Exact boundary controllability of Galerkin approximations of Navier-Stokes system for soret convection

We study the exact controllability of finite dimensional Galerkin approximation of a NavierStokes type system describing doubly diffusive convection with Soret effect in a bounded smooth domain in R (d = 2, 3) with controls on the boundary. The doubly diffusive convection system with Soret effect involves a difficult coupling through second order terms. The Galerkin approximations are introduced under certain assumptions on the Galerkin basis related to the linear independence of suitable traces of its elements over the boundary. By Using Hilbert uniqueness method in combination with a fixed point argument, we prove that the finite dimensional Galerkin approximations are exactly controllable.


Introduction
Control of fluid flows modeled by Navier-Stokes equations has received considerable attention due to its importance in practice and to the theoretical and computational challenges it poses.There is now an extensive body of literature devoted to this subject, see [13,27,11] for surveys in this area.In this paper, we consider controllability of a doubly diffusive convection with Soret effect modeled by a coupled Navier-Stokes type partial differential equations.The doubly diffusive system with Soret effect involves a difficult coupling through second order terms.Significant work has been devoted to studying the stability and physics of doubly diffusive convection with and without Soret effect (thermal diffusion), see for e.g.[3,20,26,22,24,14].These studies have reported convective flows lead to undesirable effects in certain applications.For example, thermosolutal convection is responsible for macrosegragation and can affect the uniformity and speed of growth rate in crystal growth.It is also responsible for erosion of gradient zone in solar ponds and roll-over instability (sudden over pressure) in storage and transport of gases.In spite of this, work concerning control of doubly diffusive flows is quite limited although there exists substantial work on control of thermal convection in fluid flows [15,2], for example.In [28], control of temperature in doubly diffusive flows is studied computationally using boundary heat flux ignoring Soret effect.Optimal boundary control of doubly diffusive flows with Soret effect is studied in [21].Mathematical aspects of doubly diffusive convection system such as existence and uniqueness can also be found in [21].
The doubly diffusive system under study here includes as a particular case the classical incompressible Navier-Stokes equations.Therefore it is clear that one can not expect exact controllability of this system with arbitrary target functions due to its dissipative and non-reversibility properties.The approximate controllability, despite its questionable practical utility, has been addressed in [4] for the two dimensional Navier-Stokes equations in the iso-thermal and isoconcentration cases.However, the boundary conditions in that work are assumed to be non standard (slip boundary condition or the so called Navier slip boundary condition) and the problem of approximate controllability with classical Dirichlet boundary condition is still open.In [8,12,7] local exact controllability to uncontrolled trajectories of Navier-Stokes equations is proved.In [5], global exact controllability for the two-dimensional Navier-Stokes equations in a manifold without boundary is proved.
In [17,18], exact controllability of finite dimensional Galerkin approximations of Navier-Stokes equations are proved.In the present work, we investigate the exact controllability for the doubly diffusive convection with Soret effect modeled by the Navier-Stokes system approached by Galerkin approximations.
The remainder of the paper is organized as follows.In Section 2, we present some preliminaries and study the wellposedness of the Navier-Stokes system for Soret Convection.In Section 3, we introduce Galerkin approximation of the doubly diffusive system and prove the exact boundary controllability result for this system.The proof uses the Hilbert uniqueness method to study the exact boundary controllability of linear system and a fixed point method.

Notations
This section provides, for use in later sections, a summary of notations and function spaces.Let Ω ⊂ R d (d = 2, 3) be a bounded domain with Lipschitzian boundary Γ = Γ D ∪ Γ N .As usual, L p (Ω), or simply L p denotes the linear space of all real Lebesgue measurable functions φ and bounded in the usual norm denoted by φ L p (Ω) .The inner product and norm in L 2 (Ω) are denoted by (•, •) and • , respectively.Let H s (Ω) be the usual Hilbertian Sobolev space with s derivatives in L 2 (Ω).We denote with • s the norm in H s (Ω).The closed subspace of functions in H 1 (Ω) with zero trace on Γ D will be denoted by H 1 D (Ω).The closed subspace of functions in L 2 (Ω) with zero mean on Ω will be denoted by L 2 0 (Ω).The trace space H r (Γ) consists of functions that are the restriction to the boundary of functions in H r+1/2 (Ω), r > 0. We denote the norm and inner product for functions in H r (Γ) by • r,Γ and (•, •) r,Γ , respectively.In the sequel, we denote by boldface letters R d -valued function spaces such as L p (Ω) := [L p (Ω)] d and H r (Ω) := [H r (Ω)] d .For details, see [1,10].We introduce the solenoidal spaces and We denote the dual of V by V * .If we identify H with its dual H * , then we get the following continuous and dense embedding: For a Banach space X, we denote by L p (0, T ; X) the time-space function space endowed with the norm We will often use the abbreviated notation L p (X) := L p (0, T ; X) for convenience.We also introduce the space W(0, T ) := where and We end this section by recalling some inequalities that we will use in later sections.

