Global Analysis of a Blood Flow Model with Artificial Boundaries

A theoretical model for blood flow in ramifying arteries was introduced and studied numerically (Quarteroni andVeneziani, 2003). A special experimental condition was considered on the artificial boundaries. In this paper, the aim is to analyze the well-posedness of this model, with the focus on the stilted boundary conditions. We use Brouwer’s fixed point theorem to show the existence of a solution to the stationary problem. For the evolutionary version, we use some energy estimates and Galerkin’s method to prove global existence, uniqueness, and stability of a weak solution.


Introduction
Evolution systems with artificial boundaries are difficult to analyze as a result of the complex dynamics at the boundaries and very few papers have attempted to capture these processes from an analytical point of view.This paper explores this question regarding the flow of blood in a portion of a large artery and addresses the analysis of the Navier-Stokes problem, having provided boundary conditions which can be considered as a generalization of the mean pressure drop problem investigated in [1][2][3], as they arise in bioengineering applications.Our purpose is to consider both the stationary case and the nonstationary case.In this regard, we will prove the existence of a weak solution for the stationary case based on Brouwer's fixed point theorem and afterwards, we will establish a well-posedness analysis for the nonstationary case based on a suitable energy estimate that we are going to derive as well as a well-known compactness argument.
1.1.Basic Notations.In this subsection, we summarize some notations that will occur throughout the paper.Vectors and tensors are denoted by bold-face letters The spaces (Ω),   (Ω),   0 (Ω), and  ∞ 0 (Ω) and their vector-valued analogues C(Ω), C  (Ω), C  0 (Ω), and C ∞ 0 (Ω) are defined as usual, the superscript indicating continuous derivatives to a certain order and the subscript zero indicating functions with compact support.
The space   (Ω), the Hölder space  , (Ω), and the Sobolev space    (Ω) and their vector-valued analogues L  (Ω), C , (Ω), and W   (Ω) are also defined as usual.In particular, H 1 (Ω) = W 1  2 (Ω).For functions depending on space and time, for a given space V of space-dependent functions, we define (for some  > 0)  2 (0, ; V) = {k : (0, ) → V | k is measurable and with norm ‖k‖  2 (0,;) = (∫  0 ‖k()‖ 2  ) . When considering functions which depend only on time, we define the space L ∞ (0, ) = {z : (0,) → R 3 | ess sup ∈(0,) |z()| < ∞} endowed with the norm ‖z‖  ∞ (0,) = ess sup ∈(0,) |z ()| . (2) 1.2.Preliminaries.In this subsection, we recall some assumptions used in [2,4], and we are going to make use of them throughout our analysis.In what follows, Ω is a bounded domain of R 3 with boundary Γ sufficiently smooth.Γ consists of the artery wall denoted by Γ wall and some artificial sections.The velocity is required to be zero on the artery wall.To account for homogeneous Dirichlet boundary conditions on the artery wall, we define Note that Poincaré's inequality          ≤  Ω     ∇     (4) holds for  ∈ X [2,4].The artificial sections consist of the upstream section on the side of the heart and the downstream sections on the side of the peripheral vessels.Rather than giving serious thought to the artificial sections boundary conditions, in seeking a variational formulation, the test space is left free on these portions of the boundary.Accordingly, we introduce as the test space.To prove an existence theorem for a Navier-Stokes problem, either steady or nonsteady, it is convenient to construct the solution as a limit of Galerkin approximations in terms of the eigenfunctions of the corresponding steady Stokes problem.This use of the Stokes eigenfunctions originated with Prodi and was further developed by Heywood [5].
To define the corresponding Stokes operator, we introduce V ⋆ as the completion of V with respect to the norm of L 2 (Ω).
Then for every f ∈ V ⋆ , there exists exactly one k ∈ V satisfying Moreover, for each k ∈ V, there exist at most one f ∈ V ⋆ such that (6) holds.In this way, a one-to-one correspondence can be defined between elements f of V ⋆ and functions k belonging to an allowing suitable subspace of V that we denote by ( Δ).The Stokes operator is defined setting so that ( 6) is satisfied.The inverse operator Δ−1 is completely continuous and self-adjoint.Therefore it possesses a sequence of eigenfunctions {a  } ∞ =1 , which are complete and orthogonal in both V and V ⋆ .
In what follows,   ( = 1, 2, . ..) will denote generic constants, not necessarily the same at different places.The inequalities are satisfied for every k ∈ ( Δ), provided that Ω is a bounded domain (see [4] page 178).
Assume  is a bounded open subset of R  , with a  1 boundary.Suppose 1 ≤  < .Then for each 1 ≤  <  ⋆ , where As a result of that, we have Theorem 3 (trace inequality for solenoidal functions).Let u be a solenoidal function defined on .Then the following inequality holds: Theorem 4 (weak compactness).Let  be a reflexive Banach space and suppose that the sequence {u  } ∞ =1 ⊂  is bounded.Then there exists a subsequence In other words, bounded sequences in a reflexive Banach space are weakly precompact.In particular, a bounded sequence in a Hilbert space contains a weakly convergent subsequence.

