Well-Posed and Ill-Posed Boundary Value Problems for PDEView this Special Issue
A Note on the Right-Hand Side Identification Problem Arising in Biofluid Mechanics
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.
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) . 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 ).
1.1. Problem Formulation
Blood flow over the EC inside the arteries is modeled in two regions (see ). 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 ): 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
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 .
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  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 ). 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:
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 ) 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 ) it follows that Since , with , the estimate (see ) yields Hence,