A review on reproductivity and time periodicity for incompressible fluids

Bol. Soc. Esp. Mat. Apl. no41(2007), 101–116 A REVIEW ON REPRODUCTIVITY AND TIME PERIODICITY FOR INCOMPRESSIBLE FLUIDS B. CLIMENT-EZQUERRA∗, F. GUILL´ EN-GONZ´ ALEZ∗ AND M. ROJAS-MEDAR† ∗Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Espa˜na. †Departamento de Ciencias B´asicas, Universidad del B´ıo-B´ıo, Chile. [email protected] [email protected] [email protected] Abstract In this article, our aims is to review some of the results that are currently available concerning the existence, uniqueness and regularity of reproductive and time periodic solutions of the Navier-Stokes equations and some variants. By the way, we present some open problems. Key words: Reproductive and time periodic solutions, Navier-Stokes type equations, regularity of solutions AMS subject classifications: 35B10 35Q35 76D03 1 Introduction We study some problems related with time periodic solutions for models of incompressible fluids. We start recalling the main ideas to prove the existence of reproductive weak solutions (i.e. weak solutions defined in the time interval (0, T) taking the same initial and final values in time) for the Navier-Stokes equations and some variants where these ideas are applicable, such as Boussinesq, micropolar and magneto-micropolar models. This proof relies on the obtention of time periodic Galerkin approximations via Leray-Schauder point fixed argument. Moreover, in the case of 2Ddomains, using the uniqueness of weak solutions, the regularizing property of the system and the existence of global regular solutions when data are regular, one has that the periodic in time weak solutions defined as extension of reproductive solutions to the whole time interval (0,+∞) will be regular solutions. An extension of these results to the 3Dcase is possible imposing small enough external force, using the so called “weak/strong uniqueness” and the global strong solutions for small enough data (see Section 5). Partially supported by DGI-MEC (Spain), Grant MTM2006–07932 and CGCI MECDDGU Brazil/Spain Grant 117/06 101 102 B. Climent-Ezquerra, F. Guill´en-Gonz´alez y M. Rojas-Medar Also, we study in Section 4 some coupled models for velocity and pressure dynamic variables with another variable where the maximum principle holds, such as the generalized Boussinesq model (with temperature-dependent viscosity) and a nematic liquid crystal model with a Ginzburg-Landau penalization. In these cases one has, thanks to an adequate reformulation of the problem by truncation, existence of reproductive weak solutions as limit of time periodic Galerkin approximations. It is important to remark that Galerkin approximations do not verify the maximum principle but their limit does. Finally, we will see that, for these models related with the maximum principle, the argument to prove regularity of reproductive solutions in the Navier-Stokes framework (see Section 5 below) are not valid in general. The particular case of generalized Boussinesq model with Neumann boundary condition for the temperature can be solved with other arguments, but the case of nematic liquid crystal model remains as an open problem. 