Formulation of the Problem
Let Ω ⊂ R 3 be the artery portion where we aim at providing a detailed flow analysis.For each x ∈ Ω and at any time  > 0, we denote by u(x, ) and (x, ) the blood velocity and pressure, respectively.In larger vessels, it is reasonable to assume that blood has a constant viscosity, because the vessel diameters are large compared with the individual cell diameters and because shear rates are high enough for viscosity to be independent of them.Hence, in these vessels the non-Newtonian behavior becomes insignificant and blood can be considered to be a Newtonian fluid ([4] and references therein).In what follows, we assume that the vessels are large enough; blood density and blood viscosity are assumed to be constant.Under these assumptions, blood flow can be described by the Navier-Stokes equations where T = T = I − (∇u + (∇u)  ).The equations are expressions of balance of linear momentum and incompressibility.We are neglecting the presence of any external forces.Then (18) is obtained by substitution of the divergence-free constraint into the expression for the stress in (17) as follows In fact, For the sake of simplicity, we normalize  to 1.

Initial and Boundary Conditions.
The initial condition requires the specification of the flow velocity at the initial time; for example, for  0 = 0; where the given initial velocity field u 0 is divergence-free.The system (18) has to be provided with boundary conditions.In this respect, we split the boundary Γ into different parts.In the present work, we are assuming that the wall is rigid so that no-slip boundary condition holds.The other parts of Γ are the artificial boundaries which bound the computational domain.For the sake of clarity, we distinguish the upstream section on the side of the heart denoted by Γ 1 and the downstream sections on the side of the peripheral vessels denoted by Γ 2 and Γ 3 .As introduced in [4], the following boundary conditions are provided: where   is a suitable nonnegative constant,   =   () is the prescribed mean pressure on each artificial section Γ  , and n represents the outward normal unit vector on every part of the vessel boundary.These conditions can be considered as a generalization of the mean pressure drop problem investigated in [2] in the sense that when   = 0 ( = 1, 2, 3), we recover the usual Neumann or natural conditions associated with (18).The physical justification for the case in which   ̸ = 0 is provided in ([4] Figure 4.1).The mathematical formulation of the problem is therefore described by the system of differential equations: Next we define the following bilinear and trilinear forms: (k (1) , k (2) ) =  (∇k (1) , ∇k (2) ) : 1) , k (2) , k (3) ) = ((k (1) ⋅ ∇) k (2) , k (3) ) .
The former is a consequence of the trace inequality for solenoidal functions [2] and the proof of (29) is given in [4].

The Stationary Problem
Our purpose is to establish the existence of a weak solution to the stationary version of problem (23).We are going to consider the inner product with  ∈ V in order to formulate the weak formulation.

Weak Formulation.
Assume that u is a solution of the boundary value problem described above and  is a smooth solenoidal vector-valued function defined on Ω.The function u satisfies the following identities: We notice that | Γ wall = 0 since  ∈ V.
We calculate to obtain It is clear that (33) does not depend on the pressure ; therefore, the weak formulation of the problem reads as follows.
Given a divergence free velocity field u 0 ∈ V and nonnegative constants   and   ( = 1, 2, 3), find u ∈ V such that holds for all  ∈ V, where V, defined in (5), is the space of test functions.

Galerkin Approximations.
The inverse operator Δ−1 of the Stokes operator Δ is self-adjoint and possesses a sequence of eigenfunctions {a  } which are orthogonal in V. Following [6] we fix a positive integer .Galerkin approximations are defined as solutions of the finite system of equations ( = 1, . . ., ) as follows: This is a system of linear equations for constant unknowns    ( = 1, . . ., ).The identity (37) for u  is obtained by multiplying (36) through by    and summing over  = 1, . . ., : Together with (4), we make use of (28) to obtain where For each integer  = 1, 2, . .., there exists a function u  of the form (35) satisfying (36) such that Proof.Owing to Poincaré's inequality (4), ‖∇‖ is a norm equivalent to ‖‖ 1 for all  ∈ V; therefore, (45) defines a closed ball in span{a 1 , . . ., a  }.To prove the solvability of the finite-dimensional problem (36), we follow Fujita in using Brouwer's fixed point theorem (see Theorem 1.1 in [7]), applying it to the continuous mapping k → w defined by the linear problem ( = 1, . . ., ): Equation ( 46) is a system of  linear equations.These linear equations are uniquely solvable if k lies in the ball defined by (45), because then w = 0 is the only solution of the corresponding homogeneous equation (  =   = 0, 1 ≤  ≤ 3).In fact, if k satisfies (45) and w satisfies (46) with   =   = 0, we have Together with Poincaré's inequality (4), this imply that w = 0.
To see that the mapping k → w takes the ball defined by (45) into itself, suppose that k satisfies (45).Then, similarly to (43), we obtain that and therefore, Thus, (46) defines a continuous mapping k → w of the closed ball into itself.The map has at least one fixed point, and any such fixed point is a solution of (36).u  is chosen to be any one of these fixed points.Hence, Brouwer's fixed point has been applied and has given the existence of Galerkin approximations satisfying . (52)

