Short-time existence of a quasi-stationary fluid–structure interaction problem for plaque growth

1.1 Model description

In this article, we consider a quasi-stationary fluid–structure interaction problem for plaque growth, which describes the formation of plaque during the reaction–diffusion and transport of different cells in human blood and vessels. The problem is set up in a smooth domain Ω t ⊂ R 3 , with three disjoint parts Ω t = Ω f t ∪ Ω s t ∪ Γ t , where Γ t = ∂ Ω f t , Ω f t ̄ ⊂ Ω t and Ω f t , Ω s t denote the domains for the fluid and solid, respectively. Γ s t = ∂ Ω t stands for the outer boundary of Ω t , which is also a free boundary. Given T > 0 , let

As in [2,3,43], the blood is described by the incompressible Navier-Stokes equation:

where T f ≔ − π f I + ν s ( ∇ v f + ∇ ⊤ v f ) denotes the Cauchy stress tensor. v f : R 3 × R + → R 3 , π f : R 3 × R + → R are the unknown velocity and pressure of the fluid. ρ f > 0 stands for the fluid density and ν s represents the viscosity of the fluid.

Compared to the problems in [2,3,43], where the evolutionary neo-Hookean material was employed, the vessel in this article is assumed to be quasi-stationary since it moves far slower than the blood from a macro point of view. Thus, we model the blood vessel by the equilibrium of a nonlinear elastic equation. To mathematically describe the elasticity conveniently, the Lagrangian coordinate was commonly used, see, e.g., [18,19]. Thus, we set reference configuration as the initial domain, which is defined by Ω ≔ Ω t ∣ t = 0 , as well as Ω f = Ω f 0 , Ω s = Ω s 0 and Γ = Γ 0 , then Ω = Ω f ∪ Ω s ∪ Γ . Let X be the spatial variable in the reference configuration. Now we introduce the Lagrangian flow map

with

for all X ∈ Ω and x ( X , 0 ) = X , where

denotes the displacement for fluid or solid. In the sequel, without special statement, the quantities with a hat will indicate those in Lagrangian reference configuration, e.g., u ˆ ( X , t ) = u ( x ( X , t ) , t ) , while the operators with a hat means those act on the quantities in Lagrangian coordinate. Then the tensor field

with F ˆ f ( X , 0 ) = I and F ˆ s ( X , 0 ) = I + ∇ ˆ u ˆ s 0 is referred to be the deformation gradient and J = J ˆ ≔ det ( F ˆ ) denotes its determinant. For the blood vessels, since the growth is taken into account, we impose the so-called multiplicative decomposition for the solid deformation gradient F ˆ s as follows:

with J ˆ s = J ˆ s , e J ˆ s , g , where F ˆ s , e is the pure elastic deformation tensor and F ˆ s , g denotes the growth tensor, which will be specified later. For more details about the decomposition, see, e.g., [18,22,35,43]. Inspired by Goriely [18, Chapter 11–13] and Jones-Chapman [22, Section 3.2], a general incompressible hyperelastic material is considered for solid as follows:

where T s ≔ − π s I + J s , e − 1 DW ( F s , e ) F s , e ⊤ stands for the Cauchy stress tensor. π s : R 3 × R + → R is the unknown pressure of the solid. The scalar function W : R 3 × 3 → R + is called the strain energy density function (also known as the stored energy density), which needs some general assumptions for the sake of analysis.

Besides (1.5), the balance of mass should hold as well together with the growth, namely,

where ρ s > 0 is the solid density. v s ≔ ∂ t u s denotes the velocity of the solid. The function f s g ≔ γ f s r with γ > 0 on the right-hand side of (1.6) stands for the growth rate, which comes from the cells reaction assigned later.