2 Navier-Stokes equations The modern theory of the Navier-Stokes equations began in the 1930s with Leray’s pioneering work ([10]). Let Ω ⊂IRd(d= 2 or 3) a bounded and regular enough domain filled by the fluid, and [0, T] the time interval. We denote Q= (0, T)×Ω and Σ = (0, T)×∂Ω. In the case where the fluid is subject to the action of a body force f, the Navier-Stokes equations can be written as follows ∂u ∂t + (u· ∇)u−ν∆u+∇p=f,div u= 0 ,(1) where u=u(x, t) is the velocity field evaluated at the point x∈Ω and at time t∈[0, T], p=p(x, t) is the pressure field and ν > 0 is the coefficient of kinematical viscosity (which is taken constant). This system can be completed with several boundary conditions. For simplicity, we fix the following non-slip boundary conditions: u(t, x) = 0,x∈∂Ω, t > 0 (2) Finally, supplementary conditions in time must be considered. The more classical is the initial condition: u(0,x) = u0(x),x∈Ω (3) Other possibility is to change this initial condition by the following time-periodic condition: u(0,x) = u(T, x),x∈Ω.(4) Mathematical properties for system (1) have been deeply investigated over the years and are still the object of profound researches. We introduce some space functions. Let Vthe vectorial space formed by all fields v∈C∞ 0(Ω)dsatisfying ∇ · v= 0. We consider the Hilbert spaces H Reproductivity and time periodicity for incompressible fluids 103 (respectively V) as the closure of Vin L2(respectively H1). Furthermore, one has H={u∈L2;∇ · u= 0,u·n= 0 on ∂Ω}, V={u∈H1;∇ · u= 0,u=0on ∂Ω} We denote L2 0(Ω) = ½p∈L2(Ω) : ZΩ p dx = 0¾. 2.1 Main classical results for the initial-boundary problem Definition 1 Given u0∈Hand f∈L2(0, T;H−1(Ω)), it will said that uis a weak solution of the problem (1),(2), (3) in (0, T ), if u∈L2(0, T;V)∩L∞(0, T;H), and verifies (3) and the variational formulation ZT 0ZΩn−u(t)v′(t) + ∇u(t) : ∇u(t)−(u(t)· ∇)v(t)u(t)−f(t)v(t)odxdt = 0, for all v∈C1([0, T ]; H)∩C([0, T ]; V), with compact support contained in (0, T). In addition, if u0∈Vand f∈L2(0, T;L2(Ω)) any weak solution will be a strong solution if u∈L2(0, T;H2∩V)∩L∞(0, T;V),ut∈L2(0, T ;H), p ∈L2(0, T;H1∩L2 0(Ω)) and verifies the system (1) pointwise a.e. in (0, T)×Ω. Remark 1 The previous definition can be extend to the case of final time T=∞changing the regularity L2(0, T)by L2 loc(0,+∞). The following results hold. Theorem 1 [22] For any u0∈Hand f∈L2(0, T;H−1(Ω)), the problem (1)- (2) has (at least) a weak solution. If Ω⊂IR2, one has uniqueness of weak solutions. Theorem 2 [22] For any u0∈Vand f∈L∞(0,∞;L2(Ω)), the problem (1)- (2) has a unique strong solution (u, p)local in time, defined in (0, T⋆)with T⋆>0small enough. In fact, if a solution has the strong regularity, it coincides with any weak solution associated with the same data (this property is called weak/strong uniqueness). Moreover, this strong solution is global in time, defined in the whole time interval (0,∞)if either Ω⊂IR2or Ω⊂IR3and data (u0,f)are small enough in their respective spaces V×L∞(0,∞;L2(Ω)). 104 B. Climent-Ezquerra, F. Guill´en-Gonz´alez y M. Rojas-Medar 2.2 On the time-periodic weak solutions Theorem 3 [8] For any f∈L2(0, T;H−1(Ω)), there exists a weak solution of (1)-(2) and (4), (i.e. the weak solution uhas the so-called reproductive property: u(0, x) = u(T, x)). Notice that the time periodic extension, e u, of any weak reproductive solution uto the whole time interval (0,+∞) is a periodic weak solution of (1)-(2) corresponding to the data, ef, defined as the time periodic extension of f. Main ideas of the proof of Theorem 3 Let ukthe unique approximate solution of the Galerkin initial-boundary problem of Navier-Stokes in the finite-dimensional subspace Vk, spanned by the first kelements of the “spectral” basis of V(orthogonal in Vand orthonormal in H), associated to a initial discrete data uk 0∈Vk. Since V֒→H, there exists a Poincar´e constant c1>0 such that c1kukk2 L2≤ k∇ukk2 