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An approximation of the axially symmetric flow through a pipe-like domain with a moving part of a boundary
Boundary Value Problems volume 2013, Article number: 241 (2013)
The purpose of this work is to study the existence of solutions for approximation of an unsteady fluid-structure interaction problem. We consider a perturbed Navier-Stokes equation in the cylindrical coordinate system assuming axially symmetric flow. A priori unknown part of the boundary (that interacts with the fluid) is governed with a linear equation of fifth order. We prove the existence of at least one weak solution as long as the boundary does not touch the axis of symmetry. An explicit expression for a class of divergence-free functions is given.
MSC:35Q30, 35Q35, 35D05, 74F10, 76D03.
This paper is devoted to the perturbed Navier-Stokes equations in a cylindrical coordinate system assuming rotationally symmetric flow
with an artificial equation for the pressure
() in an a priori unknown domain
with the given
and ρ, μ, L, , T, , are given positive constants, , .
Unknown are the velocity field , the pressure p and the interface η on . We shall require from η to satisfy the following conditions:
for , ,
for any and, for convenience, also
Here a, b, c, f are given nonnegative numbers and d, e, E are supposed to be positive. Moreover, is a given function. Of course, (1.1) is to be complemented with additional boundary and initial conditions. We shall consider
for any , ,
for any and , ,
for any , ,
for any , ,
for any , and, finally,
for any , . Here and are given functions of two variables for and , respectively, .
The aim of this paper is to extend the existence result of  concerning the two-dimensional fluid-structure interaction problem to the radially symmetric case expressed in terms of the cylindrical coordinate system. Since we have not been able yet to handle this nonlinear problem modelling the flow of an incompressible viscous fluid in a cylinder with deformable wall, here nonlinear perturbed Navier-Stokes equations coupled with the artificial equation (1.2) and assuming in the equation of the wall (1.6) are studied. As it turns out, we have not been able to let
up to now due to the lack of regularity in the time variable.
We can give a justification of this limit, but only for a linearised perturbed Navier-Stokes equations and by including the additional term
in (1.6). Let us illustrate some arguments for the presence of this term in (1.6). To prove our existence result, we shall transform for h given by (1.4) on a fixed domain by setting
The incompressibility condition takes then the form
In the final stage of the proof, we are to send in an integral identity that includes also terms
solve an appropriate collection of approximating problems and are test functions. The only way to overcome the lack of regularity for is to consider divergence-free test functions, i.e.
This motivates us to find an explicit expression for a class of functions satisfying (1.18). For , let us define
and the desired equality (1.18) is valid for . But now another difficulty arises, for, by inserting (1.19) into (1.17), the term
appears. The problem is that we cannot deduce more than weak convergence of the velocity gradient in . What saves us are just the strong convergences of
This example of implies that we in general need more regularity for , and therefore we will assume (1.15). Section 7 provides a detailed analysis of a limit process as in the linearised perturbed NS equations.
Our study was originally initiated by the papers of Quarteroni [2–4] and it is in some sense a continuation of the paper . Due to the fact that our main contribution is in introducing the cylindrical coordinate system to the fluid-structure interaction-like problem, we do not provide a representative list of references for 2-D problems. For a full summary of the research on solvability of the fluid-structure interaction problem, we refer the reader to the introduction in  and references therein. See also [6–8] and . The methods that we apply borrow some material from [5, 10–12] and ; nevertheless, their application to our problem seems to be not straightforward.
The paper is organised as follows. In Section 2 we regularise once more our problem by considering it on a domain without the axis of symmetry (see, e.g. ) and then we transform it on the fixed domain. The main result of this section is the identity (2.21). Sections 3 and 5 provide necessary a priori estimates including the delicate identity (3.12). Sections 4 and 6 present main results of the paper. In Section 4, using the Schauder fixed point theorem, we prove the existence of the solution for the unknown interface, and in Section 6 we go to a limit with our regularised parameters. Finally, in Section 7 we let in (1.2) but only for the linearised version of our problem, see (7.2).
Throughout the paper we will always assume that there are positive constants α, K, , such that
for all . Note that at the end we have to prove the validity of (1.20).
Our approach is to consider an approximate problem first. Given numbers
and given a function satisfying (1.20), consider the system
equipped with the following boundary and initial conditions:
for any , ,
for any and , ,
for any , ,
for any , ,
for any , ,
for any , ,
for any , ,
for any , ,
for any and, finally,
for any . Note that in this section h is given and , p and η are to be found (h and η are therefore not related by (1.4)).