There are two essential assumptions when either the density or the volume does not change with growth. Generally, a constant-density type growth is assumed for the incompressible tissue, see, e.g., [22,35], in this article, we assume this kind of growth as in [2,3,43]. Then by [18, (13.10)], one obtains

As in [2,3,43], the plaque grows when the macrophages in the vessels accumulate a lot from the monocytes in the blood flow and turn to be the foam cells. Thus, denoting by c f , c s , and c s * the monocytes, the macrophages, and the foam cells, respectively, we introduce

where D f ⁄ s > 0 are the diffusion coefficients in the blood and the vessel, respectively. f s r ≔ β c s with β > 0 stands for the reaction function, modeling the rate of transformation from macrophages into foam cells. Noticing that the foam cells are supposed to only move with vessels motion, one has (1.10).

To close the system, one still needs to impose suitable boundary and initial conditions. For this purpose, we follow a similar setting as in authors’ previous work [2]. On the free interface Γ t , one has the continuity of velocity and normal stress tensor for the fluid–structure part, while for the cells, the concentration flux is continuous and the jump of the concentration is determined by the permeability and flux across the vessel wall.

where n Γ t represents the outer unit normal vector on Γ t pointing from Ω f t to Ω s t . For a quantity f , 〚 f 〛 denotes the jump defined on Ω f t and Ω s t across Γ t , namely,

Here ζ >0 denotes the permeability of the sharp interface Γ t with respect to the monocytes, which basically should depend on the hemodynamic stress T f n Γ t . It is supposed to be a constant for simplicity. Moreover, Γ s t is assumed to be a free boundary as well due to the physical compatibility, see, e.g., [2, Remark 1.3]. Then we give the boundary conditions on Γ s t ,

Finally, the initial values are prescribed as follows:

1.2 Previous work

Our main goal is to investigate the short time existence of strong solutions to (1.1)–(1.16). Before discussing the technical details, let us recall some literature related to our work.

In the case of 3d–3d fluid–structure interaction problem with a free interface, the strong solution results can be tracked back to Coutand and Shkoller [12], who addressed the interaction problem between the incompressible Navier-Stokes equation and a linear Kirchhoff elastic material. The results were extended to fluid-elastodynamics case by them in [13], where they regularized the hyperbolic elastic equation by a particular parabolic artificial viscosity and then obtained the existence of strong solutions by delicate a priori estimates. Thereafter, systems consisting of an incompressible Navier-Stokes equation and (damped) wave/Lamé equations were continuously investigated in [20,21,23,34] with further developments. Besides the aforementioned models, one refers to [8] for a compressible barotropic fluid coupled with elastic bodies and to [37] for the magnetohydrodynamics (MHD)-structure interaction system, where the fluid is described by the incompressible viscous non-resistive MHD equation and the structure is modeled by the wave equation of a superconductor material.

In the context of 3d–2d/2d–1d models, various kinds of models and results were established during the last 20 years. The widely focused case is the fluid-beam/plate systems where the beam/plate equations were imposed with different mechanical mechanism (rigidity, stretching, friction, rotation, etc.), and readers are referred to [9,16,31,42] for weak solution results and to [5,14,17,25,28,30] for strong solutions. Besides, the fluid–shells interaction problems were studied as well in [6,24] for weak solutions and in [10,11,27] for strong solutions. It is worth mentioning that in recent works [14, 28], a maximal regularity framework, which requires lower initial regularity and less compatibility conditions compared to the energy method, was employed.

Recently, inspired by [43], the authors considered the local well-posedness of a fluid–structure interaction problem for plaque growth in a smooth domain [2] and a cylindrical domain [3], respectively. By the maximal regularity theory, one obtains the well-posedness of linearized systems and then derives the existence of a unique strong solution to the nonlinear system via a fixed-point argument. However, concerning the origin of the model, which consists of blood flow and blood vessels, it is reasonable that the movement of vessels is supposed to be far slower than the blood, namely, the time scales of motions of the fluid and the solid are distinguished and the kinetic energy of the vessel is neglected. Thus, in this article, we take an incompressible quasi-stationary hyperelastic equation into account, i.e., for every time t the elastic stresses are in equilibrium. To the best of our knowledge, this is the first result concerning strong solutions to a quasi-stationary fluid–structure interaction problem with growth.

