On a fluid-structure interaction problem for plaque growth

We study a free-boundary fluid-structure interaction problem with growth, which arises from the plaque formation in blood vessels. The fluid is described by the incompressible Navier-Stokes equation, while the structure is considered as a viscoelastic incompressible neo-Hookean material. Moreover, the growth due to the biochemical process is taken into account. Applying the maximal regularity theory to a linearization of the equations, along with a deformation mapping, we prove the well-posedness of the full nonlinear problem via the contraction mapping principle.

1. Introduction 1.1.The free-boundary fluid-structure interaction model.In this paper, we consider a free-boundary fluid-structure interaction problem with growth, which is used to describe the plaque formation in a human artery.The blood is assumed to be the incompressible Navier-Stokes equation and the artery is modeled by an elastic equation with viscosity.Based on [46], where the model was proposed and simulated in a cylindrical domain, we analyze such problem in a bounded domain Ω t =⊂ R n , n ≥ 2. See Figure 1.Here, Ω t = Ω t f ∪ Ω t s ∪ Γ t , where Ω t is divided by the interface Γ t into two disjoint parts, fluid domain Ω t f and solid domain Ω t s .Γ t s denotes the outer boundary of Ω t , which is also a free boundary.
Before giving a precise description of the model, we introduce the setting of Lagrangian coordinate.For convenience, we define the moving domain at initial time t = 0 as Ω = Ω f ∪ Ω s ∪ Γ, where Ω f = Ω 0 f , Ω s = Ω 0 s and Γ = Γ 0 .From the viewpoint of material deformation (see e.g.[14,23]), we set the so-called reference configuration at t = 0 and the deformed configuration at time t.Moreover, we denote the spatial variable at t = 0 by the Lagrangian variable X, resp., by the Eulerian variable x the spatial variable at t.The velocities of deformations are v(X, t) and v(x, t) respectively.In the sequel, without special statement, the quantities or operators with a hat "•" will indicate those in Lagrangian reference configuration.To formulate the model, we define the deformation as (See Figure 2) and x| t=0 = ϕ(X, 0) = X.have the inverse deformation gradient by F = F −1 .In the following, quantities in fluid and structure domain will be distinguished by subscript "f " and "s" respectively, quantities without subscript are defined both in fluid and structure domain.Now, we are in the position to describe the whole system.For a finite time T > 0, 0 < t < T , we model the motion of the fluid by the classical incompressible Navier-Stokes equations, which is where ρ f is the known constant fluid density.
denotes the Cauchy stress tensor, π f is the unknown fluid pressure and ν f represents the fluid viscosity.The equations for the solid are written as: where ρ s is the solid density.σ s = σ e s + σ v s is the stress tensor of the solid where π s is the unknown solid pressure.Moreover, µ s denotes the Lamé coefficient and ν s represents the solid viscosity, which are all positive constant.The first equation is the balance equation of linear momentum.σ e s is given by the constitutive relation of an incompressible Neo-Hookean material, which is hyperelastic, isotropic and incompressible.This relationship was widely used to describe blood vessel wall by many investigators, see e.g.[44,46].The tensor F e s is the inverse elastic deformation gradient under the assumption of growth and will be assigned later in Section 1.3.We consider not only the elastic stress σ e s , but also the viscoelastic stress σ v s , which could be deduced by linearizing the Kelvin-Voigt stress tensor, see Mielke and Roubíček [35].The second equation of (1.3) is due to the mass balance, where f g s is called growth function and represents the rate of mass growth per unit volume due to the formation of plaque, see e.g.[7,27,46].Remark 1.1.For short time existence, the Kelvin-Voigt viscous stress σ v s we introduced brings the parabolicity to the solid equation, which dominates the regularity of solutions.Moreover, after linearization one obtains a two-phase Stokes type problem, which ensures us to get the solvabilities and regularities of fluid and solid velocities by maximal regularity theory.In a recent work [9], a similar stress tensor of solid part was also considered to investigate weak solutions of the interaction between an incompressible fluid and an incompressible immersed viscous-hyperelastic solid structure.
Remark 1.2.In [46,44], some numerical simulations are carried out by considering that µ s depends on the concentration of some chemical species, and hence varies from healthy vessel to plaque area.In the case of viscoelasticity, ν s may also vary over the solid domain.However, to simplify the model for the analysis, we assume that these coefficients are constant over the solid domain.
The interaction between the fluid and solid is modeled by transmission conditions on the interface Γ t , which consists of the continuity of velocity and the balance of normal stresses: where n Γ t stands for the outer unit normal vertor on Γ t pointing from Ω t f to Ω t s .For a quantity f , f denotes the jump defined on Ω t f and Ω t s across Γ t , namely, f (x) := lim Moreover, to ensure the compatibility with growth and incompressibility, the boundary condition on Γ t s is assumed to be the so-called "stress-free" boundary condition: where n Γ t s is the unit outer normal vector on Γ t s = ∂Ω t .Remark 1.3.The "stress-free" boundary condition (1.6) is set due to physical reality.Since we consider the growth of solid part and both fluid and solid part are incompressible, we can not impose some types of boundary conditions.For example, the clamped condition v s = 0 on Γ t s (correspondingly, v s = ∂ t u s = 0 on Γ t s ), will destroy the property of incompressibility.
