))k2 L2(Ω) ≤e−νt αkcurl(um(0) −α∆um(0))k2 L2(Ω) +2 α(kum(0)k2 L2(Ω) +αk∇um(0)k2 L2(Ω))+ 2P2 ν2KkfkL∞(0,T ;L2(Ω)) +2α νkfkL2(0,T ;H(curl;Ω)). 4. Proof of the Theorem 1.1 In this section, we prove the Theorem 1.1. To this end, at first we prove the existence of a sequence of Reproductive Galerkin solutions, by following the ideas given in [3], which converges to the reproductive solution of the grade two system fluid. Vol. 27, No. 1, 2009] 22 L. Friz, F. Guillén-González, & M. A. Rojas-Medar We define the operator Lm(t) : [0, T ]→Rmas Lm(t) = (cm 1(t), cm 2(t),...,cm m(t)),(15) where cm j(t)are the coefficients of the expansion of um. For every (ξ1, ξ2,...,ξm)∈Rm, we define the following equivalent norms: k(ξ1, ξ2,...,ξm)k2 a,Rm:= kuk2 L2(Ω) +αk∇uk2 L2(Ω), k(ξ1, ξ2,...,ξm)k2 b,Rm:= kcurl(u−α∆u)k2 L2(Ω), where u=ξ1w1+ξ2w2+···+ξmwm. We define the operator Φm:Rm→Rmin the following manner: Given Lm 0∈Rm, we define Φm(Lm 0) = Lm(T), where Lm(t)is defined in (15). It is clear that Φmis continuous and we want to prove that it has a fixed point. In order to prove this result, we will use the Leray-Schauder Theorem. Indeed, it suffices to show that for all λ∈[0,1], the possible solution Lm 0(λ)of the equation Lm 0(λ) = λΦm(Lm 0(λ)) (16) are bounded independently of λ. Since Lm 0(0) = 0, we will consider λ∈(0,1]. In this case, (16) can be written as Φm(Lm 0(λ)) = 1 λLm 0(λ). Thus, by definition of Φmand Lemma 3.2, we obtain  1 λLm 0(λ) 2 a,Rm ≤e−νKT kLm 0(λ)k2 a,Rm+P2 νZT 0 eνKskf(s)k2 L2(Ω)ds, which implies that kLm 0(λ)k2 a,Rm≤ P2 νZt 0 eνKskf(s)k2 L2(Ω)ds 1−e−νKT =M0. Now, from Lemma 3.3 and definition of Φm, we have that  1 λLm 0(λ) 2 b,Rm ≤e−νT αkLm 0(λ)k2 b,Rm+2 αkLm 0(λ)k2 a,Rm +2P2 ν2KkfkL∞(0,T ;L2(Ω)) +2α νkfkL2(0,T ;H(curl;Ω)), [Revista Integración Reproductive solution for grade-two fluid model in two dimensions 23 then, we deduce that kLm 0(λ)k2 b,Rm≤ 2M0 α+2P2 ν2KkfkL∞(0,T ;L2(Ω)) +2α νkfkL2(0,T ;H(curl;Ω)) 1−e−νT α =M1, for each λ∈(0,1]. This last estimate is independent of λ∈[0,1] and m∈ N. Consequently, Leray-Shauder Theorem implies the existence of al least one fixed point of Φm, and then the existence of reproductive Galerkin solution um. Moreover, since the previous estimates do not depend on m∈ N and by Lemma 3.3, there exists M∈R independent of msuch that kum(t)kV2≤M, (17) for each t∈[0, T ], it means that (um)m≥1is bounded in L∞(0, T ;V2). By Lemma 2.1, we can write kum(t)kH3(Ω) ≤M, for each t∈[0, T ]. It remains to pass to the limit with respect to m. This is a standard argument and we have only to prove that (u′ m)m≥1is bounded in L∞(0, T ;V′ 2). In order to prove this bound, we use the arguments given in [1], p 190. At first, we note that |b(um(t),um(t),v)| ≤ Ck∇umk2 L2(Ω)kvkV2, which implies that there exists T1 m(t)∈V′ 2)such that b(um(t),um(t),v) = hT1 m(t),vi,∀v∈V2, and by estimate (17), we have that (T1 m)m≥1is bounded in L∞(0, T ;V′ 2). In the same manner, there exists a bounded sequence (T2 m)m≥1in L∞(0, T ;V′ 2)such that b(um(t),∆um(t),v) = hT2 m(t),vi,∀v∈V2. Finally, from equation (8), we can conclude that u′ m=TmPm, where (Tm)m≥1is a bounded sequence of L∞(0, T ;V′ 2)and Pmis the projection of V2on Vm 2. This completes the proof. Acknowledgments L. Friz was supported by Fondecyt (Chile) grant 1090510. F. Guillén-González and M. A. Rojas-Medar were supported by DGI-MEC (Spain) Grant MTM2006-07932 and Fondecyt (Chile) Grant 1080628. Vol. 27, No. 1, 2009]