1.3 Technical discussions

Under the aforementioned setting, the fluid–structure part is of parabolic-elliptic type, while the cells part is similar to the ones in [2,3]. To solve the nonlinear problem, our basic strategy is still a fixed-point argument in the framework of maximal regularity theory, while more issues come up when we consider the linearized systems. More precisely, for the linearization of the fluid–structure part, it is hard to solve it directly by the maximal regularity theory due to the parabolic-elliptic type coupling, which results in the unmatched regularity between v ˆ f and u ˆ s on the sharp interface Γ . To overcome the problem, one tries to decouple the system to a nonstationary Stokes equation with respect to fluid velocity v ˆ s and a quasi-stationary Stokes-type equation with regard to the solid displacement u ˆ s . Note that one key point is to separate the kinetic and dynamic condition on the interface correctly. Specifically, we impose the Neumann boundary condition for v ˆ s and a Dirichlet boundary condition for u ˆ s , see Section 4. Otherwise if v ˆ f ∣ Γ = v ˆ s , one may face the problem that there is no regularity information about the velocity of solid v ˆ s = ∂ t u ˆ s , since the solid equation is quasi-stationary without any damping.

Another issue is the choice of function spaces for the elastic equation, i.e., how to assemble suitable function spaces for the data in the linearized solid equation so that the regularity of ∇ ˆ u ˆ s matches with ∇ ˆ v ˆ s of fluid on the interface in the nonlinear system. Our choice is

This space is motivated by the observation that the nonstationary Stokes equation (4.1) is uniquely solvable if the Neumann boundary data

hold, which imply that

as well. Here, H q s and W q s are the Bessel potential space and Sobolev-Slobodeckij space, respectively, which will be given in Section 2.2. In fact, the anisotropic Bessel potential space we assigned is a sharp regularity if one goes back to the anisotropic trace operator, see Lemma 2.7, and it is natural to equip f with the aforementioned regularity. Because of this sharp regularity setting, one can not expect the Lipschitz estimates of nonlinear terms only with small time, which gives rise to an additional smallness assumption (3.7) on the initial solid displacement and pressure. Detailed discussion can be found in Remark 3.2 and Proposition 5.7.

To solve the quasi-stationary (linearized) elastic equation, one treated it as a Stokes-type problem with respect to the displacement u ˆ s and the pressure π ˆ s , due to the incompressibility. However, as we assigned the certain regularity space for it as mentioned earlier and the Stokes operator is not a standard one (namely, div ( D 2 W ( I ) ∇ ⋅ ) ), one needs to consider a generalized stationary Stokes equation with f in L q and W q , Γ 1 − 1 , respectively, for which the maximal regularity theory of analytic C 0 semigroups is applied, as well as a complex interpolation method with a very weak solution in L q of a mixed-boundary Stokes-type equation, which can be solved by a duality argument.

For the cells part, there is a problem of the positivity for the concentrations, compared to [2]. The idea in [2] to prove it is to apply the maximum principle to the original equation and deduce a contradiction with the help of Hopf’s Lemma. However, due to the lack of regularity of v s = ∂ t u s , one can not expect it to be Hölder continuous in space-time, even continuous. To deal with this trouble, we make use of the idea of mollification, i.e., approximating v s by sufficient smooth functions v s ε such that ∫ 0 t v s ε → u s in certain space. Then arguing by a similar procedure in [2], we obtain an approximate nonnegative solution c ε . Finally, one can show that it converges to a nonnegative function c , which exactly satisfies the original equations of cell concentrations.

1.4 Structure of the paper

This article is organized as follows. In Section 2, we briefly introduce some notations and function spaces together with some corresponding properties. Moreover, a reformulation of the system is done in Section 3.1, and later we give the main result for the reformulated system. Section 4 is devoted to three linearized systems in Sections 4.1–4.3. The main results of this section are the L q -solvability for these linear problems, for which a careful analysis is carried out. In Section 5, we first introduce some preliminary lemmas in Section 5.1, which will be frequently used in proving the Lipschitz estimates in Section 5.2. Then by the Banach fixed-point theorem, we derive the short time existence of strong solutions to the nonlinear system in Section 5.3. Moreover, the cell concentrations are shown to be nonnegative, provided that the initial concentration is nonnegative. In addition, we establish the solvability of a Stokes-type resolvent problem with mixed boundary condition in Appendix A.