Remark 1.4.In this work, the fluid part is supposed to be surrounded by the solid part.In fact, if the solid is immersed in fluid domain, there will be no essential difference in our framework of analysis.Specifically, the outer boundary will still be a Neumann-type boundary, which is a "do-nothing" outer boundary for fluid.
The initial values for velocities are prescribed as (1.7) 1.2.Biochemical processes.The formation of plaque is usually due to biochemical processes in the blood flow and vessel wall.Following the descriptions in [46,47], we introduce the dynamics of monocytes in the blood flow and dynamics of macrophages and foam cells in the vessel wall, of which the concentrations are denoted by c f , c s , c * s , respectively.Convection and diffusion happen during these biochemical processes, so the motion of monocytes in the blood is given by the transport-diffusion equation where D f > 0 is the diffusion coefficient in the blood, which is assumed to be a constant since the fluid is incompressible and homogeneous.Analogously, the motion of macrophages in the vessel wall is described by where D s > 0 is the diffusion coefficient in the vessel wall.To simplify the model, we assume that the solid is a homogeneous material, and thus D s is a constant.We mention that vessel wall could be inhomogeneous, which represents different diffusion rate in healthy and diseased vessel, see e.g.[46].f r s is the reaction function, modeling the rate of transformation from macrophages into foam cells.Furthermore, since foam cells do not diffuse inside the solid material, they are accumulated only due to the convection with v s and the transformation from macrophages, which results in the equation of foam cells (1.10) The reaction term f r s is supposed to depend on the concentration of macrophages c s linearly, namely, ) where β > 0 is assumed to be a constant.In reality, it is more complicated and may depend on the concentration of other chemical species.We just assume a linear relation for the sake of analysis.Then, we give another linear dependence of f g s , which is ) with a positive constant γ. (1.12) indicates the plaque growth as mentioned in (1.3), resulting from the accumulation of foam cells.
To close the system, we still need to model the penetration of monocytes from the blood into the vessel wall, for which the transmission conditions are given on Γ t by the jump condition where ζ denotes the permeability of the interface Γ t with respect to the monocytes, which should depend on the hemodynamical stress σ f • n Γ t , however, is supposed to be a constant for simplicity.Moreover, the condition on the outer boundary Γ t s is given by D s ∇c s • n Γ t s = 0, on Γ t s .
(1.15)At initial state, the vessel is supposed to be healthy.Hence, there is no foam cell in vessel, i.e., c * s (x, 0) = 0, in Ω t s .
(1.16) The initial values for concentrations of monocytes and macrophages are given by (1.17) 1.3.Description of growth.Now, we give the description of growth.Normally, prescribing the rate of growth function f g s is not enough to capture the full effect of the tissue growth.Specifically, the real deformation and corresponding deformation gradient Fs are induced by both growth and mechanics.Hence, the deformation gradient Fs is not enough to capture all responses in the system, for example the deformation gradient in σ e s .Thus, simply transforming the system to Lagrangian coordinates, such as in [16], we can not solve the whole fluid-structure interaction problem with growth.
As in [46], Yang et al. took the idea of deformation gradient decomposition based on the theory of multiple natural configurations.In this formulation, one needs a new configuration, which is usually called natural configuration, so that one can decompose the whole process into a pure growth and a pure elastic one, see Figure 3.For more details, readers are referred to [7,27,39,46,47].In this article, we assume

Ĵe
s .Growth may happen in different ways.In applications, two assumptions were most commonly applied: constant-density, which stands for adding new material with the same density; constant-volume, by which the total mass is added and density varies.Since constant-density growth is usually coupled with the assumption of an incompressible tissue, see e.g.[27,39], we take this kind of growth into consideration in this work.Then the second equation of (1.3) reads as ρ s div v s = f g s .Moreover, we assume that plaques grows isotropically: F g s = ĝI, in Ω s , where ĝ = ĝ(X, t) is the metric of growth, a scalar function depending on the concentration of macrophages.Hence, where n is the dimension of space.As [7] mentioned, ĝ describes the deformation state of the material, either growing or resorbing, as From [7,27,46], under the assumption of constant-density growth, we deduce that This equation shows the specific dependence on ĉs of ĝ.At initial state, ĝI is supposed to be the identity, i.e., ĝ(X, 0) = 1, in Ω s , without growth or resorption of the material.1.4.Literature.During last decades, fluid-structure interaction problems attracted much attention from mathematicians due to its strong applications in various areas, e.g., biomechanics, blood flow dynamics, aeroelasticity and hydroelasticity.Studies can be divided into two types depending on the dimensions of the fluid and the solid.They are for example 3d-3d coupled and 3d-2d coupled systems, where the solid is contained in the fluid and one part of fluid's boundary respectively.