Poincaré-Friedrichs' inequality:
where λ > 0 is a constant.Young's inequality: For any a, b ≥ 0 and ǫ > 0, and q, r > 1 approximation in the body force term in the momentum equation, as with the boundary conditions [u + ǫ(−pn in Ω where u is the velocity, θ the temperature, S the concentration and p the pressure.The nondimensional parameters P r, Le, Gr θ and Gr S denote the Prandtl number, the Lewis number, the thermal Grashof number and the species Grashof number, respectively.The ratio of species buoyancy to thermal buoyancy N r is defined by N r = GrS Gr θ .In (1) 4 the first term on the right side corresponds to Soret effect.The cases α * > 0 and α * < 0 corresponds to positive and negative Soret effect, respectively.The Soret effects can have significant implications on convection in liquid mixtures, for example semi-conductor crystal growth [14].Therefore the Dufour effect has been neglected here in comparison to Soret effect as is common for flows in liquid mixture.In addition, the constant ǫ in the boundary condition on Γ N is non-negative.Note that by setting ǫ = 0 the Robin boundary conditions become the Dirichlet boundary conditions.In fact, in actual computational implementation one can develop approximations of Dirichlet control problem by allowing ǫ → 0 + , see [21] for studies related to this in the context of optimal control.
Before proceeding, we present a little motivating discussion regarding the nonlinear Robin type boundary conditions.We observe that by integration by parts and Similarly, we can show that We now define the weak solution to the initial boundary value problem ( 1)-( 4) as follows: to be a weak solution of ( 1)-( 4) if Then, there exists a solution (u, θ, S) ∈ W(0, T ) satisfying (5) and and where Le M 1 and M 3 : Proof.We employ Galerkin approximation, a priori estimates and compactness methods to prove the existence of solutions.Let {(e k (x), [19] for a proof of existence of such a basis.For each m = 1, 2, . .., we set , where (u 0m , θ 0m , S 0m ) is the L 2 − orthogonal projection of (u 0 , θ 0 , S 0 ) onto the space V m .Since ( 6) is an initial value problem for nonlinear ODEs, existence of unique local solutions in some neighborhood [0, t m ), for some t m > 0, follows by Picard-Lindelöf theorem.The a-priori estimates we will prove later in L ∞ (0, T ; L 2 (Ω))-norm show that continuation of solutions beyond t m follows.We will employ energy methods to derive those a-priori estimates.First we multiply (6) k , and add these equations for k = 1, . . ., m.Using the skew-symmetry of the trilinear forms, we get From the equation ( 7) 2 , by integration we obtain the a priori estimate By applying Cauchy-Schwarz inequality and Young's inequality in (7) 3 , we obtain Integrating this with respect time and using the fact that θ m remains bounded in a bounded set of Let us now turn to the a-priori estimate for u m .First we note that the right hand side of (7) 1 can be majorized using Young's inequality as follows Applying the Poincare-Friedrichs inequality and employing the result in (7) 1 we obtain ΓN .Integrating this with respect to time and using ( 8)-( 9), we obtain In order to obtain bounds for ∂ t u m , we first notice that by Holder's inequality and the embedding ) is bounded due to the bounds in ( 8)- (10).Similarly, we can show are bounded as well.The a-priori estimates we obtained so far allow us to extract subsequences again denoted by strongly in L 2 (0, T ; L 2 (Ω)) .

  
Here the strong convergence follows by the Aubin- . When taking limit in (6), it is convenient for us to use the trilinear forms c(•, •, •) and c i (•, •, •) involving boundary terms, i.e., However, the presence of nonlinear boundary terms require that we prove strongly.In order to prove such a convergence, we first recall the integration by parts formula [10, Equation (I.2.17), p. 28] we obtain a unique solution v ∈ L 2 (0, T ; H 1 (Ω)) such that Therefore by taking v in (11) to be the unique solution of this variational problem (12) and using the fact that The weak convergence u m → u in L 2 (0, T ; V) and trace theorem imply that ) is bounded.Therefore the required strong convergence follows from (11).
Let ψ i (t), i = 1, 2, 3, be a continuously differentiable function on [0, T ] with ψ i (T ) = 0. We multiply (6) i by ψ i , i = 1, 2, 3 and integrate with respect to time.Further, we integrate by parts in the time derivative term to move the derivative onto ψ i .Now we can take limit in (6) by using standard techniques and show (u, θ, S) is indeed a solution of (5).The a-priori estimates in the lemma follow by taking the limit on the a-priori estimates ( 8)-( 10) and using the weak lower semi-continuity of the norms .
The uniqueness of the weak solutions discussed in Proposition 1 is an open problem.We denote by (u(x, t; (g, h, f )), θ(x, t; (g, h, f )), S(x, t; (g, h, f ))) the set of all possible solutions.

