Abstract and Applied Analysis

Volume 2012 (2012), Article ID 548508, 25 pages

doi:10.1155/2012/548508

Research Article

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

Abdullah Said Erdogan

Department of Mathematics, Fatih University, Buyukcekmece 34500, Istanbul, Turkey

Received 18 March 2012; Accepted 13 May 2012

Academic Editor: Allaberen Ashyralyev

Copyright © 2012 Abdullah Said Erdogan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The inverse problem of reconstructing the right-hand side (RHS) of a mixed problem for one-dimensional diffusion equation with variable space operator is considered. The well-posedness of this problem in Hölder spaces is established.

1. 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 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]).

1.1. Problem Formulation

Blood flow over the EC inside the arteries is modeled in two regions (see [6]). Core region flow is defined through the center of capillary and porous region 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 , 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:

Here, and are unknown functions, ,   ,, and are given sufficiently smooth functions, and . Also, is a sufficiently smooth function assuming that and .

2. Main Results

2.1. Differential Case

To formulate our results, we introduce the Banach space ,   , of all continuous functions defined on with satisfying a Hölder condition for which the following norm is finite:

In a Banach space , with the help of a positive operator we introduce the fractional spaces , consisting of all for which the following norm is finite:

Positive constants will be indicated by which can be differ in time. On the other hand is used to focus on the fact that the constant depends only on , and the subindex is used to indicate a different constant.

Theorem 2.1. Let ,  , and . Then for the solution of problem (1.1), the following coercive stability estimates

hold.

Proof. Let us search for the solution of inverse problem (1.1) in the following form (see [23]):

where

Taking derivatives from (2.4) with respect to and , we get

Moreover, substituting by in (2.4), we obtain

Differentiating both sides of (2.7) with respect to , we get

From identity (2.8) and the triangle inequality, it follows that

for any . Using problem (1.1) and (2.4)–(2.7), one can show that is the solution of the following problem:

under the same assumptions on . Estimate (2.9) and the following theorem conclude the proof of Theorem 2.1.

Theorem 2.2. For the solution of problem (2.10), the following coercive stability estimate

holds.

Proof. Let us rewrite problem (2.10) in the abstract form as an initial-value problem:

in the Banach space . Here, the positive operator is defined by

with

and for every fixed , the differential operator is given by the formula

Here, is a positive constant. The right-hand side functions are defined by

where are known, and is unknown abstract functions defined on with values in ,    is unknown scalar function defined on ,   ,   ,   , and   are elements of , and is a number.

It is known that operator- generates an analytic semigroup and the following estimate holds:

where [29].

By the Cauchy formula, the solution can be written as

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

Here,

From the fact that the operators and commute, it follows that [29]

Now, we estimate for separately. Applying the definition of norm of the spaces and estimate (2.21), we get

Then, using estimate (2.17) for , we reach to

for any ,   .

Let us estimate :

By the definition of norm of the spaces , we have that

Let us estimate the first term. From the definition of norm of the spaces it follows that

Using estimate (2.17), we obtain

for any . From that it follows

Then, we get

Using definitions of norm of spaces and and estimate (2.21), we obtain that

for any .

From estimate (2.29), the estimate of is as follows:

Now, let us estimate . By the definition of the norm of the spaces , we get

Equation (2.2) yields that

Now, we consider the second term. Using (2.2), we get

for any . Then,

Combining estimates (2.35) and (2.37), we obtain

The estimate of is as follows. Since operators and commute, we can write that

Let us estimate

Since

we get

Finally combining estimates (2.23), (2.32), (2.33), (2.38), (2.39), and (2.42), we get

where .

Using Gronwall's inequality, we can write

Applying the formulas

and the triangle inequality, we can write

Using boundedness of , problem (2.12), and estimate (2.46), we have

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

Theorem 2.3 (see [29]). For , the spaces and coincide and their norms are equivalent.

2.2. Difference Case

For the approximate solution of problem (1.1), the Rothe difference scheme

where , and is constructed. Here, and are assumed. represents the floor function of .

With the help of a positive operator , we introduce the fractional spaces , consisting of all for which the following norm is finite:

To formulate our results, we introduce the Banach space ,  , of all grid functions defined on

with equipped with the norm

Moreover, is the Banach space of all grid functions defined on with values in equipped with the norm

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,

Proof. The solution of problem (2.48) is searched in the following form:

where

Difference derivatives of (2.55) can be written as

for any . At the interior grid point , we have that

Taking the difference derivative of the last equality and using the triangle inequality, we obtain

for any .

In estimate (2.60), is the solution of the following difference scheme:

where . 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

holds.

Proof. We can rewrite difference scheme (2.61) in the abstract form:

in a Banach space with the positive operator defined by

acting on grid functions such that it satisfies the condition

For every fixed , the difference operators are given by the formula

where is a positive constant and the right-hand side functions are

Let us denote . In problem (2.63), we have that

for all . By recurrence relations, we get

Then, the following presentation of the solution of problem (2.63)

is obtained. Here,

Now, let us estimate for separately. We start with . Applying the definition of norm of the spaces , we get

Using estimate

we get

for any ,   .

Let us estimate

where

From the definition of norm of the spaces , it follows that

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

In the first part, by the definition of norm of the spaces and the identity (see [29])

we deduce that

The Hölder inequality with and the definition of the gamma function yield that

By the fact that and , we get

So, we have that

In the second part, we have that

Combining estimates (2.83) and (2.84), we obtain

Let us estimate the second term. From the Cauchy-Riesz formula (see [29])

it follows that

Since , with , the estimate (see [29])

yields

Hence,

Summing the geometric progression, we get

Since the function

does not increase for , we have for all . Consequently,

for any . Hence,

Then, using estimates (2.85) and (2.94), we get

Now, let us estimate

Since

and using estimate (2.95), we obtain

can be estimated as follows:

From

it follows that

The estimations of and are as follows. By the definition of the norm of the spaces and (2.95), we get

Combining estimates (2.74), (2.96), (2.99), and (2.102)–(2.104), we get

Using the discrete analogue of Gronwall's inequality, we get

It follows from (2.63) and the triangle inequality that

for every ,   . Then, we have that

The following theorem finishes the proof of Theorem 2.6.

Theorem 2.6 (see [30]). For , the spaces and coincide and their norms are equivalent.

3. 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.

Acknowledgment

The author is grateful to Professor Allaberen Ashyralyev (Fatih University, Turkey) for his comments and suggestions to improve the quality of the paper.

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