In the case of 3d-3d model, which is exactly our consideration, let us recall some existence results of strong solutions.Well-posedness of such model was firstly established by Coutand and Shkoller [15], where they investigated the interaction problem between the Navier-Stokes equation and a linear Kirchhoff elastic material.The results were extended to the quasilinear elastodynamics case by them, where they regularized the hyperbolic elastic equation by a particular parabolic artificial viscosity and then obtained the existence of strong solutions together with the a priori estimates in [16].Thereafter, Ignatova, Kukavica, Lasiecka and Tuffaha [24,25] investigated the coupled system of incompressible Navier-Stokes equation and a wave equation from different aspects.More specifically, In [24], static damping and velocity internal damping were added in the wave equation and boundary friction was considered, by which exponential decay was obtained.Later, the boundary friction was removed in [25] by introducing the tangential and time-tangential energy estimates.The coupling of the Navier-Stokes equations and the Lamé system was analyzed by Kukavica and Tuffaha [28] with initial regularity (v 0 , ξ 1 ) ∈ H 3 (Ω f ) × H 2 (Ω s ), while Raymond and Vanninathan [38] further proved the existence and uniqueness of local strong solutions with a weaker initial regularity (v 0 , ξ 1 ) ∈ H 3/2+ε (Ω f ) × H 1+ε (Ω s ), ε > 0 arbitrarily small, with periodic boundary conditions.Lately, Boulakia, Guerrero and Takahashi [11] showed a similar result for the Navier-Stokes-Lamé system in a smooth domain with reduced demand of the initial regularity.
There are also other variants of free-boundary fluid-structure interactions models.For compressible fluid coupled with elastic bodies, we refer to [10], where Boulakia and Guerrero addressed the local in time existence and the uniqueness of regular solutions with the initial data (ρ 0 , u 0 , w 0 , w 1 ) This results was later improved by Kukavica and Tuffaha [29] with a weaker initial regularity (ρ 0 , u 0 , w 1 ) ∈ H 3 (Ω f ) × H 3/2+r (Ω f ) × H 3/2+r (Ω s ), r > 0.More recently, Shen, Wang and Yang [40] consider the magnetohydrodynamics (MHD)-structure interaction system, where the fluid is described by the incompressible viscous nonresistive MHD equation and the structure is modeled by the wave equation with superconductor material.They solved the existence of local strong solutions with penalization and regularization techniques.
For the 3d-2d/2d-1d systems where where the structure is seen as one part of the fluid's boundary, we just mention several works on the existence and uniqueness of strong solutions to be concise.The mostly investigated case is the fluidbeam/plate systems where the beam/plate equation was imposed with different mechanical mechanism (rigidity, stretching, friction, rotation, etc.), readers are refer to [8,17,21,22,30,31,34,36] and references therein.Moreover, the fluid-structure interaction problems with nonlinear shells were studied in [12,13,33].It has to be mentioned that in the recent works [17,34], a maximal regularity framework, which requires lower initial regularity and less compatibility conditions compared to the energy method, was employed.1.5.Mathematical strategy and features.The new difficulties arise from the plaque formation in the blood vessels, along with the interaction between the fluid and the solid separated by a free interface, the reaction and the diffusion of different cells and the growth of the vessel wall.Numerical computations were carried out in recent years [20,46,47] to simulate the plaque formation and test the effects of different parameters.To our best knowledge, this is the first work concerning the existence of the strong solutions to the fluid-structure interaction problems with growth.Unlike most of the literature above, which are associated with Hilbert spaces (L 2 -setting) and energy methods, we establish our local strong solutions under the framework of maximal L q -regularity for more general dimension.The method is based on the Banach fixed-point theorem, for which we rewrite the free boundary problem established with Eulerian coordinates in Lagrangian reference configuration, linearize the system at the initial configuration, construct a contraction mapping in a fixed ball and show the local existence and uniqueness of strong solutions.Throughout the proof, we point out the following features.i) We adapt the maximal L q -regularity theory to solve our problem.Hence, there will be no "regularity loss" from initial data to the solution spaces and only few compatibility conditions are needed.ii) The growth is considered to be of the constant-density type.Then under the assumption of isotropy, the growth will be indicated by the metric function ĝ.An ordinary differential equation of ĝ provides the regularity of ĝ needed for the solid velocity and the concentration of macrophages.iii) The Kelvin-Voigt viscous stress σ v s we introduced brings the parabolicity to the solid equation.For the linearization, we can use a two-phase Stokes type problem for the fluid-structure interaction problem.This makes sure that we can get the solvabilities and regularities of fluid and solid velocities by maximal regularity theory.iv) The transformed two-phase Stokes problem is endowed with a stress free (Neumann-type) outer boundary condition due to Remark 1.3.One of our aims is to obtain the solvability of such system.To this end, reduction and truncation arguments are applied.More specifically, we firstly reduce the inhomogeneous linear system to a source and initial value homogeneous problem (except the boundary terms), in order to obtain the pressure regularities.