Assume for a moment that is in fact a smooth solution of problem (2.2)-(2.6) above. Then
for , , and
for , solve the following problem:
in , where ,
for any , with the boundary and initial conditions listed below:
for any , , ,
for any , , ,
for any , , ,
for any , ,
for any , and, finally,
for any and for any .
We continue this section by making precise the meaning of the solution of problem (2.9)-(2.17). To this end, we first define
Throughout this and the next sections, we assume
The function spaces we use are rather familiar, and we adopt the notation of .
Definition 2.1 We call a weak solution of the initial boundary value problem (2.9)-(2.17) if the following two properties are fulfilled:
, , , and .
satisfies the system of differential equations, that is,(2.21)
for every , , .
Remark 2.1 Note that
for every test function with .
At the end of this section, we discuss the existence and uniqueness of weak solutions to problem (2.9)-(2.17) for given h, , and satisfying (1.20) and (2.20).
Theorem 2.1 Assume (2.1), (2.20), and let satisfying (1.7) and (1.8) be given. Hereafter denotes h defined by (1.4) with the given , and we assume that (1.20) holds. Then there exists a unique solution
of problem (2.9)-(2.17) in the sense of Definition 2.1 such that
where the function ℓ in (2.23) is defined by (2.10) for , i.e.
Proof of Theorem 2.1 By analogy with the approach taken in [, the proof of Theorem 6.3], we can construct our weak solution by the implicit time discretisation method. The paper  presents all the technicalities of the proof, therefore we omit the proof here. □
We conclude this section by some interpolation inequalities which are needed especially for estimating the nonlinear item. For this purpose, we complete the notations of function spaces (2.18) and (2.19) by appropriate weighted spaces. Let
Proposition 2.1 (i) Let φ be any function in such that on or on . Then, for any and for any number θ with
there exists a constant such that
Moreover, if such that on or on for almost all , then the following holds for any :
Let now and fulfil the condition
Then there exists a constant such that
If, moreover, such that on and on for almost all , then the following holds for any :
Proof (i) The form of Nirenberg-Gagliardo inequality (2.26) can be found, e.g. in [, Theorem 2.2]. Then (2.27) follows from (2.26) for by integration over .
(ii) To prove (2.28), we define and transform the integrals on the domain D into integrals on the cylinder . Then the weighted interpolation inequality (2.28) is equivalent to an unweighted interpolation inequality in for given on , which yields the assertion (cf. also [, Theorem 19.9] for ). Inequality (2.29) again follows by integration of (2.28) over with . □
3 The first a priori estimate
Our ultimate goal is to show that a subsequence of solutions of the approximate problems (2.9)-(2.17) converges to a weak solution of (1.1)-(1.14). For this we will need some uniform estimates. Let us start with the following theorem.
Theorem 3.1 Let the hypotheses of Theorem 2.1 be satisfied, and let be the corresponding solution of problem (2.9)-(2.17).
Then there exists a constant C such that
for a.e. time , where
The constant C does not depend on α, K, ε, κ and δ.
Before proving Theorem 3.1, we pause to introduce some of its consequences.
Theorem 3.2 Let and . According to the energy estimates (3.1), we see that the set
for any , the set
and the set
for all ε, κ, δ. Moreover, we obtain that
for any fixed and all ε, κ, δ satisfying (2.1).
Proof of Theorem 3.2 It remains to prove (3.5) and it suffices to prove the assertion for the first component of u since for it follows from estimate (3.1). For a moment, let us fix x and t and set , . As , we have
Now we shall distinguish two cases. In order to derive (3.5) in the case , let us multiply (3.6) by and integrate it over . Due to (3.7), this gives the relation
and (3.5) follows by integration over . If , then
and (3.5) follows easily. □
Note that (3.8) is a special case of the Hardy inequality [, Theorem 6.2]. Moreover, we remark that this argument also applies to q, which in analogue to (3.5) yields that
for any fixed .
Proof of Theorem 3.1 Fix now a positive and substitute into the identity (2.21). This is legitimate since . Then, with the assistance of (2.23), we write the resulting expression as
Our plan now is to check the ellipticity condition that follows from the following two propositions.