Exact controllability of Galerkin approximations
In this section, we introduce a Galerkin approximation of (1) and, for this finite dimensional system, we establish the exact controllability.
Theorem 1.The Galerkin approximation ( 14) is exactly controllable in the sense of (15).Moreover, the cost of control J (g, h, f ) is bounded independently of the nonlinearity.
Proof.The proof of this theorem uses a fixed point argument.In order to show and make explicit that the cost of control can be bounded independent of nonlinearity, we introduce a family of state equations with the boundary conditions and initial conditions u(x, 0) = u 0 (x) , θ(x, 0) = θ 0 (x) , and S(x, 0) = S 0 (x) in Ω, where α, β, γ ∈ R.
The result we just obtained allows us to define a functional G : L 2 (0, T ; E) → R by where is the solution to (18) that satisfies (15)} is the set of admissible controls.We will use a duality argument to prove that G(U ) is bounded by a constant independent of U, α, β and γ.That is, where C is a constant independent of U,α β and γ . ( where g := (g, h, f ).
Let us also define two functionals F 1 (g, h, f ) and F 2 (g 1 , g 2 , g 3 ) by ΓN ×(0,T ) Then the functional G(U ) can be written as and by the duality theorem of Fenchel and Rockafellar, see for example [23], we have that where ǧ := (g 1 , g 2 , g 3 ) and L * : 3 is the adjoint of L, and F * 1 and F * 2 are the Fenchel conjugate of F 1 and F 2 , respectively.It follows easily from (22) that the adjoint L * is given by Moreover, the Fenchel conjugate F * 1 and F * 2 can be shown to be given by ΓN ×(0,T ) Therefore (24) becomes where ǧ := (g 1 , g 2 , g 3 ) .But, in view of the assumptions on the bases for E 1 , E 2 and E 3 , for some constants c 1 and c 2 depending only on E 1 , E 2 and E 3 .Therefore (25) can be written as Ω×(0,T ) ) where ǧ := (g 1 , g 2 , g 3 ) .From (20), we obtain by the skew symmetry properties of the trilinear forms that Integrating the preceding differential equations with respect to time from t to T yields Adding the last three equations and integrating with respect to t from 0 to T yields + T 0 t (P r 2 Gr θ µi 3 , ζ) + α * Le (∇ξ, ∇ζ) + (P r 2 Gr θ N r µi 3 , ξ) ds .
(27) Notice by the finite dimensionality of the spaces E 1 , E 2 and E 3 , we have for some constant C > 0 that depends only on E 1 , E 2 and E 3 are finite dimensional.Also by the finite dimensionality of E 1 , E 2 and E 3 , we have (29) for some constant C > 0 depending only on E 1 , E 2 and E 3 .Using ( 28) and ( 29) in ( 27) yields for some constant depending only on E 1 , E 2 and E 3 , P r, Le, α * , Gr θ and N r.Employing (30) in (26), we have where ǧ := (g 1 , g 2 , g 3 ) .Therefore, we have from which it follows that G(U ) ≤ C, where C is a constant independent of U , α and β.Let us finally consider the nonlinear system (16).For a given U in L 2 (0, T ; E 1 ), let (g, h, f ) be the unique element such that 1 2 ΓN ×(0,T ) This defines a continuous mapping U → (g, h, f ) from L 2 (0, T ; E 1 ) into L 2 (Γ N × (0, T )) 3 .Let us denote by (u(U ), θ(U ), S(U )) the solution of (18) with control (g, h, f ) = (g(U ), h(U ), f (U )) .It follows from (18) by the skew symmetry properties of the trilinear forms that 1 2 for some constant C. In view of the uniform estimate ( 23), we have from (33) that, when U varies in L 2 (0, T ; E 1 ), (u, θ, S) remains in a bounded subset S 1 × S 2 × S 3 ⊆ L 2 (0, T ; E 1 ) × L 2 (0, T ; E 2 ) × L 2 (0, T ; E 3 ).Let us next prove that ∂ t u remains bounded in a bounded set of L 1 (0, T ; E 1 ) when U varies in S 1 .To this end, notice that from ( 18 This proves that ∂ t u remains bounded in a bounded set of L 1 (0, T ; E 1 ) when U varies in S 1 .Let us now define a mapping Q from S 1 to S 1 by U → u(U ).Then the range of Q is relatively compact in S 1 by Aubin-Simon's lemma.Schauder fixed point theorem now implies that Q has a fixed point in S 1 .Therefore since (18) is exactly controllable in T > 0, we have that ( 17) is exactly controllable.
Exact boundary controllability of Galerkin approximations of Navier-Stokes system for soret convection 45 we have the embeddings V ֒→֒→ H ֒→ V * and H 1 D (Ω) ֒→֒→ L 2 (Ω) ֒→ H 1 D (Ω) * .Moreover, we have the following convergence results: Simon compactness lemma [25, Corollary 4, p.85] as Huntsville, which he joined in 1999.Prior to this appointment, he was an NRC research fellow in the Flow Modeling and Control Branch at NASA Langley Research Center.Previous to that, he was a visiting assistant professor in the Center for Research in Scientific Computation at North Carolina State University.As principal investigator of various research grants, he has conducted research for agencies such as the National Science Foundation, DOD, NASA Langley Research Center and NASA Marshall Space Flight Center.He has to his credit numerous refereed publications in prestegeous journals.He has also given invited lectures in France, Austria, Spain, India, Canada and the United States, and at professional societies such as SIAM, IEEE, AIAA, and ASME.His scientific expertise has been recognized by over 1400 citations of his publications and, by invitations to consult by industry and government labs and to serve in the Editorial Boards of a number of international journals.