Existence of a Weak Solution
There exists a weak solution to the stationary version of problem (23).

ISRN Applied Mathematics
Proof.By Poincaré's inequality [3], the fact that the sequence {‖∇u  ‖} ∞ =1 is bounded implies that the sequence {u  } ∞ =1 is bounded in V.As a result, Theorem 4 yields the existence of a subsequence {u Next we show that the weak limit u is in fact a weak solution.
In this respect, we are going to show that for each  ∈ V. Fix an integer , ( = 1, 2, . ..).From (36), we have the identity We recall (54), to find upon passing to weak limits that for each  ∈ V.
Note that, in taking this limit, there is no difficulty with the nonlinear term.In fact, we have that (59) Also, a  ∈ H 2 (Ω) is an eigenfunction of the inverse of the stokes operator Δ and H 2 (Ω) ⊂ C(Ω).This implies that a  ∈ C(Ω) [6].It follows that (see [5], page 651).

The Nonstationary Problem
The mathematical analysis of the Navier-Stokes problem is based on its weak formulation.

Weak Formulation.
Assume that u is a solution to problem (23) and  is a smooth solenoidal vector-valued function defined on Ω. u satisfies the following identities: We calculate to obtain where , , and  are defined by ( 24), (25), and (26), respectively.We notice that (63) does not depend on the pressure .Therefore, the weak formulation of the problem reads as follows.

Galerkin Approximations.
We are going to follow the same approach as in [6].We will construct the weak solution of the initial boundary value problem by first solving for a finite dimensional approximation.We recall that the inverse operator Δ−1 of the Stokes operator Δ is self-adjoint and possesses a sequence of eigenfunctions {a  } which are orthogonal in V and orthonormal in L 2 (Ω).Fix a positive integer  and write where we intend to select the coefficients    () (0 ≤  ≤ ,  = 1, . . ., ) to satisfy Theorem 7.For each integer  = 1, 2, . .., there exists a unique function u  of the form (65) satisfying (66)-(67).
Proof.Assuming that u  is given by (65), we observe using the fact that {a  } ∞ =1 is an orthonormal basis of L 2 (Ω) that where (   )  () is the derivative of    () with respect to .Furthermore, from (24) and the fact that  is a bilinear form, we have that where    = (a  , a  ).From (25) and recalling that  is a trilinear form, we have that where   , = (a  , a  , a  ).Moreover, we set It follows that (67) becomes the system of ODEs and also Moreover, It follows that We now make use of (28) to obtain which leads us to the required energy estimates.

Local Existence of a Weak Solution.
In this subsection, we use Galerkin method to build up a local weak solution of the initial/boundary-value problem.We have already constructed the Galerkin approximations sequence {u  } ∞ =1 in Section 4.2.Our goal now is to extract from this sequence a subsequence that converges to the weak solution.In this respect, we are going to show that this sequence is bounded and thereafter, we will make use of a compactness result.

Lemma 9. Let 𝑌 be a nonnegative function satisfying the inequality
and let (0) =  0 be a strictly positive real number; then there exists a time interval (0,  ⋆ ) where () is bounded by a positive constant  and  depends only on  0 ,  and .
Theorem 10.Let  be such that there is a time interval (0,  ⋆ ) on which a weak solution of (23) exists. Proof where  > 0 and  > 0 are some constants depending on the data.Note that for each integer , the function   () = ‖∇u  ‖ 2 verifies the requirements of Lemma 9 with   (0) = ‖∇u  ‖ 2 since the initial condition is the same for all functions u  ,  = 0, 1, 2, . ... Consequently, the sequence {‖∇u  ‖ 2 } ∞ =1 is bounded by a real number  which depends only on the initial data.We make use of Poincaré's inequality (4) to find that for each integer ,     u Therefore, the sequence {‖u  ‖  2 (0, ⋆ ;V) } ∞ =1 is bounded.Furthermore, the space  2 (0,  ⋆ ; V) being a Hilbert space, Theorem 4 can be applied to the sequence In fact, the sequence {u  } ∞ =1 is bounded in  2 (0,  ⋆ ; V).Consequently, there exists a subsequence {u Next, we show that the weak limit u is in fact a weak solution.
In this respect, we are going to show first that and, thereafter, (1) Fix an integer  and choose a function w ∈ C(0,  ⋆ ; V) having the form where {  }  =1 are given functions and {a  } is the basis of V. We choose  ≥ , multiply (67) by   (), sum  = 1, . . ., , and then integrate with respect to  to find that We recall (107) to find upon passing to weak limits (see remark 3.1 and [5]) that Equality (112) then holds for all functions w ∈ L 2 (0, ; V), as functions of the form (110) are dense in this space [5].Hence, in particular, for each  ∈ V and a.e. ≥ 0.

