VISCOUS-INVISCID COUPLED PROBLEM WITH INTERFACIAL DATA

The work presented in this article shows that the viscous/inviscid coupled prob- lem (VIC) has a unique solution when interfacial data are imposed. Domain decomposition tech- niques and non-uniform relaxation parameters were used to characterize the solution of the new system. Finally, some exact solutions for the VIC problem are provided. These type of solutions are an improvement over those found in recent literatures.


VISCOUS-INVISCID COUPLED PROBLEM WITH INTERFACIAL DATA Sonia M. Ramirez
Abstract.The work presented in this article shows that the viscous/inviscid coupled problem (VIC) has a unique solution when interfacial data are imposed.Domain decomposition techniques and non-uniform relaxation parameters were used to characterize the solution of the new system.Finally, some exact solutions for the VIC problem are provided.These type of solutions are an improvement over those found in recent literatures.
2000 Mathematics suject classification: 35A07, 35A15 1. Introduction.The Navier-Stokes equation is the primary equation of computational fluid dynamics describing the flow/motion of fluids in R n , (n = 2, 3).These types of equations are often used in computations of aircraft and ship design, weather prediction, and climate modelling.By appropriate assumptions, it has been generalized to a system of equations known as the incompressible Navier-Stokes equations, see [8].This important system has been studied for centuries by mathematicians, engineers and other scientists to explain and predict the behavior of the system under consideration, but still the understanding of the solutions to this system remains minimal.The challenge is to make substantial progress toward a mathematical theory which will solve the puzzle behind the Navier-Stokes equations.To make contributions to this mathematical theory, scientists have studied and derived many other systems from it.Among them is the viscous/inviscid coupled problem (VIC) introduced first by Xu Chuanju in his Ph.D dissertation [15].
The work presented in this paper involves Xu's [17] problem and focuses on three main objectives.The first one is to show the existence and uniqueness of the solution for the system, which results from the viscous/inviscid coupled problem when interfacial data (VIC-ID) are imposed.The second objective is to prove that the solution of this system can be obtained as a limit of solutions of two subproblems defined in different subdomains of the domain by using non-uniform relaxation parameters.Xu [17] used a similar techniques for the VIC problem but using lifting operators and uniform relaxation parameter.Finally, the last objective is to provide new exact solutions when all boundary conditions are satisfied in at least one of the subdomains (weaker boundary conditions) of the viscous/inviscid coupled problem, these solutions are an inprovement over those found in recent literatures.The new improvements presented in this paper demostrate progress towards the existing theory of the VIC problem and therefore for the Navier-Stokes equations.
We end this section by introducing some notation, definitions and very well known result from P.D.E that can be found [1] and [14] .Along this article we use boldface letters to denote vectors and vectors functions.
Let Ω be a bounded, connected, open subset of R We define the boundaries Γ + and Γ − of the subdomains Ω + and Ω − respectively, as follows and the interface is given by we assume Γ = ∅.Let n denote the unit normal on ∂Ω to Ω, and n + , n − are the unit normals to ∂Ω + , and ∂Ω − , respectively.See figure 1 for a typical decomposition of the domain Ω.We denote by L 2 (Ω) the space of real functions defined on Ω that are squareintegrable over Ω in the sense of Lebesgue measure dx = dx 1 dx 2 .This is a Hilbert space with the scalar product We then define the constrained space L 2 0 (Ω) as so L 2 0 (Ω) consists of all square integrable functions having zero mean over Ω.Let D(Ω) be the linear space of functions infinitely differentiable and with compact support on Ω.Then set For any integer m, we define the Sobolev space H m (Ω) to be the set of functions in L 2 (Ω) whose partial derivatives of order less than or equal to m belong to L 2 (Ω); i.e where The set H m (Ω) has the following properties: (ii) H m (Ω) is a Hilbert space with the scalar product Since we are dealing with 2-dimensional vector functions, we use the following notation and assume that these product spaces are equipped with the usual product norm (or any equivalent norm).We defined the space H 1 0 (Ω) as the closure of D(Ω) for the norm .m, Ω .In order to study more closely the boundary values of functions of H m (Ω), we assume that Γ, the boundary of Ω, is bounded and Lipschitz continuous, i.e.Γ can be represented parametrically by Lipschitz continuous functions.Let dσ denote the surface measure on Γ and let L 2 (Γ) be the space of square integrable functions on Γ with respect to dσ, equipped with the norm Theorem 1 (Trace Theorem).If ∂Ω is bounded and Lipschitz continuous, then there exist a bounded linear mapping γ : H (ii) γv = v| ∂Ω for all v ∈ D(Ω).