Then by choosing a cutoff function (see (3.13)) which is supported in a subset U ⊆ Ω and imposing an artificial vanishing Dirichlet boundary on Γ s = ∂ Ω, one obtains the solvability of the linear system since the two-phase Stokes problem with Dirichlet boundary is solved in Appendix A.1.
1.6.Outline of the paper.In Section 2 we briefly introduce some notations and function spaces along with several preliminary results.Transformation from the deformed configuration to the reference one is shown in the last subsection, as well as the main theorem for the transformed system.Section 3 is devoted to the analysis of the underlying linear problems, where three separated parts of analysis are proceeded.The main results of this section are the maximal L q -regularities for these linear problems.The first one is the two-phase Stokes problems with Neumann boundary condition, to which reduction and truncation (localization) arguments are applied.The second problem consists of two reaction-diffusion systems with Neumann boundary condition due to the decoupling of the transmission problem, while the last one is an ordinary differential equations for growth and foam cells.In Section 4, we firstly give some estimates related to the deformation gradient, which are of much importance when proving that the constructed nonlinear terms are well-defined and endowed with the property of contraction in the next subsection.Then the full nonlinear system is shown to be well-posed locally in time via Banach fixed-point theorem.Moreover, the cell concentrations are showed to be always nonnegative, provided that the initial data is nonnegative.Additionally, we introduce some maximal L q -regularity results of several linear systems in Appendix A and establish a uniform extension of the Sobolev-Slobodeckij spaces in Appendix B.

General settings and main results
2.1.Mathematical notations.For matrices A, B ∈ R n×n , let A : B = tr(B ⊤ A) and corresponding induced modulus of A as |A| = √ A : A. The set of invertible matrices in R n×n is GL(n, R).For a differentiable A : R + → GL(n, R), we have two useful formulas as which can be found in [19,23].Furthermore, for a vector function u and a tensor matrix T , we give an identity which will be used later (see e.g.[23, (3.20)]): For metric spaces X, B X (0, r) represents the open ball with radius r > 0 around x ∈ 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 paper represents generic positive constant which may change its value from line to line or even in the same line, unless we give a special declaration.

Function spaces. If
denotes the usual Lebesgue space and • L q (M ) its norm, as well as the mean value zero Lebesgue space with |M| < ∞.Moreover, L q (M; X) denotes its vector-valued variant of strongly measurable q-integrable functions/essentially bounded functions, where X is a Banach space.If M = (a, b), we write for simplicity L q (a, b) and L q (a, b; X).By simple computation, we have Let Ω ⊆ R n be a open and nonempty domain, W m q (Ω) denotes the usual Sobolev space with m ∈ N and L q (Ω) = W 0 q (Ω).Moreover, we set we consider the standard definition of the Besov spaces by real interpolation of Sobolev spaces (see Lunardi [32]) , where s = (1 − θ)k + θk ′ , θ ∈ (0, 1).In the special case q = p, we also have Sobolev-Slobodeckij spaces , which is endowed with norm The multiplication property of such space is given in the next lemma.

Lemma 2.1 (Multiplication).
Let Ω be a bounded Lipschitz domain.For f, g ∈ W s q (Ω) and sq > n with s > 0, we have the multiplication property, which is , where M q is a constant depending on q.
Proof.For the case s ∈ N + , we refer to [45,Theorem 1].For the other cases, since W s q = B s q,q for every s ∈ R + \N, then [26, Theorem 6.6] implies this.Next, for an interval I ⊂ R and a Banach space X, we recall the definition of vector-valued Sobolev-Slobodeckij space as Then we define 0 W s q (0, T ; X) with 0 < T ≤ ∞ to be a vector-valued space having a vanishing trace at t = 0, i.e., 0 W s q (0, T ; X) := u ∈ W s q (0, T ; X) : u| t=0 = 0 .In addition, we introduce one embedding result from Simon [43,Corollary 17].
Lemma 2.2.Suppose 0 < r ≤ s < 1 and 1 ≤ p ≤ ∞.Then W s q (I; X) ֒→ W r q (I; X) and, for all f ∈ W s q (I; X), for bounded I, For r, s ≥ 0, the anisotropic Sobolev-Slobodeckij spaces W r,s q is defined as Based on the trace method interpolation at time zero [37, Section 3.4.6]and [6, Chapter III, Theorem 4.10.2],we give some useful embeddings, which will be employed later.