2.1 Notations

Let us introduce some notations. For a vector u ∈ R d , d ∈ N + , we define the gradient as ( ∇ u ) i j = ∂ i u j . For a matrix A = ( A i j ) i , j = 1 d ∈ R d × d , we introduce the divergence of a matrix row by row as ( div A ) i = ∑ j = 1 d ∂ j A i j . Given matrices A = ( A i j ) i , j = 1 d , B = ( B i j ) i , j = 1 d ∈ R d × d , we know ( A B ) i k = ∑ j = 1 d A i j B j k . Then we denote by A : B ≔ tr ( B ⊤ A ) = ∑ j , k = 1 d B j k A j k the Frobenius product and ∣ A ∣ ≔ A : A the induced modulus. For a differentiable and invertible mapping A : I ⊂ R → R d × d , we have

see, e.g., [18,19]. Moreover, we define R sym + d × d as the space of all positive-definite symmetric matrices and

as the set of all proper orthogonal tensors.

Generally, B X ( x , r ) denotes the open ball with radius r > 0 around x in a metric space X . For normed spaces X , Y over K = R or C , the set of bounded, linear operators T : X → Y is denoted by ℒ ( X , Y ) and in particular, ℒ ( X ) = ℒ ( X , X ) .

As usual, the letter C in the present article denotes a generic positive constant, which may change its value from line to line, even in the same line, unless we give a special declaration.

2.2 Function spaces and properties

For an open set M ⊂ R d with Lipschitz boundary ∂ M , d ∈ N + , we recall the standard Lebesgue space L q ( M ) and Sobolev space W q m ( M ) , m ∈ N , 1 ≤ q ≤ ∞ , as well as follows:

where q ′ is the conjugate exponent of q satisfying 1 ⁄ q + 1 ⁄ q ′ = 1 . Furthermore, we set the Sobolev spaces associated with Γ ⊂ ∂ M as follows:

The vector-valued variants are denoted by L q ( M ; X ) and W q m ( M ; X ) , where X is a Banach space. In particular, L q ( M ) = W q 0 ( M ) , L q ( M ; X ) = W q 0 ( M ; X ) .

Moreover, for 1 < q < ∞ and 1 ≤ p ≤ ∞ , the standard Besov space B q , p s and Bessel potential space H q s coincide with the real and complex interpolation of Sobolev spaces, respectively (see, e.g., Lunardi [26], Runst and Sickel [36], and Triebel [41])

where s = ( 1 − θ ) k + θ m , θ ∈ ( 0 , 1 ) , k , m ∈ N , k < m . In particular, W q l = H q l for l ∈ N and setting p = q above yields the Sobolev-Slobodeckij space:

Then we define the seminorm of W q s ( M ) as follows:

for f ∈ W q s ( M ) with 0 < s < 1 , 1 < q < ∞ .

For an interval I ⊂ R , 0 < s < 1 , 1 < q < ∞ , we recall the Banach space-valued spaces K q s ( I ; X ) , K ∈ { W , H } with norm

For convenience, with T > 0 , we define the corresponding space with vanishing initial trace at t = 0 as follows:

Now we recall several embedding results and properties for Banach space-valued spaces that will be frequently used later.

Adapting from Meyries and Schnaubelt [29, Proposition 3.2] and Prüss and Simonett [33, Section 4.5.5], we use the following time-space embedding lemma.

Now we include results on multiplication and composition.

In the following, an anisotropic trace lemma is introduced for a fractional power space.

3.1 System in Lagrangian coordinates

In this section, we transform (1.1)–(1.16) in the deformed domain Ω t to the reference domain Ω , whose definitions are given in Section 1.1. Let ϕ ⁄ ϕ ˆ be any scalar function in Ω t ⁄ Ω and w ⁄ w ˆ be any vector-valued function in Ω t ⁄ Ω . Then one can easily derive the relations between derivatives in different configurations as follows: