1 Introduction

Mathematical modelling is very useful in many practical applications. For example, in medicine, it can be helpful by choosing the optimal strategy of medical treatment. In such modelling, very important issues are multiscale mathematical models of the blood circulation in a network of vessels. The full 3D computations are nowadays very time consuming and may be applied only for small parts of the blood circulation system. Therefore, a new trend is related to the creation of hybrid dimension models that combine the 1D reduction in the regular zones (mostly in straight vessels) with 3D zooms in small zones of singular behaviour. This method of asymptotic partial decomposition of a do- main was proposed by Panasenko (see [12]) and developed in [13–16]. It mathematically justifies a size of zoomed areas and prescribes asymptotically exact junction conditions. These hybrid-dimension models require significantly smaller computational resources.

The 1D Poiseuille-type flows in straight vessels play a very important role in hybrid- dimension models.

The steady-state Poiseuille flow in an infinite straight pipe of constant cross-section σ was invented by Jean Louis Poiseuille in 1841 (see [2,11,24]). The Poiseuille flow is described by the fact that the associated velocity field has only one nonzero component directed along the x_n-axis of Π, which depends only on , and the pressure function is linear. The Poiseuille-type solutions can be also defined in the nonstationary case (see [7, 17–21, 23, 26]). Moreover, in [14] the behaviour of the nonstationary Poiseuille flow was studied in a thin cylinder (with the cross-section of radius ε), and the asymptotics of it as was found.

In the time-periodic case, such flow is usually called Womersley’s flow (see [28]). The time-periodic Poiseuille-type solutions were studied in [3] and [8]. The time periodic solutions for the full Navier–Stokes problem were considered in many papers (see, e.g., [4–6]). Notice that the time-periodic case is very important because of applications to hemodynamics.

In mentioned above papers the Poiseuille-type solutions were studied in the case when data is sufficiently regular. However, in real applications, one usually does not have data defined by smooth functions, and it is important to study the case of minimal regularity of data. The nonstationary Poiseuille-type solution with a prescribed initial condition and given flow rate F(t) belonging to was studied in [22], where a new class of weak solutions was introduced, and the unique existence of the solution in such class was proved. The goal of the present paper is to extend the result obtained in [22] to the case of time-periodic Poiseuille-type solutions.

Let us consider the time-periodic Navier–Stokes problem describing the motion of a viscous incompressible fluid in the infinite cylinder II:

where u is the fluid velocity, p is the pressure function, and is the constant kinematic viscosity of the fluid.

We look for the solution u of (1) in Π satisfying the additional condition of prescribed flow rate (flux) F(t):

where

The solution of problem (1) has the following expression:

with an arbitrary function p₀(t). Putting (2) into (1), we obtain the following problem on the cross-section σ

where and q(t) are unknown functions, is the Laplace operator with respect to .

The Poiseuille flow can be uniquely determined either prescribing the pressure drop q(t) or the flow rate F(t). In the first case the problem is reduced to the standard time- periodic problem for the heat equation for unknown velocity with time- periodic forcing q(t). Problems of such type are well studied (see, e.g., [9]). However, in the real word applications the pressure is unknown, and only the flow rate (flux) of the fluid is given. Therefore, it is necessary to prescribe the additional condition

In this case the solution of problem (3), (4) is a pair of functions , and one has to solve for and q(t) more complicated inverse parabolic problem: for given F(t), to find a pair of functions solving problem (3) with satisfying the flux condition (4).

Thus, in the second case the relation between q(t) and F(t) depends on the solution of the inverse problem (in the stationary case the flux F and the pressure gradient q are proportional, and the problem remains very simple). The solvability of the time-periodic problem with the assumption that the flux F(t) is from the Sobolev space was proved by Beirão da Veiga [3], and in [8] an elementary relationship between the pressure drop q(t) and the flux F(t) was found. However, in applications and numerical computations, the data usually is not regular. Therefore, in this paper, we study problem (3), (4) assuming only that .

Problem (3), (4) can be reduced to the case when all involved functions have zero mean values. Let us denote by the mean value of a function H. Let be a solution of the following problem on σ (the stationary Poiseuille solution corresponding to the flux ):

The solution of (5) can be represented in the form , where

and

Let us represent the solution in the form

Then, obviously, , and is the solution of the problem

where . So, without loss of generality, we assume , that is .

Below, we deal with a weak solution of problem (3), (4). The reasoning about the reduction to the case of functions with zero mean values remains valid for weak solutions as well, and we will study only this case.

The rest of the paper is organized in the following way. In Section 2, function spaces are defined and the main result is formulated. In Section 3 the Galerkin approximations of the solution are constructed, and in Section 4 a priori estimates for these approximations are proved. In Section 5 the main result of the paper, that is the existence and the uniqueness of the solution, is proved.