Proposition 3.1 Suppose
see (2.19). Then
Proof of Proposition 3.1 Let us write the left-hand side of (3.12) as a sum of two integrals
and we deduce that it equals
Now, let us focus on the first term in the second line of (3.13).
Lemma 3.3 Assume , then
Proof of Lemma 3.3 Assume for a moment that . After integrating by parts, we obtain
and (3.14) follows easily. The validity of (3.14) for is a consequence of the approximation argument. □
We apply now Lemma 3.3 to the first term on the second line of (3.13) and the validity of (3.12) can be verified by direct computations. Let us note, however, that it is rather tedious. □
The proof of the next proposition can be found in [, Lemma 4.3].
Proposition 3.2 Let the hypotheses of Theorem 2.1 be satisfied, and let , then
We would now like to finish the proof of Theorem 3.1. Returning to the relation (3.11), we estimate its second line with the assistance of (3.12), (3.15) and its right-hand side by Hölder’s inequality to find
for . Next, observe that for , the last two terms on the right-hand side of (3.16) are bounded by
for any and therefore also for . Consequently,
for a.e. . Thus Gronwall’s inequality yields the estimate
that due to (3.16) and (3.17) implies (3.1). □
4 A free boundary value problem
So far we have studied problem (2.9)-(2.17) for with the given function , and Theorem 2.1 gives the solution . For this section, keeping still ε, κ, δ fixed, we turn our attention to the problem of finding a function η such that
Our plan is hereafter to formulate it as a fixed point problem.
Thus, let be the corresponding solution of problem (2.9)-(2.17). Then the first a priori estimate (3.1) suggests that bounds (1.20) should be valid for if positive T or the norms of outer pressure are chosen sufficiently small provided satisfies (1.20). Indeed, let us denote
and for ,
In view of Theorem 3.1 and the embeddings
(note that is increasing in T), we have the following theorem.
Theorem 4.1 Suppose that the data satisfy (2.20), and assume such small that
Fix any point such that satisfies (1.20) with α and K given above. Let us recall that is the solution of problem (2.9)-(2.17) with in the sense of Theorem 2.1. If we recall the notation of (2.8), then
provided is sufficiently small. Moreover, satisfies (1.20) with the given α and K.
Note that smallness of (cf. Theorem 3.1) can be achieved by small outer pressure or a small time interval, but from now on, let T be fixed.
Proof It remains to prove that satisfies (1.20). Because of and , the following holds:
The second estimate in (1.20) obviously follows from (4.4) by the definition of h and the choice of K. □
Denote now and
Then is compact and convex. Now we define a mapping
in the following way: We start with and define corresponding to formula (1.4), i.e. by
Now, let be the solution of problem (2.9)-(2.17) according to Definition 2.1 with , and let us define a mapping .
Then our fixed point operator is defined by , i.e.
Corollary 4.2 is well defined on and maps into itself.
Proof Note that fulfils (1.20) if in view of Theorem 4.1. Then Theorem 2.1 implies that the mapping is well defined and unique. Moreover, due to Theorem 3.1, belongs to with the finite norm defined by (4.2). Finally, estimate (4.4) in Theorem 4.1 yields . □
An application of the Schauder fixed point theorem requires the continuity of our mapping . This means that we need continuous dependence of solutions on the data.
Theorem 4.3 Assume (2.1). Let and be weak solutions of the initial boundary value problem (2.9)-(2.17) in the sense of Definition 2.1 with given functions , , , and , , , , respectively, and suppose that both , satisfy (1.20). Let, moreover, , , with and for all . Then there exist positive constants and such that
for almost all , where , , and ω is a positive function as . Moreover, for any , there is such that
Unfortunately, ι depends on the regularising parameter δ in such a way that as .
Proof of Theorem 4.3 Due to tedious calculations, we postpone the proof to the Appendix of this paper. □
Consider next the special case , , , , , and remember , then for , estimate (4.5) gives
Together with the embedding inequality (4.3) this yields continuity of the operator on . Hence, we arrive at the following result.
Theorem 4.4 Suppose that the data satisfy (2.20) and assume such small that
In addition, let our regularised parameters be given and satisfy (2.1). Moreover, let
Then has a fixed point , i.e. there is a solution to problem (2.9)-(2.17) with
on the time interval . This solution is unique.