Lemma 2.3.Let X 1 , X 0 be two Banach spaces and X 1 ֒→ X 0 .Define X T = L q (0, T ; X 1 ) ∩ W 1 q (0, T ; X 0 ) for all 1 < q < ∞ and 0 < T < ∞, then where (Ω) and together with Lemma 2.4.Let Σ be a compact sufficiently smooth hypersurface.For 1 < q < ∞, where 3. An equivalent system in Lagrangian reference configuration.In this section, we transform the free-boundary fluid-structure problem with growth from deformed configuration (Eulerian) to a fixed reference configuration (Lagrangian) and state the main result.For quantities in different configurations, we define v(X, for all x = ϕ(X, t), X ∈ Ω and t ≥ 0. Then one can easily deduce the derivatives between quantities in different configurations as ) ) where φ/ φ is any scalar function in Ω/ Ω and u/ û is any vector-valued function in Ω/ Ω.From [14], we know that the Piola transform establishes a correspondence between tensor field defined in deformed and reference configurations, which is where T is the first Piola-Kirchhoff stress tensor.Moreover, the following property of the Piola transformation will be useful: Lemma 2.5 ([14, Theorem 1.7-1]).For a stress tensor σ(x, t) in the deformed configuration Ω, and the corresponding first Piola-Kirchhoff stress tensor T (X, t) in reference configuration Ω, we have: For the fluid part, it follows from (2.1) that For the solid part, since the deformation from natural configuration Ω g s to the deformed configuration Ω t s conserves mass, incompressibility yields Ĵe s = 1 and hence, Ĵs = Ĵg s = ĝn , in Ω s .
Now combining formulas (1.18), (2.7)-(2.12)and Lemma 2.5, we rewrite the fluid-structure interaction problem (1.2)-(1.17) in the reference configuration Ω. ) where the corresponding stress tensors are For the maximal L q -regularity setting, we assume where we define Ω = Ω f ∪Ω s .D q := D 1 q ×D 2 q will be the initial space for velocities and concentrations.Moreover, we introduce the compatibility conditions for q > n + 2, which were also used in e.g.Abels [1], Prüss and Simonett [37], Shibata and Shimizu [41], Shimizu [42]: and where (•) τ denotes the tangential part on the surface, namely, ( Besides this, we define the solution space for (v, π, ĉ, ĉ * s , ĝ) as 1.These spaces are constructed from the problem and the maximal regularity theory, endowed with the natural norms.In particular, π and π| Γs are determined by the regularities of the Neumann trace of v on Γ and Γ s respectively.Hence, we add the norm of π W 1−1/q,(1−1/q)/2 q (Γ×(0,T )) and π| Γs W 1−1/q,(1−1/q)/2 q (Γs×(0,T )) in Y 2  T -norm correspondingly.One can easily verify that all spaces are Banach spaces.Now the main theorem is given as follows.
Theorem 2.1 (Main theorem).Let q > n + 2. Assume that Γ t is a hypersurface of class C 3 , (v 0 , ĉ0 ) ∈ D q such that the compatibility conditions (2. 19) and (2.20) hold, then there is a positive T 0 = T 0 ( (v 0 , ĉ0 ) Dq ) < ∞ such that there exists a unique strong solution (v, π, ĉ, ĉ * s , ĝ) ∈ Y T 0 to system (2.13)-(2.18).Moreover, ĉ ≥ 0 and ĉ * s , ĝ > 0, if ĉ0 ≥ 0. Remark 2.2.In this work, the boundary of domain is supposed to be C 3 .We remark here that if the boundary is not smooth enough, for example, C 0,1 Lipschitz domain, it will encounter the contact line problems with a contact angle.As far as we know, it is still an open problem.The authors considered the similar model with a ninety degree contact angle in [3] recently.
The proof of Theorem 2.1 relies on the Banach fixed-point theorem.To this end, we need to linearize the nonlinear system (2.13)-(2.18).Since we consider a nonzero initial reference configuration, a standard perturbation method is applied to (2.13)-(2.18),for which we rearrange the system at the initial deformation and move all perturbed terms to the right-hand side, namely, v| t=0 = v0 in Ω, (2.25) ) where S(v, π) = −πI + ν ∇v + ∇⊤ v in Ω and Hence, G possesses the form (2.36) Remark 2.4.In generic, the system (2.26)-(2.30)for concentrations of monocytes and macrophages can be considered as a transmission problem in Ω f and Ω s with a common boundary Γ.However, if we use the concentration and stress jump condition as boundary condition on Γ, we will meet the regularity problem due to the high order term D s ∇ĉ s • nΓ in (2.28) 2 .More precisely, in our further perturbation argument, all perturbated or unrelated terms will be removed to right-hand side of the equation and the regularities of both sides should coincide with each other.The point is that in such argument, the right-hand side of (2.28) 2 contains D s ∇ĉ s • nΓ , which leads to a lower regularity, provided the same regularity of ĉ on the both side.
