A Note on the Right-Hand Side Identification Problem Arising in Biofluid Mechanics

and Applied Analysis 3 Theorem 2.1. Let φ ∈ ◦ C 2α 2 0, L , F1 ∈ C 0, T , ◦ C 2α 0, L , and ρ′ ∈ C 0, T . Then for the solution of problem 1.1 , the following coercive stability estimates ‖ut‖ C 0,T , ◦ C 2α 0,L ‖u‖ C 0,T , ◦ C 2α 2 0,L ≤ M(x∗, q)∥∥ρ∥∥C 0,T M ( a, δ, σ, α, x∗, q, T ) × ( ∥ ∥φ ∥ ∥ ◦ C 2α 2 0,L ‖F1‖ C 0,T , ◦ C 2α 0,L ∥ ∥ρ ∥ ∥ C 0,T ) , ∥ ∥p ∥ ∥ C 0,T ≤ M ( x∗, q )∥ ∥ρ′ ∥ ∥ C 0,T M ( a, δ, σ, α, x∗, q, T ) [ ∥ ∥φ ∥ ∥ ◦ C 2α 2 0,L ‖F1‖ C 0,T , ◦ C 2α 0,L ∥ ∥ρ ∥ ∥ C 0,T ] 2.3 hold. Proof. Let us search for the solution of inverse problem 1.1 in the following form see 23 : u t, x η t q x w t, x , 2.4


Introduction
It is known that many applied problems in fluid mechanics, other areas of physics, and mathematical biology were formulated as the mathematical model of partial differential equations of the variable types 1-3 .A model for transport across microvessel endothelium was developed to determine the forces and bending moments acting on the structure of the flow over endothelial cells ECs 4 .Computational blood flow analysis through glycocalyx on the EC is performed as a direct problem previously under smooth and nonsmooth initial conditions see 5-7 .But it is known that, due to the lack of some data and/or coefficients, many real-life problems are modeled as inverse problems 8-11 .In this paper, the well-posedness of the inverse problem of reconstructing the right side of a parabolic equation arisen in computational blood flow analysis is investigated.The importance of well-posedness has been widely recognized by the researchers in the field of partial differential equations 12-16 .Moreover, the well-posedness of the RHS identification problems for a parabolic equation where the unknown function p is in space variable and in time variable is well investigated 17-27 .As it is known, well-posedness in the sense of Hadamard means that there is existence and uniqueness of the solution and the solution is stable.In this study, we deal with the stability analysis of the inverse problem of reconstructing the right-hand side.The existence of a solution for two-phase flow in porous media has been studied previously for instance, see 28 .

Problem Formulation
Blood flow over the EC inside the arteries is modeled in two regions see 6 .Core region 0 < x < l flow is defined through the center of capillary and porous region l < x < L flow is through the glycocalyx.RHS function includes the pressure difference along the microchannels under unsteady fluid flow conditions.When the pressure difference is an unknown function of t, we reach a new model, and, by overdetermined additional conditions derived from an observation point, the solution of this problem can be obtained.The model can be considered as the mixed problem for one-dimensional diffusion equation with variable space operator:

1.1
Here, u t, x and p t are unknown functions, a x , b t, x , f t, x ,g t, x , ρ t , and ϕ x are given sufficiently smooth functions, and a x a > 0. Also, q x is a sufficiently smooth function assuming that q 0 q L 0 and q x * / 0.

Differential Case
To formulate our results, we introduce the Banach space , of all continuous functions φ x defined on 0, L with φ 0 φ L 0 satisfying a H ölder condition for which the following norm is finite: In a Banach space E, with the help of a positive operator A we introduce the fractional spaces E α , 0 < α < 1, consisting of all v ∈ E for which the following norm is finite: Positive constants will be indicated by M which can be differ in time.On the other hand M i α, β, . . . is used to focus on the fact that the constant depends only on α, β, . .., and the subindex i is used to indicate a different constant.
• C 2α 0, L , and ρ ∈ C 0, T .Then for the solution of problem 1.1 , the following coercive stability estimates and for every fixed t ∈ 0, T , the differential operator B is given by the formula Here, σ is a positive constant.The right-hand side functions are defined by where f t f t, x , g t g t, x , b t b t, x are known, and w t w t, x is unknown abstract functions defined on 0, T with values in E • C 0, L , w t, x * is unknown scalar function defined on 0, T , q q x , q q x , ϕ ϕ x , and a a x are elements of E • C 0, L , and q * q x * is a number.It is known that operator-A generates an analytic semigroup exp{−tA} t > 0 and the following estimate holds: where t, δ, M > 0 29 .
By the Cauchy formula, the solution can be written as

2.18
Then, the following presentation of the solution of abstract problem 2.12 exists:

2.20
From the fact that the operators R, exp{−λA} and A commute, it follows that 29

2.21
Abstract and Applied Analysis 7 Now, we estimate G k t for k 1, 2, . . ., 5 separately.Applying the definition of norm of the spaces E α and estimate 2.21 , we get Then, using estimate 2.17 for α 0, we reach to

2.24
By the definition of norm of the spaces E α , we have that

2.25
Let us estimate the first term.From the definition of norm of the spaces E α it follows that t 0 M 2 a, σ, α, x * , q, T .

8
Abstract and Applied Analysis Using estimate 2.17 , we obtain for any λ > 0. From that it follows

2.28
Then, we get Using definitions of norm of spaces E and E α and estimate 2.21 , we obtain that 31 for any t ∈ 0, T .
From estimate 2.29 , the estimate of G 3 t is as follows:

2.33
Now, let us estimate G 4 t .By the definition of the norm of the spaces E α , we get

2.35
Now, we consider the second term.Using 2.2 , we get

2.37
Combining estimates 2.35 and 2.37 , we obtain

2.38
The estimate of G 5 t is as follows.Since operators A and e −tA commute, we can write that

2.39
Let us estimate G 6 t : Aw s E α ds.

2.40
Since we get Aw s E α ds.

2.44
Applying the formulas and the triangle inequality, we can write w z E α dz .

2.46
Using boundedness of B, problem 2.12 , and estimate 2.46 , we have w z E α dz .

2.47
So, Gronwall's inequality and the following theorem finish the proof of Theorem 2.3.

Difference Case
For the approximate solution of problem 1.1 , the Rothe difference scheme where p k p t k , q n q x n , x n nh, and t k kτ is constructed.Here, q s / 0 and q 1 − q 0 q M 0 are assumed.x represents the floor function of x.
With the help of a positive operator A, we introduce the fractional spaces E α , 0 < α < 1, consisting of all v ∈ E for which the following norm is finite:

2.49
To formulate our results, we introduce the Banach space with φ 1 − φ 0 φ M 0 equipped with the norm

2.51
Abstract and Applied Analysis 13 Moreover, C τ E C 0, T τ , E is the Banach space of all grid functions φ τ {φ t k } N−1 k 1 defined on 0, T τ {t k kτ, 0 ≤ k ≤ N, Nh T } with values in E equipped with the norm

2.52
Then, the following theorem on well-posedness of problem 2.48 is established.
Theorem 2.4.For the solution of problem 2.48 , the following coercive stability estimates hold.Here,

2.54
Proof.The solution of problem 2.48 is searched in the following form: where Difference derivatives of 2.55 can be written as At the interior grid point s x * /h , we have that

2.58
Taking the difference derivative of the last equality and using the triangle inequality, we obtain In estimate 2.60 , {w h k } N k 0 is the solution of the following difference scheme:

2.61
where x n nh, t k kτ.Therefore, estimate 2.60 and the following theorem finish the proof of Theorem 2.5.
Theorem 2.5.For the solution of problem 2.61 , the following coercive stability estimate Proof.We can rewrite difference scheme 2.61 in the abstract form:

in a Banach space E
• C 0, l h with the positive operator A x h defined by acting on grid functions u h such that it satisfies the condition For every fixed t ∈ 0, T , the difference operators B x h t are given by the formula where σ is a positive constant and the right-hand side functions are

2.67
Let us denote R I τA x h −1 .In problem 2.63 , we have that By recurrence relations, we get

2.69
Then, the following presentation of the solution of problem 2.63 is obtained.Here,

2.71
Now, let us estimate J k r for r 1, 2, . . ., 6 separately.We start with J k 1 .Applying the definition of norm of the spaces E α , we get

2.72
Using estimate Abstract and Applied Analysis we get Let us estimate J k 2 : where a a q n 1 − 2q n q n−1 q s h 2 .

2.76
From the definition of norm of the spaces E α , it follows that

2.77
Let us estimate each term separately.We divide first term into two parts:

2.78
In the first part, by the definition of norm of the spaces E α and the identity see 29

2.80
The H ölder inequality with p 1/α, q 1/ 1 − α and the definition of the gamma function yield that

2.81
By the fact that Γ n n − 1 !and Γ n n − 1 Γ n − 1 , we get

2.82
So, we have that

2.83
In the second part, we have that

2.85
Let us estimate the second term.From the Cauchy-Riesz formula see 29 adz.

2.90
Summing the geometric progression, we get

2.94
Then, using estimates 2.85 and 2.94 , we get

2.96
Now, let us estimate J k 3 :

2.97
Since and using estimate 2.95 , we obtain

2.99
J k 4 can be estimated as follows:

2.100
From

2.102
The estimations of J k 5 and J k 6 are as follows.By the definition of the norm of the spaces E α and 2.95 , we get

2.108
The following theorem finishes the proof of Theorem 2.6.
Theorem 2.6 see 30 .For 0 < α < 1/2, the spaces E α C 0, L h , A x h and C 2α 0, L h coincide and their norms are equivalent.

Conclusion
Since artery disease caused by atherosclerosis is one of the most important causes of the death in the world, investigation of the effect of flow over the glycocalyx takes an important place.The flow equations can be formulated as an inverse problem.Here, our aim is to give more detailed understanding of the flow phenomena.Therefore, the well-posedness of the inverse problem of reconstructing the right side of a parabolic equation was investigated.Further, a new computer code regarding the flow analysis for the unknown pressure difference will be written.

Theorem 2.2. For the solution of problem 2.10 , the following coercive stability estimate
Proof.Let us search for the solution of inverse problem 1.1 in the following form see 23 :