For proofs see [3] and [11].Both theorems can be extended to vector-valued functions.The range space of the mapping γ, denote by H 1/2 (∂Ω), is a Hilbert space with the norm µ 1/2,∂Ω = inf Let H −1/2 (∂Ω) be the corresponding dual space of H 1/2 (∂Ω) , with norm given by μ −1/2,∂Ω = sup where , denotes the duality between H −1/2 (∂Ω) and H 1/2 (∂Ω).For any vector function v ∈ L 2 (Ω), we consider the pair of functions v − = v| Ω− and v + = v| Ω+ .We define the following inner products in Ω + and Ω − respectively as follows: and for any Ψ and Φ ∈ L 2 (Γ) as: The scalar product on which coincides with the usual one on L 2 (Ω).Consider the following space: Theorem 3 (Divergence Isomorphism Theorem).The divergence operator is an isomorphism from V ⊥ onto L 2 0 (Ω) and satisfies For a proof see Girault [7].This theorem plays an important role in the proof of uniqueness and existence of the VIC-ID problem.
2. Viscous-Inviscid Coupled Problem with Interfacial Data.In this article we show that is it is possible to impose further conditions on the interfacial data and still have a solution to the problem.These conditions are expressed in the form of membership to certain function spaces.Specifically, we study the VIC-ID problem.For f ∈ L 2 (Ω), ĝ0 ∈ H −1/2 (Γ), p0 + ∈ L 2 (Γ), α and ν positive constants, find two pairs (u − , u + ), (p − , p + ), defined in (Ω − , Ω + ), satisfying the following conditions where the domain Ω satisfies the properties mentioned previously, and the function ĝ0 satisfies the condition Γ ĝ0 = 0.
In fluid mechanics this conditions is generally known as compatibility condition, see [5] and [10].From above, the first three equations correspond to the viscous part, from fourth to sixth to the inviscid part, and the last four corresponds to the interface data.Next we prove existence and uniqueness of the solution.For that we use the saddle point theory which involves: (i) finding the weak or variational formulation of the VIC-ID problem, to describe the spaces where the interfacial data have sense.(ii) rewriting the weak formulation to find the saddle point problem; which involves the definition of two bilinear forms a, b and must satisfy the following conditions: for some positive constant c.
(iv) Inf-sup condition for the bilinear form b: inf q∈M sup v∈X b(v, q) v X q M ≥ β, for some positive constant β.
Above conditions guarantee the existence and uniqueness of the solution for the given problem, for more details see [3] and [4].By showing the following steps we can find the weak formulation of (2).
(i) From the first and fourth equations of (2) the following inner product equations are obtained (ii) From the second and the fifth equations of (2) we get the two pair of inner product equations and adding equations ( 5) and ( 6), we have Using the following well known identities for vector functions: adding equations ( 3) and ( 4), and by appropriate substitutions, We have the following weak formulation of VIC-ID problem: find (u, p) where X, M are the two Hilbert space, defined by with respective norms The VIC-ID problem is well posed in the sense that its corresponding weak formulation admits a unique solution.The statement of the theorem and proof is given below, which is one of the main results of these work.
Proof.The second part of the theorem is trivial.In order to prove the well posedness of (8), we need to apply the saddle point theory.This can be achieved by reorganizing the terms of the weak formulation by defining two bilinear forms a, b as follows: By appropriate substitutions in (8), the saddle point problem is given as: find b(v, q) = ĝ0 , q + Γ ∀q ∈ M. To show the above forms a, b satisfy the conditions mentioned previously: (ii) The form a is coercive, by definition of the inner product, we have choosing min(α, ν) = d 0 , and applying Schwartz inequality we get , and adding appropriate terms to complete the norms for the above form, we obtain (iv) The form b satisfies the inf-sup condition in the space X × M : The objective is to decompose the general vector v into v + and v − in such a way that inf-sup condition is satisfied.Let q ∈ M , and q − can be decomposed as where q 0 − ∈ L 2 0 (Ω − ) and r − is a constant.The decomposition of q − is justified by the following calculation where k 1 is a constant,and set Using theorem 3 it follows that there exists a positive constant β − and a function Choosing a function g ∈ X such that where To establish the existence of w we apply a special case of the saddle point theory when only one bilinear form is given.Consider the following bilinear form b : that satisfies b(w, q) = (∇•g, q) − ∀q ∈ L 2 o (Ω − ).The norms for this spaces are .
Therefore b satisfies the inf-sup condition for β = 1.To guarantee surjectivity of b it is necessary to show that for every q ∈ L 2 o (Ω − ) there exists v ∈ H 1 0 (Ω − ) such that b(v, q) = 0.By choosing v = −∇q such a way so that it satisfies the required condition, otherwise ∇q 2 0, Ω− = 0 occurs, if and EJQTDE, 2005 No. 4, p. 9 only if q is zero.Therefore, the existence of w is proved.Since we prove the existences of w, then the following statement is true By setting v − = v 0 − − r − v− , and using relationships (11) , (12), and ( 14) it follows that by linearity of the inner product, we have Using the properties of q 0 − , ∇v − and divergence theorem, it follows which can be equivalently expressed in the following form By applying the same procedure as in equation ( 11) to decompose q + in the subdomain Ω + as q + = q 0 + + r + , where q 0 + ∈ H 1 (Ω + ) ∩ L 2 0 (Ω − ) and r + is a constant.Let v 0 + = ∇q 0 + , then (∇q Using the fact that ∇q + ∈ L 2 (Ω + ), we can choose v + = v 0 + , so (∇q and the vector v is characterized as follows: Therefore, v ∈ X and from equations (15), and ( 16), we can write