Therefore, to avoid such awkward situation, we rewrite the transmission conditions as two Neumann type boundary conditions.Then the transmission problem can be decoupled into two separated parabolic system, which are both imposed with Neumann boundary and defined in Ω f and Ω s respectively.This is why we treat the boundary conditions on Γ as the form shown in (2.28). 5) with suitable regularities, existence and uniqueness of (v, π, ĉ, ĉ * s , ĝ) in associated spaces will be obtained by the well-poesdness of linear systems in the next section.

Analysis of the linear systems
As seen in (2.21)-(2.34), the linearized system can be seen as a two-phase type Stokes problem (2.21)-(2.25),two separated reaction-diffusion systems (2.26)-(2.30)and two ordinary differential equations (2.31)-(2.34)(equation for foam cells and growth, respectively).In this section, thanks to the maximal L q -regularity theory, we establish the existence for strong solutions to these systems with prescribed initial data and source terms in appropriate spaces.(2.22).Then we get the problem addressed in this subsection.

Two-phase Stokes problems with Neumann boundary condition
Now, we will prove the following theorem, namely, existence of unique solution to a two-phase Stokes problem with outer Neumann boundary condition.
Theorem 3.1.Let q > n + 2, T > 0, Ω a bounded domain with Γ s ∈ C 3 , Γ a closed hypersurface of class C 3 .Assume that (k, g, h 1 , h 2 ) are known functions contained in Z v T with initial value zero and v0 ∈ D 1 q with compatibility conditions Then the Stokes problem (3.1) admits a unique strong solution (v, π) in Y v T .Moreover, there exist a time T 0 > 0 and a constant C = C(T 0 ) > 0 such that for 0 < T ≤ T 0 , where endowed with norms + tr Γ (g) + tr Γs (g) To simplify the proof of Theorem 3.1, we reduce (3.1) to the case (k, g, v0 ) = 0. First of all, we define v as the solution of the parabolic transmission problem with k ∈ L q ( Ω × (0, T )) and v0 ∈ D 1 q .Since the Lopatinskii-Shapiro conditions are satisfied, (3.6) is uniquely solvable in W 2,1 q ( Ω × (0, T )), thanks to [37, Theorem 6.
Consequently, ∆φ ∈ Y 2 T .3.1.2.Proof of Theorem 3.1.As stated in the last section, we analyze the reduced system of (3.1) with (k, g, v0 ) = 0. Due to the outer Neumann boundary condition, the proof is proceeded by a truncation (localization) argument, based on the results given in Appendix A. More precisely, with suitable cutoff function, we decompose the system into a two-phase Stokes problem with Dirichlet boundary conditions and a one-phase nonstationary Stokes problem, which are uniquely solvable as in Section A.1 and Abels [2, Theorem 1.1] respectively.

Heat equations with Neumann boundary condition. From (2.26)-(2.30), we have two decoupled systems with given functions
According to the maximal L q -regularity results we introduced in Appendix A.2, we immediately have following theorem.
Theorem 3.2.Let q > n + 2, Ω a bounded domain with Γ s ∈ C 3 , Γ a closed hypersurface of class C 3 .Assume that (f 1 , f 2 , f 3 ) are known functions contained in Z c T and ĉ0 ∈ D 2 q with compatibility conditions Then the parabolic problems (3.18) and (3.19) admit unique strong solutions ĉf and ĉs in Y 3 T respectively.Moreover, there exist a constant C > 0 and a time T 0 > 0 such that for 0 < T < T 0 , where T with Z 5 T := L q (0, T ; L q ( Ω)),

3.3.
Ordinary differential equations for foam cells and growth.Given functions (f 4 , f 5 ) in Ω s , we have T := L q (0, T ; W 1 q (Ω s )).Moreover, there exists a constant C independent of T such that ĉ *

Local in time existence
This section is intended to prove Theorem 2.1.
Lemma 4.1 (Estimates on deformation gradient).Let q > n, n ≥ 2. F (v) is a deformation gradient defined in (1.1) corresponding to a function v ∈ Y 1 T .Then for every R > 0, there are a constant C = C(R) > 0 and a finite time 0 < T R < 1 depending on R such that for all 0 < T < T R , F −1 exists and where r = q 2 n .Proof.Recalling from (1.1) the definition of F that where M q is the constant of multiplication of W 1 q ( Ω), see Lemma 2.1.According to the Neumann series (see [5,Section 5.7]), F −1 does exist and Consequently, if follows from (2.2) and Lemma 2.1 that where C = C(R) depends on R.These estimates prove first two statements.
For the third and fourth statements, we have which can be used to deduce and therefore from (2.4) and the definition of Sobolev-Slobodeckij space, For the rest statements, we notice from (1.1) that Then for all 0 < T < T R , it follows from the multiplication property of W 1 q ( Ω) again that for all 0 < T < T R , Moreover, Hence From the embedding (2.6), we know that for v ∈ Y 1 T , The Gagliardo-Nirenberg inequality tells us ∇v For r = q 2 n > q, we obtain Then, and also, for û Consequently, with W 1 q ( Ω) ֒→ L ∞ ( Ω) for q > n, Similarly, Thus, Moreover, we can also conclude from (4.1) that Therefore, for all 0 < T < T R , Again with the help of (2.4) and the definition of Sobolev-Slobodeckij space, one obtains the last statement.This completes the proof.