2 Notation and formulation of main result

2.1 Function spaces

Below, we will use the following notation. If G is the domain in means, as usual, the set of all infinitely differentiable functions in G, and is the subset of functions from with compact supports in G. We use the usual (see [1,9]) notation for Lebesgue and Sobolev spaces: , and . The norm of an element u in the function space V is denoted by is the Bochner space of functions u such that for almost all , and the norm

is finite

Let us consider the set of smooth periodic functions defined on the interval . Let be a Lebesgue space on the interval . We extend the functions from to the whole line R by putting for any t. To emphasize that functions are periodically extended to R, we used notation . Let . Clearle is a closure of in

- norm, and it is a proper subspace of . Let be the closure of the set in -norm. Since function f from coincides with a continuous function on a set whose complement is of measure zero, we may assume that . Let be dual of .

For any function , denote by its primitive:

Clearly, .

If , then . Moreover,

and is a period function:

Thus, . Note that functions defined by (9) with various t₀ differ from each other by a constant.

Note that the L²-limit of a sequence ) is not necessary a primitive of some function from . We will prove that an element possesses a primitive in the distributional sense.

Lemma 1.Any functional can be represented in the form

with the uniquely determined .

Proof. Obviously, the functional given by formula (10) obeys the estimate

and hence, .

Let us take an arbitrary functional and show that it can be represented in the form (10). Consider the operator due to periodicity, . Since for any , the equality holds with

we have , and the operator ∂ is an isomorphism from to , where the bounded operator is given by (11).

For , define the functional . Clearly

Hence, there exists a uniquely defined such that

Thus

and is represented in the form (10).

Remark 1. Note that if the functional h can be represented in the form = with and arbitrary , then H(t)= . Therefore, also in the case of a general functional h, for the distributional primitive H, we use the notation .

2.2 Formulation of main result

Definition of a weak solution

Let . By a weak solution of problem (8) we understand a pair such that satisfies the flux condition

and the pair satisfies the integral identity

for any test function such that .

For a regular solution , taking into account that , identity (13) can be easily obtained multiplying equation (8)1 by η, integrating over σ and over the interval and integrating by parts with respect to and t. On the other hand, by uniqueness of such weak solution (see Theorem 1 below) it follows that for , the solution coincides with the regular one, that is . Thus, the proposed definition is an extension of the concept of weak solutions.

Theorem 1.Let . Then problem (8) admits a unique weak solution (V, s). Then there holds the estimate

where the constant c depends only on σ.

Remark 2. Since and all primitives of the function differ from each other by a function independent of t, the integral identity (13) remains valid for any primitive function S_V, and we can assume, for example, that S_V is taken to be zero at the point

Theorem 1 will be proved applying some version of Galerkin approximations (see Sections 3 and 4). Notice that in order to get estimates of approximate solutions , we have used primitive functions defined over the integrals with specially chosen points and . Thus, estimate (14) is valid only for the primitive function S_V obtained as a limit of the sequence

3 Construction of Galerkin approximations

Let and be eigenfunctions and eigenvalues of the Laplace operator:

Note that and . The eigenfunctions are orthogonal in , and we assume that are normalized in . Then

Moreover, is a basis in and .

We look for an approximate solution of problem (8) in the form

The coefficients and the function are obtained by solving the following linear problems

which, in virtue of the orthonormality of functions , are equivalent to ordinary differential equations for the functions :

where . Note that .

The solution of problem (17) has the form

where

It is easy to see that .

Substituting expression (18) into (15), we obtain

Now the flux condition yields

Thus, for the function , we derived Fredholm integral equation of the first kind:

It is well known (see, e.g., [10, 25]) that such equations, in general, are ill-posed in L2 setting. In order to regularize equation (19), we consider the following Fredholm integral equation of the second kind:

where α later will tend to 0, i.e., instead of problem (16), (19), we study the regularized problem

where

Lemma 2.Let . Then equation (20) admits a unique solution .

Proof. First, we show that equation (20)is well defined in the space .Obviously, if F is periodic and is the solution of (20), then also is a periodic function. Assume that mean value . Then

Since

the second term in (22) is equal to and from (22) it follows that

and thus, .

From this it follows that the mean value also vanishes: .

It is well known that Fredholm integral equations of the second kind satisfy the Fredholm alternative (e.g., [27]). So, it is enough to prove the uniqueness of the solution to (20). Let . By construction

and the homogeneous equation (20) gives

Multiplying (21) by , summing by k from 1 to N and integrating over the interval yield

Integrating by parts with respect to t and taking into account the time-periodicity of , we obtain

Thus, for a.a. , and the lemma is proved.

4 A priori estimates of Galerkin approximations

Let the pair be the solution of problem (21) and be the solution of problem (6). Consider the integral . Since the mean value , we have

Therefore, there exists such that . By periodicity we also have

Let define the notation , define the notationSince the mean value of F nabishes, we hace . Moreover,

Lemma 3. The following estimate

holds with a constant c independent of α and N.

Proof. Define. Multiplying (21) by summing the obtained relation from k = 1 to k = N and integrating them over the interval , we obtain

Using the relation

and (21), we derive

Therefore, (24) can be written as

Then

Calculating similarly two integrals J₁ and J₂ on the left-hand side of (25), we derive that

Hence, relation (25) takes the form

Now multiply (21) by , sum from k = 1 till k = N and integrate over the interval :

Evaluating each of the three integrals in (27) and having in mind that mean values of participating functions are zero, we obtain

From (26) and (28) it follows that

and

Let us estimate the integral . Let be a solution of problem (6). Remind that the flux of U₀ is nonzero, (see (7)). Since is a basis in , U₀ can be expressed as a Fourier series in :

Let us multiply relations (21) by ak and sum them over k. This gives

On the other hand, multiplying (6) by and integrating by parts in σ, we obtain

Substituting (32) into (31) yields

i.e.,

Integrating (33) with respect to t from τ to , we obtain

Here we have used the choice of the point t_∗, that is

and hence,

From (34) it follows that

Substituting (35) into (29) yields

and choosing ε sufficiently small, we obtain

Estimates (36) and (35) give

Finally, from (30) and (37) it follows that

The constants in (36)–(38) are independent of α and N.

5 Convergence of Galerkin approximations. Proof of Theorem 1

5.1 Proof of existence

The constructed approximate solutions satisfy equalities (21). Multiplying these relations by arbitrary functions such that , summing over k from k = 1 to , integrating with respect to t and then integrating by parts, we obtain the integral identity

for test functions η having the form .

Recall that obey a priori estimate (23) with a constant c independent of α and N.

Since all functions in (23) are 2π-periodic, inequality (23) is equivalent to

Let us fix N and choose a subsequences and such that l converges weakly in to some converges weakly in to S_V(N)⁵, while converges weakly in to s^(N). The last convergence means that

where , and , and is a primitive of s^(N) in the distributional sense.

Obviously, for the limit functions V^(N) and S_s(N), estimate (40) remains valid with a constant c independent of N. In (39), taking and passing to the limit as , we get

Let us show that satisfy the flux condition

Integrating equation (21) for from t to 2π yields

Obviously, the sequence is bounded in . So we may assume, without loss of generality, that is weakly convergent to in . Then the sequence of primitives for all , and hence, . From (43) we have

Therefore,

and differentiating this equality with respect to t, we get (42).

Now we choose a subsequence such that converges weakly in to some converges weakly in ; to S_V and converges weakly in to s. In (41) ,passing to the limit over the subsequence as yields

Exactly as above, we can prove that satisfies the flux condition (12). However, the integral identity (44) is proved, up to now, only for test functions η, which can be represented as the sums: with such that . After we have passed to the limit in (41) as , the subscript M in these sums can be arbitrary large natural number. Such sums are dense in the space. This can be proved exactly in the same way as it is done in the book [9] for the case of an initial boundary value problem. Thus, they also are dense in the subspace , and therefore, (41) remains valid for all This proves that satisfies the integral identity (13), and thus, it is a weak solution of problem (8). Estimate (14) for follows from estimate (40) for the approximate solutions.

5.2 Proof of uniqueness

Assume that . Since , we take

Obviously, . Putting this η into identity (13), we obtain

Since

by Fubini’s theorem and the time-periodicity of the function we have

Therefore, from (45) it follows that

Then, and hence, . Integrating over σ, we get

Thus, , and since, we conclude that, that is for a.a. and t.

This implies . From identity (13) it follows that

In (46), we take , where is arbitrary, and is the solution of problem (6). Recall that . Then (46) takes the form

Thus, S_s(t)=const. Since ,i.e. the mean value of S_s(t) is equal to zero, we get that S_s(t) = 0. Therefore, the functional s=0.

References

1. R.A. Adams, Sobolev Spaces, Academic Press, New York, San Francisco, London, 1975.

2. G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge Univ. Press, Cambridge, 2002.

3. H. Beirão da Veiga, Time-periodic solutions of the Navier–Stokes equations in an unbounded cylindrical domains. Leray’s problem for periodic flows, Arch. Ration. Mech. Anal., 178:301– 325, 2005, https://doi.org/10.1007/s00205-005-0376-3.

4. G.P. Galdi, On the problem of steady bifurcation of a falling sphere in a Navier–Stokes liquid, J. Math. Phys., 61(8), 2020, https://doi.org/10.1063/5.0011248.

5. G.P. Galdi, On the self-propulsion of a rigid body in a viscous liquid by time-periodic boundary data, J. Math. Fluid Mech., 22(61), 2020, https://doi.org/10.1007/s00021-020- 00537-z.

6. G.P. Galdi, T. Hishida, Attainability of time–periodic flow of a viscous liquid past an oscillating body, J. Evol. Equ., 2021, https://doi.org/10.1007/s00028-020-00661-3.

7. G.P. Galdi, K. Pileckas, A. Silvestre, On the unsteady Poiseuille flow in a pipe, Z. Angew. Math. Phys., 58:994–1007, 2007, https://doi.org/10.1007/s00033-006-6114-3.

8. G.P. Galdi, A.M. Robertson, The relation between flow rate and axial pressure gradient for time-periodic Poiseuille flow in a pipe, J. Math. Fluid Mech., .:215–223, 2005, https://doi.org/10.1007/s00021-005-0154-x.

9. O.A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Springer, New York, 1985.

10. P.K. Lamm, Surveys on Solution Methods for Inverse Problems, Springer, Vien, 2000.

11. L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon Press, Oxford, New York, 1987.

12. G. Panasenko, Asymptotic expansion of the solution of Navier–Stokes equation in a tube structure, C. R. Acad. Sci., Paris, Sér. II, Fasc. b, Méc. Phys. Astron., 326(12):867–872, 1998, https://doi.org/10.1016/S1251-8069(99)80041-6.

13. G. Panasenko, Multi-Scale Modelling for Structures and Composites, Springer, Dordrecht, 2005.

14. G. Panasenko, K. Pileckas, Asymptotic analysis of the nonsteady viscous flow with a given flow rate in a thin pipe, Appl. Anal., 91(3):559–574, 2012, https://doi.org/10.1080/00036811.2010.549483.

15. G. Panasenko, K. Pileckas, Asymptotic analysis of the non-steady Navier–Stokes equations in a tube structure. I: The case without boundary layer-in-time, Nonlinear Anal., Theory Methods Appl., 122:125–168, 2015, https://doi.org/10.1016/j.na.2015.03.008.

16. G. Panasenko, K. Pileckas, Asymptotic analysis of the non-steady Navier–Stokes equations in a tube structure. II: General case, Nonlinear Anal., Theory Methods Appl., 125:582–607, 2015, https://doi.org/10.1016/j.na.2015.05.018.

17. K. Pileckas, On the behavior of the nonstationary Poiseuille solution as t→∞ , Sib. Math. J., 46:707–716, 2005, https://doi.org/10.1007/s11202-005-0071-5.

18. K. Pileckas, Existence of solutions with the prescribed flux of the Navier–Stokes system in an infinite cylinder, J. Math. Fluid Mech., 8:542–563, 2006, https://doi.org/10.1007/ s00021-005-0187-1

19. K. Pileckas, The Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s problem, in S. Friedlander, D. Serre (Eds.), Handbook of Mathematical Fluid Dynamics, Vol. 4, North-Holland, 2007, pp. 445–647, https://doi.org/10.1016/S1874-5792(07) 80012-7.

20. K. Pileckas, Solvability in weighted spaces of the three-dimensional Navier–Stokes problem in domains with cylindrical outlets to infinity, Topol. Methods Nonlinear Anal., 29(2):333–360, 2007.

21. K. Pileckas, Global solvability in W 2,1-weighted spaces of the two-dimensional Navier–Stokes problem in domains with strip-like outlets to infinity, J. Math. Fluid Mech., 10(2):272–309, 2008, https://doi.org/10.1007/s00021-006-0232-8.

22. K. Pileckas, R. Cˇ iegis, Existence of nonstationary Poiseuille-type solutions under minimal regularity assumptions, Z. Angew. Math. Phys., 71(192), 2020, https://doi.org/10. 1007/s00033-020-01422-5.

23. K. Pileckas, V. Keblikas, Existence of nonstationary Poiseuille solution, Sib. Math. J., 46: 514–526, 2005, https://doi.org/10.1007/s11202-005-0053-7.

24. J.L.M. Poiseuille, Recherches Expérimentales sur le Mouvement des Liquides dans les Tubes de Très-Petits Diamètres, Imprimerie Royale, Paris, 1844.

25. W.W. Schmaedeke, Approximate solutions for the Volterra equations of the first kind, J. Math. Anal. Appl., 23(3):604–613, 1968, https://doi.org/10.1016/0022-247X(68) 90140-6.

26. S.S. Sritharan, On the acceleration of viscous fluid through an unbounded channel, J. Math. Anal. Appl., 168(1):255–283, 1992, https://doi.org/10.1016/0022-247X(92) 90204-Q.

27. F.G. Tricomi, Integral Equations, Intersience, New York, 1957.

28. J.R. Womersley, Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known, J. Physiol., 127(3):553–563, 1955, https://doi.org/10.1113/jphysiol.1955.sp005276.