By using
q − 2 0, Ω− = (q − , q − ) − = (q 0 − + r − , q 0 − + r − ) − , and applying the linearity of the inner product in the above equation, we obtain By using (17) in the bilinear form b, we have Therefore, the bilinear form b satisfies The next step is to show that the components of the vector v are bounded.
Using the definition of v − and the relationships obtained in ( 12), (13), and ( 14), we have the following estimates Since w ∈ H 1 0 (Ω − ), there exist constants c and ĉ such that Therefore, By knowing that r − 0, Ω− = q − − q 0 − 0, Ω− , we can find a constant c 1 such that By combining ( 19), ( 20), (21), and ( 22), it follows that v − satisfies the following inequality: where c 2 = 1+ĉc1 β− .Using the definition of v + we get the following estimates: By making use of ( 23) and ( 24), it follows that Using ( 18) and ( 25), we obtain and setting β = β− c1+β− , in the above inequality we get By taking inf-sup of (26), we get the following inequality which completes the proof.
3. The iteration-by-subdomain procedure and its convergence.The purpose of this section is to prove that the solution of VIC-ID problem can be obtained as a limit of solutions of two subproblems in the subdomains Ω − and Ω + , respectively, of Ω.Let {p m + } be a sequence of functions in L 2 (Γ) such that pm + −→ p + , for some p + ∈ L 2 (Γ).We define the sequence of function pairs (u m − , p m − ) m≥1 by solving for each m the following viscous interfacial data problem in Ω − : We have to find the variational problem corresponding to the viscous interfacial data problem using the saddle point theory.For that we consider the first and the second equation of ( 27) to get the pair of inner product equations as follows: Using the well known identities for vector functions, by combining the equations in (28), and making the appropriate substitutions we can state the following weak formulation of the viscous interfacial data problem: and A − is defined by The next step is to make use of the following theorem, from the Navier-Stokes equations literature( for more details see [2]) that guarantees the existence and uniqueness of the solution for the weak formulation of the viscous interfacial data problem.
By the theorem we can define a sequence of function pairs (u m − , p m − ) to state the inviscid interfacial data problem in Ω + as follows: The functions ϕ m are defined by: 3.1 Convergence of the iteration-by-subdomain procedure.To prove convergence we use the fact that the relaxation parameter in ϕ m is non-uniform and the properties of the norms imposed on the interface Γ and the spaces X + and X − .The next step is to prove the sequence {ϕ m } converges to u − n − with respect to the dual norm of H 1 2 (Γ) .For that we have the following estimates: Using Schwartz inequality, we have the following By taking sup and dividing by µ 1/2,Γ , we obtain By using the following estimate (see [1] and [11] for further details), and equation (30) in theorem 5, , we obtain the following Therefore, Since t m −→ 0 as m −→ ∞, and pm The next objective is to prove that u m + − u + −→ 0 in Ω + .By taking f + = 0, and using the variational form A + as discussed in section 4.1 for the inviscid problem, we have and (34) from theorem 6, we get and by (30) in theorem 5, we get Using the hypothesis pm which concludes the proof of convergence of the iteration-by-subdomain procedure.4. Exact solutions with weaker boundary conditions.During the investigation of the viscous/inviscid coupled problem we found few exact solutions with weak boundary conditions.For computational purposes it is always advantageous to have as many test functions as possible, and the visualization for analyzing and characterizing the behavior of the problem under investigation, see [12].Also with the purpose of designing newer algorithms and testing the ones suggested by others.We briefly describe two kinds of solutions with weak boundary conditions found in earlier literatures.The first kind of solutions (u, p) are those that do not satisfy some of the boundary conditions in both subdomains, but satisfy the interface conditions.In such solutions the vector field component, u(x, y) = (u 1 (x, y), u 2 (x, y)), has a particular form, in such a way that one of the component, either u 1 (x, y) = 0 or u 2 (x, y) = 0.In other words, the graph of the vector field u is embedded in R 3 .Almost all the solutions found in earlier articles are of this kind, see [18]).
The second kind of solutions (u, p) are those that do not satisfy any of the boundary conditions but satisfy the interface conditions.In this case the graph of the vector field component is in R 4 .In fact only one of this kind of solution was found by Xu in his articles [16].New examples of these solutions are found in Ramirez's dissertation [13].Exact solutions with weaker boundary conditions are those that satisfy all the boundary conditions in at least one of the subdomains, and the interface conditions.In fact these solutions are an improvement over the exact solutions with weak boundary conditions found in earlier literatures.Here we present two types of solutions, which are exponential and polynomial in nature.Given the domain Ω as follows: which is decomposed into the two subdomains Ω + and Ω − as Example 1: Exact solution with weaker boundary condition in Ω + .Consider the following data: (i) the external force in the subdomain Ω − is of the following form + e 1−y 2 (e 1−y 2 − 1)[−4 + 8y 2 ]] − π sin(πy) sin(πx)), and (ii) the external force in the subdomain Ω + is of the following form Then the vector field component u of the exact solution in the entire domain Ω is given by u(x, y) = (4ye 1−y 2 (e 1−y 2 − 1)(e 8+x 3 − 1) 2 , 6x 2 (e 1−y 2 − 1) 2 e 8+x 3 (e 8+x  For more information of these solutions see [13].