Lemma 4.2.Under the assumption of Lemma 4.1, there exist a constant C = C(R) > 0 and a finite time T R > 0 depending on R such that for all 0 < T < T R and for two arbitrary functions f (X, t) ∈ L q (0, T ; W 1 q ( Ω)) and f ∈ L q (0, T ; Proof.The key point to deduce these estimates is to use the multiplication property of W 1 q ( Ω) with q > n, which was given in Lemma 2.1.Then Lemma 4.1 implies these results.
Proof.The lemma can be easily proved by using the argument in [1, Lemma 4.2], where a layer-like domain with C 1,1 boundary is considered.Besides, it can be seen as a corollary of Lemma 2.2.

4.2.
Proof of Theorem 2.1.In this subsection, we prove Theorem 2.1 by applying the strategy of a fixed-point procedure.
Proposition 4.1.Let L be defined as in Proof.As L ∈ L(Y T , Z T × D q ), it suffices to show that L is bijective, thanks to the bounded inverse theorem.To employ the contraction mapping principle to (4.3), we then investigate the dependence and contraction of (K, G, , π, ĉ, ĉ * s , ĝ).To this end, we define where the elements are given by (2.35).Then it is still needed to show that M (w) : Y T → Z T is well-defined in terms of (v, π, ĉ, ĉ * s , ĝ) ∈ Y T and to verify M (w) possesses the contraction property.Proposition 4.2.Let q > n and R > 0. Assume w = (v, π, ĉ, ĉ * s , ĝ) ∈ Y T with ĝ| t=0 = 1 and w Y T ≤ R, then there exist a constant C = C(R) > 0, a finite time T R > 0 depending on R and δ > 0 such that for 0 < T < T R , M (w) : Y T → Z T is well-defined and bounded along with the estimates: Moreover, for Proof.Firstly, we prove the second part.To this end, for w i Y T ≤ R, i = 1, 2 we estimate the following terms respectively 35), with the help of Lemma 2.1, 4.1 and 4.2, we derive that Kf (w 1 ) − Kf (w 2 ) L q (0,T;W 1 Let ĝ ∈ W 1 q (0, T ; W 1 q (Ω s )) with ĝ| t=0 = 1.Now we claim that there exists a time T R > 0 such that for 0 < T < T R , ĝ ≥ 1 2 > 0. Let ĝ be such function with ĝ W 1 q (0,T ;W 1 q (Ωs)) ≤ R for some R > 0. Then for 0 < t < T , where we choose T R > 0 small enough such that For K s = div Ks + Kg s , the first part can be estimated similarly that Ks (w 1 ) − Ks (w 2 ) L q (0,T;W 1 q (Ωs)) The second part follows from (2.4), Lemma 4.1 and 4.2 that Kg s (w 1 ) − Kg s (w 2 ) L q (0,T ;L q (Ωs) L q (0,T ;L q (Ωs)) n×n =: From the definition of σs and ĝ ≥ 1/2, where Then we get . From the definition of Z 2 T given by (3.4), we need to verify that G(w 1 ) − G(w 2 ) is contained both in L q (0, T ; W 1 q ( Ω)) and W 1 q (0, T ; W −1 q ( Ω)), as well as the trace regularity The first regularity follows easily from (2.35), Lemma 4.1 and 4.2 that G(w 1 ) − G(w 2 ) L q (0,T;W 1 q ( Ω)) From approximation argument in [4, Page 15], we know that weak derivative with respect to time does exist for G. Hence, substituting G by the form (2.36), using integration by parts, we have for every φ ∈ W 1 q ′ ,0 ( Ω), where •, • X×X ′ denotes the duality product between a pair of dual space X and X ′ .Then according to (2.2), the Sobolev embedding W 1 q ( Ω) ֒→ C 0,1−n/q ( Ω) ֒→ L ∞ ( Ω) and Lemma 4.1, one obtains L q (0,T ;L q (Ωs)) L q (0,T ;L q (Ωs)) s L q (0,T ;L q (Ωs)) Then we are in the position to prove tr Γ (G(w 1 ) − G(w 2 )) ∈ W 1−1/q,(1−1/q)/2 q (Γ × (0, T )).Recalling the definition of such mixed space (2.5), we first write the explicit norm.