Global Existence of a Weak Solution for Small Data.
In this section, we make use of Galerkin's method to establish the global existence of a weak solution.We are going to show that under certain circumstances, the weak solution u is defined at any time .In this regard, we are going to consider the energy estimate and the following lemma.
(1) We first show that ( ⋆ ) = .Set  = { ∈ R + , () > } and choose any  > 0; we have and so  ⋆ −  does not belong to the set .It follows that ( ⋆ − ) ≤ , and this is true for each  > 0; hence On the other hand, for each natural number , we have  ⋆ + (1/) >  ⋆ .We make use of the fact that  ⋆ = inf() to see that there exists   ∈  such that where   is not necessarily unique.We choose one value that   may take and we denote it by   .This defines a real sequence {  } ∞ =1 .For each natural number , since   ∈ , we have that (  ) > .The fact that  ⋆ ≤   ≤  ⋆ +(1/) implies that the sequence {  } ∞ =1 converges to  ⋆ .We make use of the continuity of  to see that the sequence converges to  ( ⋆ ) . (122) Since   ∈  for each , we have (  ) > , and then (  ) → ( ⋆ ) implies that It follows that ( ⋆ ) ≤  and ( ⋆ ) ≥ ; therefore, (2) Next we show that   ( ⋆ ) > 0. Suppose that   ( ⋆ ) ≤ 0; then there exists a nonnegative natural number  such that  decreases on the interval ( ⋆ ,  ⋆ + (1/)).Also we have that this implies that But from part (1), we have ( ⋆ ) =  and (  ) > ; because   ∈ , it follows that Hence, ( ⋆ ) ≥ (  ) and ( ⋆ ) < (  ), which is impossible.Therefore, Finally that we show that   ( ⋆ ) ≤ 0 and this contradicts the result obtained in (2).From (117), we have Set The energy estimate (132) takes the form where Let  > 0 be any positive real number, Thus, for each nonnegative real number , the sequence {‖u  ‖  2 (0,;V) } ∞ =1 is bounded.We make use of Lemma 9 to extract a subsequence {u  } ∞ =1 that converges to an element u of  2 (0, ; V), and we use the same steps as we did for the local existence to show that u is a weak solution of problem (23).
Furthermore, the fact that the sequence {∇u  } ∞ =1 is bounded by √  implies that the subsequence {∇u  } ∞ =1 is also bounded by √ .
Therefore, since ∇u is the limit of this subsequence, it follows that (141)

Uniqueness of the Solution
Theorem 13.There exists a time interval  1 where the solution of problem (23) is unique.

Stability of the Solution
In this section, we are going to prove that there exists a time interval of continuous dependence on the data.We start with the boundary conditions and afterward we move on to the initial condition.

Boundary Conditions Stability
Theorem 14.There exists a time interval where the solution of problem (23) depends continuously on the prescribed data as the real functions   ( = 1, 2, 3) are varied.

Initial Condition Stability
Theorem 15.There exists a time interval where the solution of problem (23) depends continuously on the prescribed data as the initial condition u 0 (x) is varied.
Hence, a small change in the given data produces a correspondingly small change in the solution.

x
= ( 1 ,  2 ,  3 ): location of fluid particle, u: velocity field of the flow, : pressure, ∇: the pressure gradient, I: identity tensor, T: symmetric Cauchy stress tensor, : dynamic viscosity, : fluid mass density, R: the set of real numbers, | ⋅ |: the absolute value of R and correspondingly, the norm of R 3 , Ω: a bounded domain in R 3 , Γ: the boundary of Ω, ∇u: the gradient of u, ∇ ⋅ u: the divergence of u, Δu: the Laplacian of u.
We make use of Lemma 11 to find that   = ‖∇u  ‖ 2 is bounded by  for all  > 0 and the bound does not depend on .Poincaré's inequality (4) implies that ‖w‖ 2 +  6 ‖w‖ ‖∇w‖ 2 .