5.
Conclusions.We have presented in this article (i) existence and uniqueness of the viscous-inviscid coupled problem with interfacial data, when suitable conditions are imposed on the interface (see theorem 4).(ii) convergence of the algorithm-subdomain procedure without using lifting operators (see section 3.2).(iii) exact solutions with weaker boundary condition in at least one of the subdomains for the viscous-inviscid coupled problem (see section 4), an improvement over those exact solutions with weak boundary conditions found in earlier literatures.
Further extensions of this work will be focused in several objectives: (i) Approximate the solutions of the viscous/inviscid coupled problem using finite element methods, with non-uniform relaxation parameters found along this work to improve convergence.(ii) Investigate the existence and uniqueness theorem for general case of the Navier-Stokes equations and apply to special applications in the field of fluid dynamics, see [6] and [9].(iii) Extend the viscous-inviscid coupled problem with interfacial data to the unsteady Navier-Stokes equations by applying a similar methodology as Xu did in his work.EJQTDE, 2005 No. 4, p. 22

Figure 1 :
Figure 1: Decomposition of the domain Ω for the scalar product (∇u , ∇v) = ∂ui ∂x j ∂vi ∂x j .Thus we have the following divergence isomorphism theorem.EJQTDE, 2005 No. 4, p. 4 for some positive constant c.EJQTDE, 2005 No. 4, p. 5 (ii) Coerciveness of the bilinear form a: a(v, v) d v 2 X , for some positive constant d. (iii) Continuity of the bilinear form b: |

H 1 0
= .0, Ω− and .L 2 o = | .| 1,Ω− , respectively.Since the continuity of b is obvious from the definition itself.Next step is to show that the form b satisfies the inf-sup condition as follows: taking v = −∇q, and substituting in the form b we have the following b

Figure 2 Figure 2 : 2 x 2 + y 2 .Figure 3 Figure 3 :
Figure 2 below shows the behavior of the velocity vectors close to the boundary in Ω and on the interface Γ.In other words, the graph below shows that the velocity vectors have an uniform behavior in the viscous part, contrary to the inviscid part where the behavior of the velocity vectors change dramatically close to the boundary, this happens because some of the boundary conditions are not satisfied in the inviscid part.