Proof of Theorem 2.1.Since L : Y T → Z T × D q is an isomorphism as showed in Proposition 4.1, from the estimates in Theorem 3.1, we set a well-defined constant For w i Y T ≤ R, i = 1, 2, we take T R > 0 small enough such that where C(R) is the constant in (4.5).Then for 0 < T < T R , we infer from Theorem 4.2 that which implies the contraction property.From (4.7) and (4.8), we have a closed subset of Y T .Hence, L −1 N : M R,T → M R,T is well-defined for all 0 < T < T R and a strict contraction.Since Y T is a Banach space, the Banach fixedpoint Theorem implies the existence of a unique fixed-point of L −1 N in M R,T , i.e., (2.21)-(2.34)admits a unique strong solution in M R,T for small time 0 < T < T R .
In the following, we prove the uniqueness of solutions in Y T by the continuity argument.Let w 1 , w 2 ∈ Y T be two different solutions of (2.21)-(2.34)and . Now we argue by contradiction.Define T as With the regularity of v, c and the embedding theorem, we know that f ∈ C α,α/2 loc (U T ) for some 0 < α < 1.By the local regularity theory for parabolic equations, one obtains ).The continuous of c can be derived directly from the Lemma 2.3, especially (2.6) with Now, given a nonnegative initial value c 0 (x) ≥ 0, x ∈ Ω 0 .Define c λ := e −λt c where λ > 0 is a constant which will be assigned later.Adding cc λ to the both sides of (4.9), we have the equation for c λ Taking λ sufficiently large such that By the weak maximum principle for parabolic equations, we have min where c − (x, t) := − min{c(x, t), 0}.Since c 0 (x) ≥ 0, now we claim that c(x, t) ≥ 0 for all (x, t) ∈ (Γ t ∪ Γ t s ) × [0, T ].To this end, we argue by contradiction.Assume that for some t 0 ∈ (0, T ], there exist a point x 0 ∈ Γ t 0 ∪ Γ t 0 s , such that c(x 0 , t 0 ) = − max which implies that x → min{c(x, t 0 ), 0} attains a negative minimum at x 0 , i.e., x → c(x, t 0 ) attains a negative minimum at x 0 .Case 1: x 0 ∈ Γ t 0 .For both Ω t 0 f and Ω t 0 s , since Γ t 0 is assumed to be a C 3− interface, we infer from Hopf's Lemma that Hence, D∇c • n Γ t 0 (x 0 ) < 0, which contradicts to (1.13).
In summary, c(x, t) ≥ 0 for all (x, t) Then there exists a unique solution u ∈ W 2,1 q (Ω × (0, T )) of (A.6).Moreover, , where C does not depend on T ∈ (0, T 0 ] for any fixed 0 < T 0 < ∞. Proof.This proposition can be easily shown by means of Prüss and Simonett [37,Theorem 6.3.2], for which we need to extend the right-hand sides just as in the proof of Proposition A.1 and construct a solution solving (6.45) in [37].This can be done since we established general extension theorems in Appendix B. (A.7) Here, we denote the inner domain by Ω − , resp.outer domain by Ω + and the unit normal vector on Σ = ∂Ω − by ν.
The second result concerns the strong solutions.
Proof.The proof is divided into three cases, namely, s = 0, 1 q < s < 1 and s = 1.Case 1: s = 0.In this situation, W s q (0, T ; X) is just the Lebesgue space L q (0, T ; X), which does not contain any time regularity.Hence for any function u ∈ L q (0, T ; X), we can take the zero extension.Case 2: s = 1.With u| t=0 = 0, we apply an even extension to u in [0, T ] around T to [0, 2T ] and zero extension for T > 2T such that the extended function ū is weakly differentiable with Then we have ū W 1 q (0,∞;X) = 2 1 q u W 1 q (0,T ;X) .
Case 3: 1 q < s < 1.With the same extension as in Case 2, we define the same function ũ.Now we are in the position to show ũ ∈ W s q (0, ∞; X), for which we only need to prove [ũ] W s q (0,∞;X) ≤ C [u] W s q (0,T ;X) , where C is independent of T .From the definition of Sobolev-Slobodeckij space, [ũ] q W s q (0,∞;X)  It is clear that W s q (0,T ;X) .|t − h| 1+sq dhdt = 2 [u] q W s q (0,T ;X) .
Next, we give an extension theorem for general functions.

Figure 1 .
Figure 1.Domain of the problem.

Figure 2 .
Figure 2. Deformation ϕ mapping from Ω into Ω.Subsequently, we denote by F the deformation gradient F = ∂ ∂X ϕ(X, t) = ∇ϕ(X, t) = I + t 5.1].Now, we are in the position to reduce g.To this end, we introduce a elliptic transmission problem with Dirichlet boundary ∆φ = g − div v =
T as the initial value for (2.21)-(2.34).Repeating the argument above, we see that there is a timeT ∈ ( T , T ) such that w 1 | [ T , T ] = w 2 | [ T , T ], which contradicts to the definition of T .In conclusion, (2.21)-(2.34)admits a unique solution in Y T .For the nonnegativity of ĉ, we show it in Eulerian coordinate.Let U T