L2, thus, from energy inequality, we have d dtkukk2 L2+c1kukk2 L2≤Ckfk2 H−1,(5) or equivalently d dt(ec1tkukk2 L2)≤C ec1tkfk2 H−1. Integrating from 0 to T, we have ec1Tkuk(T)k2 L2≤ kuk(0)k2 L2+CZT 0 ec1tkf(t)k2 H−1dt. (6) Now, we define the operator Lk: [0, T]→Rkas follows Lk(t) = (ck 1(t),...,ck k(t)) where ck i(t), i = 1,...,k, are the coefficients of the expansion of uk(t) in Vk. Note that kLk(t)kRk=kukkL2, because we have choose the (orthonormal in L2) spectral basis in V. We define the operator Φk:Rk→Rkas follows: Given Lk 0∈Rk, we define Φk(Lk 0) = Lk(T), where Lk(t) are the coefficients of the Galerkin solution with initial value with coefficients Lk 0. It is easy to see that Φkis continuous and we want to prove that Φkhas a fixed point. For this, thanks to the Leray-Schauder Theorem, it suffices to show that for all λ∈[0,1], the possible solutions of the equation Lk 0(λ) = λΦk(Lk 0(λ)),(7) Reproductivity and time periodicity for incompressible fluids 105 are bounded independently of λ. Since Lk 0(0) = 0, it suffices to consider λ∈(0,1]. In this case, (7) is equivalent to Φk(Lk 0(λ)) = 1 λLk 0(λ). Moreover, by the definition of Φkand (6), one obtains ec1T|| 1 λLk 0(λ)||2 Rk≤ kLk 0(λ)k2 Rk+cZT 0 ec1Tkf(t)k2 H−1dt, which implies kLk 0(λ)k2 Rk≤cRT 0ec1Tkf(t)k2 H−1dt ec1T−1=M, for each λ∈(0,1]. This bound is independent of λ∈[0,1] and k. Consequently, Leray-Shauder Theorem implies the existence of at least one fixed point of Φk, that is the existence of reproductive Galerkin solution. Thus, since previous estimates are independent of k, one has the same estimates for these reproductive Galerkin solutions. Finally, the convergence of a subsequence to a reproductive solution of (1),(2), (4) hold. 2.3 Relation between weak periodic solutions and global solutions Assume f: [0,+∞)→H−1(Ω) and T-time periodic. Navier-Stokes 2D One has (see Theorem 1) uniqueness of weak solution for the initial-boundary problem (associated to any initial data u0). Consequently, given a reproductive solution uassociated to u(0) = u(T) := u0, then uis the (unique) solution of the initial-boundary problem associated to the initial data u0, which is defined for all time t∈(0,∞). Moreover, this solution is T-periodic, because in (T, 2T) must be equal to the reproductive solution defined as ¯ u(t) = u(t−T) (which verifies u(T) = u(2T) = u0) and so on. Finally, using regularity of solution ufor strictly positive times (see [5]), it is easy to prove that every periodic solution is regular. Navier-Stokes 3D Since uniqueness of weak solution is not known, it is possible that the reproductive solution uand the global weak solution ˜ uassociated to the initial data u0:= u(0) = u(T) are different in (0, T), although they coincide locally in time, near of the initial time t= 0. 2.4 Open problems Navier-Stokes with large Reynolds number and a reaction term adding energy Previous arguments of the proof of reproductive solutions are based on (exponential) decreasing of energy (thanks to dissipative terms). Naturally, the 106 B. Climent-Ezquerra, F. Guill´en-Gonz´alez y M. Rojas-Medar same argument, is applicable to models with energy strictly decreasing in finite time. But this is not always possible. For instance, we consider the following Navier-Stokes system with large Reynolds number and a reaction term adding energy: ∂tu−ε∆u−u+∇p=f,∇ · u= 0, u(0) = u(T),u|Σ= 0.(8) The energy inequality is ∂tkuk2 L2+εk∇uk2 L2≤C(kfk2 H−1+kuk2 L2). Assuming εsmall enough such that kuk2 L26< εk∇uk2 L2, the strictly decreasing in time of kuk2 L2is not clear. Consequently, the existence of time-periodic weak solutions of (8) remains as an open problem. Exterior domains Assume Ω is an exterior domain where the Poincar´e inequality is not true. Then, to show the existence of reproductive solutions one could use the “embedding domain technique” together with the Galerkin Method, obtaining reproductive solutions in a sequence of (bounded) truncated domains, see for instance [6, 18, 17]. However, since Poincar´e imbedding is not applicable, it is not clear the controll to the pass to the limit from truncated domains to the whole domain. Some partial results are known. For example, the existence of strong periodic solutions for the Navier-Stokes equations in the following unbounded domains, either Ω is the whole space Rnor the half-space Rn +has been investigated by Kozono and Nakao [9] and Taniuchi [21] using the semigroup approach. By using potential theory, Maremonti [15] 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 [16]. Kozono and Nakao [9], making use of Lp−Lrestimates for the semigroup generated by the Stokes operator, constructed time-periodic solutions for small time-periodic forces and the stability of these solutions was considered in [21]. Yamazaki [23] analyzed the same problem of [9] in Morrey spaces. 3 Some variants of Navier-Stokes equations We can apply the argument to find reproductive solutions done for Navier-Stokes in the precedent Section, for some variants: 3.1 Boussinesq equations The Boussinesq system of hydrodynamics equations (see Joseph [7]) arise from zero order approximation to the coupling between the Navier-Stokes equation and the thermodynamic equation. Such a mathematical model reads: Reproductivity and time periodicity for incompressible fluids 107 Find the field u:Q→R3, the scalar functions (θ, p) : Q→R2which satisfy the system of equations: ∂u ∂t −ν∆u+ (u· ∇)u+∇p=αθg+fin Q, ∇ · u= 0 in Q, (9) ∂θ ∂t −χ∆θ+ (u· ∇)θ= 0 in Q. with ∂nθ= 0 on ∂Ω and RΩθ= 0 Here u, p, θ denote the velocity, the pressure and the temperature, respectively. gdenotes the gravitational field, α > 0 is a constant associated to the coefficient of volume expansion and fis a field of external forces. Again, ν > 0 is the viscosity coefficient. Finally, χ > 0 is the thermal conductivity coefficient. This system is completed with the boundary conditions (for instance) u|Σ= 0, θ|Σ= 0 and the time-periodic conditions u(0) = u(T), θ(0) = θ(T) in Ω. By taking uand θas test function in the u-system and θ-equation of (9) respectively, adding the resulting equalities considering an adequate balance (in order to eliminate the term that contains g), we obtain d dtkuk2 L2+βd dtkθk2 L2+νk∇uk2 L2+βχk∇θk2 L2≤ kfk2 H−1,(10) where βis a big enough number depending on αand kgkL∞. This together with the Poincar´e inequality gives an inequality of type (5). Indeed, it suffices to consider a Galerkin approximation for both variables, velocity and temperature, and to follow the proof of Theorem 2.3, changing ukby (uk, θk). Another boundary conditions are possible: Neumann, mixed, etc, whenever an inequality for (u, θ) similar to (5) holds. 3.2 Micropolar equations The equations that describes the motion of a incompressible viscous and micropolar fluids in Qare given by (see [12]) ∂u ∂t + (u· ∇)u−(ν+νr)∆u+∇p= 2νrrot w+f, div u= 0,(11) ∂w ∂t + (u· ∇)w−(ca+cd)∆w−(c0+cd−ca)∇div w+ 4νrw = 2νrrot u+g. The functions u:Q→R3,w:Q→R3and p:Q→Rdenote the liner velocity, the angular velocity (of rotation of particles) and the pressure of the fluid, respectively. The functions f:Q→R3and g:Q→R3denote external 108 B. Climent-Ezquerra, F. Guill´en-Gonz´alez y M. Rojas-Medar sources of linear and angular momentum, respectively. The positive constants ν, νr, c0, caand cdare viscosities, such that c0+cd> ca. This system is completed with the boundary conditions u|Σ= 0, w|Σ= 0 (for instance) and the time-periodic conditions u(0) = u(T), w(0) = w(T) in Ω. By taking uand was test function in the u-system and w-system of (11) respectively, adding the resulting equalities, taking into account that 2νr( rot w,u)+2νr( rot u,w) = 4νr( rot u,w) and |∇u|2=|rotu|2, we obtain d dtkuk2 L2+kwk2 L2+νk∇uk2 L2+ (ca+cd)k∇wk2 L2+ (c0+cd−ca)kdiv wk2 L2 ≤C(kfk2 H−1+kgk2 H−1). Starting from this inequality, the argument follows as in previous section. 3.3 Other models Other fluid models where one has existence of reproductive solutions are: magnetohydrodynamic model [14], Magneto-micropolar fluid motion [20], a convection-diffusion model describing binary alloy solidification processes [3], etc. 4 Reproductivity and maximum principle Given u:Q→R3such that ∇ · u= 0 in Qand u·n= 0 on ∂Ω, we consider the (reproductive) diffusion-advection problem for the unknown c:Q→R(a concentration): ∂tc−∆c+u· ∇c= 0, c|Σ=cΣ, c(0) = c(T), where 0 < c ≤cΣ≤con Σ, for some constants cand c. In particular, ∂t(c−c)−∆(c−c) + (u· ∇)(c−c) = 0 in Q. Multiplying by (c−c)+and integrating in Ω (notice that (c−c)+= 0 on Σ), one has d dt ZΩ |(c−c)+|2+ZΩ |∇(c−c)+|2≤0. Integrating in t∈(0, T) and using the periodic condition c(0) = c(T), one arrives at ZT 0 k∇(c−c)+k2 L2= 0. Hence c≤cin Qhold. Similarly c≥cin Qhold. Therefore, one has the following conclusion: The reproductive solution conserve the maximum principle. In the following models, the maximum principle has an important role. Reproductivity and time periodicity for incompressible fluids 109 4.1 Generalized Boussinesq system, with diffusion depending on temperature When the viscosity and heat conductivity are temperature dependent funtions in the Boussinesq system, one has the following system:        ∂tu− ∇ · (ν(θ)∇u) + (u· ∇)u+∇p=αθg+f, ∇ · u= 0, ∂tθ− ∇ · (k(θ)∇θ) + (u· ∇)θ= 0, (12) where ν: IR →IR+and k: IR →IR+are strictly positive continuous functions (the kinematic viscosity and the thermal conductivity respectively). The problem is to find a regular solution {u, θ, p}of (12) in Ω ×[0, T ], together the following boundary Dirichlet data: u= 0, θ =θ∂Ωon ∂Ω×[0, T),(13) and time-periodic conditions: u(0) = u(T), θ(0) = θ(T) in Ω.(14) We define θmin = min θ∂Ωθmax = max θ∂Ω. Thanks to the maximum principle, one has θmin ≤θ≤θmax in Q. Then, there exists νmin >0, kmin >0, νmax >0 and kmax >0 such that νmin ≤ν(s)≤νmax and kmin ≤k(s)≤kmax,∀s∈[θmin, θmax]. One can proves the existence of reproductive solution in the same way that in the classical Boussinesq case (see Section 3.1), considering the equivalent problem that result changing νby eνand kby ek, where eν(θ) =        ν(θmin) if θ < θmin, ν(θ) if θmin ≤θ≤θmax, ν(θmax) if θ > θmax, ek(θ) =        k(θmin) if θ < θmin, k(θ) if θmin ≤θ≤θmax, k(θmax) if θ > θmax. 4.2 Penalized Nematic liquid crystal model We assume the following nematic liquid crystal model in (0, T)×Ω, where Ω⊂RNfor N= 2 or 3 is an open bounded domain: (∂tu+ (u· ∇)u−µ∆u+∇p=−λ∇ · (∇dt∇d),∇ · u= 0, ∂td+ (u· ∇)d=γ(∆d−fε(d)).(15) 116 B. Climent-Ezquerra, F. Guill´en-Gonz´alez y M. Rojas-Medar [17] A.C. Moretti, M.A. Rojas-Medar and M.D. Rojas Medar. Reproductive weak solutions for generalized Boussinesq models in exterior domains. 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