Proof Because of Theorem 4.1, Corollary 4.2 and Theorem 4.3, the operator is continuous on and maps the convex and compact subset into itself. Then the existence of follows from the Schauder fixed point theorem. This fixed point is unique due to Theorem 4.3. Indeed, let be two functions with and , and let
Choose now . Since the solution is unique on , we have for . Hence, estimate (4.5), the fact that , together with (4.3) imply
where denotes the norm (4.2) with integration over instead of . For small enough such that , finally, we obtain , which is a contradiction to the choice of τ. □
Note that our solution with depends on δ, ε, κ; however, the maximal time interval given by (4.7) does not depend on these regularising parameters.
5 The second a priori estimate
The aim of this section is to show compactness of our set with respect to δ and κ. Throughout this section let the assumptions of Theorem 4.4 be fulfilled. We start with some obvious estimates concerning the weight based on condition (1.20):
For compactness in time, the following second a priori estimate is essential.
Theorem 5.1 Let be the solution of free boundary value problem (2.9)-(2.17) given by Theorem 4.4. Then there is a constant independent of κ and δ such that for all the inequality
Before starting the proof of the theorem, let us prepare some estimations of the nonlinear items.
Lemma 5.2 Let
then for every fixed , we have
with generic constants C independent of u and δ. If, moreover,
under the same condition on λ.
Proof of Lemma 5.2 Denote , then and , hence
Then the interpolation inequality (2.27) with implies
because of (5.5) if , i.e. . This yields (5.6) since
due to assumption (5.5) if , i.e. . Inequality (5.8) follows from Hölder’s inequality
where the first integral of the product on the right-hand side is bounded owing to (5.9) and the second integral is bounded by our assumption. □
Proof of Theorem 5.1 Let be a fixed exponent that will be specified later, let be fixed and be the characteristic function of the subinterval . Replace for a moment the integration variable with respect to time in (2.21) by s instead of t and test it with , where is independent on s. Then the first line of (2.21) passes into
(this follows by approximation from (2.22)). The idea how to derive the estimate of the theorem is now to choose
and integrate the resulting relation over . This leads to
where denotes the Steklov average of on the interval . Note that for any normed space X and , the estimate
holds. Owing to estimates (5.1) and (5.2), for , the difference of the left-hand side of (5.4) and the first two lines of (5.10) may be estimated by cτ; hence it remains to estimate the items in the other lines of (5.10) also by cτ. Since these are tedious calculations, we restrict ourselves to some exemplary items.
In order to demonstrate the technique, we start with the first item occurring in the third line of (5.10) which is
In view of (5.2), (5.1) and , we may estimate
for any . In the last steps here, we have used inequality (5.11) and boundedness of the norm due to Theorem 3.1.
We proceed with the estimation of the integral arising from the nonlinear items, cf. the forth line of (2.21). Recall the definition (2.10) of , after inserting into (2.21) and integration over , the items to be integrated are(5.12)
Observing (5.1), (5.3) and, moreover,
we have to show boundedness of integrals of the form(5.14)
where and is the Steklov average defined after (5.10). Because of , we can assume . Moreover, due to inequality (5.11), we may omit the Steklov averages if the factors under the integrals are separated by Hölder’s inequality. Just this is the case in the proof of Lemma 5.2. Hence, we can apply Lemma 5.2 to the integrals (5.14). Since the assumptions of Lemma 5.2 are fulfilled due to Theorem 3.1, the integrals over the items (5.12) and (5.13) are estimated for by cτ with a constant c independent of κ, δ, ε.
The integrals over the items and are estimated in a similar way as in (ii). We consider now(5.15)
if . Hence, in this case, we need boundedness of the integrals(5.17)
This may be obtained for again by means of Hölder’s inequality, (5.11) and the estimates of Theorem 3.1. However, since
the integrals over items (5.15) and (5.16) can only be estimated by .
Since the integrals over the derivatives of z are obviously bounded due to the estimates of Theorem 3.1, finally we have a look at the last item on the left-hand side of (2.21),(5.18)
where , , , if . Hence, we obtain
due to Theorem 3.1 with c independent of κ. Since the estimation of the right-hand side of (2.21) is obvious, this concludes the proof. □
Note that (5.4) depends on ε also if we test with and the corresponding integral in (5.4) disappears. The reason is that it remains to prove boundedness of the integral over item (5.16) which we are not able to estimate independent on ε. Define now
Then from Theorems 3.1 and 5.1 we can derive the following estimates, which yield compactness of w in