Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}-theory for a fluid–structure interaction model

We consider a fluid–structure interaction model for an incompressible fluid where the elastic response of the free boundary is given by a damped Kirchhoff plate model. Utilizing the Newton polygon approach, we first prove maximal regularity in Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}-Sobolev spaces for a linearized version. Based on this, we show existence and uniqueness of the strong solution of the nonlinear system for small data.

The unknowns in the model are the velocity u, the pressure q, and the interface Γ. We denote by ν the exterior unit normal field at Γ, by V Γ the velocity of the boundary Γ, and by e j the j-th standard basis vector in R n , i.e., e n = (0, . . . , 0, 1). The function φ Γ describes the elastic response at Γ which is given by a damped Kirchhoff-type plate model. Throughout the paper, we assume that Γ is given as a graph of a function η : R + × R n−1 → R, that is Γ(t) = (x , η(t, x )); x ∈ R n−1 , t ≥ 0, (1.2) and that Γ(t) is sufficiently flat. Thus, Ω(t) is a perturbed upper half-plane. In these coordinates, the elastic response is given as for α, γ > 0, β ∈ R, where Δ stands for the Laplacian in R n−1 . Finally, the initial configuration and velocity of the interface resp. the initial fluid velocity are given by Γ 0 and V 0 resp. u 0 = (u 0 , u n 0 ). Note that in addition to the initial position Γ 0 of the boundary, also its initial velocity V 0 has to be specified as the equation is of second order with respect to time on the boundary. We remark that in the formulation of the boundary conditions in lines 3 and 4 of (1.1), one has to take into account that the Kirchhoff plate model is formulated in a Lagrangian setting, whereas for the fluid an Eulerian setting is used. This is discussed in more detail in the beginning of Sect. 2.
The symbol of m(∂ t , ∂ ) is given as m(λ, ξ ) = λ 2 + α|ξ | 4 + β|ξ | 2 + γλ|ξ | 2 , λ ∈ C, ξ ∈ R n−1 , For γ > 0, the roots of m(·, ξ ) lie in some sector which is a subset of {λ ∈ C : Reλ < 0}. This indicates that the term −γ∂ t Δ η in φ Γ parabolizes the problem. Physically, one also speaks of structural damping of the plate. We notice that basically the same results as proved in this note can be expected by considering layer like domains or rectangular type domains with periodic lateral boundary conditions. For simplicity, however, we restrict the approach given here to the just introduced geometry.
Model (1.1) was introduced in [23] in connection to applications to cardiovascular systems. In the 2D case, this system was investigated in [3] in the L 2 -setting. In fact, in [3,Proposition 3.12] it is proved that the linear operator associated with (1.1) generates an analytic C 0 -semigroup in a suitable Hilbert space setting. This exhibits the parabolic character of the problem. Therefore, it is reasonable to consider an L p -theory for the system (1.1) which is the main purpose of this note. Alternative approaches to system (1.1) in the L 2 -setting also for the hyperbolic-parabolic case, i.e., γ = 0, are given, e.g., in [6,10,16,17,21], concerning weak solutions and, e.g., in [4,7,18,19] concerning (local) strong solutions. A more recent approach in a two-dimensional L 2 -framework concerning global strong solutions is presented in [11]. Recently, in [20] the interaction between an incompressible fluid and a damped beam (which relates to the case of a one-dimensional boundary) was studied in the L p -L q -setting.
In the present paper, we develop an L p -approach in general dimension for system (1.1). In order to formulate the main result, for k, ∈ N 0 non-cylindrical spaces are defined as The space L p (J;Ḣ 1 p (Ω(t))) for the pressure is defined accordingly. We show the existence of strong solutions for small data and give a precise description of the maximal regularity spaces for the unknowns. More precisely, we prove the following main result for (1.1). Theorem 1.1. Let n ≥ 2, p ≥ (n + 2)/3, T > 0, and J = (0, T ). Assume that for some κ > 0, where Γ 0 = graph(η 0 ) and V 0 = {(0, η 1 (x )); x ∈ R n−1 } in (1.1). Then, there exists a unique solution (u, q, Γ) of system (1.1) such that Γ = graph(η) and such that provided that κ = κ(T ) is small enough and that the following compatibility conditions are satisfied: The solution depends continuously on the data.
The compatibility conditions (1)-(3) are natural in the sense that they are also necessary for the existence of a strong solution. Condition (3) appears in a similar way for the two-phase Stokes problem, see, e.g., [22], Section 8.1. Note that the regularity for η * in (3) does not follow from η * ∈ E η . (b) We remark that the maximal regularity space E η for η describing the boundary is not a standard space. It is given as an intersection of three Sobolev spaces. This is due to the fact that the symbol of the complete system has an inherent inhomogeneous structure, and therefore the Newton polygon method is the correct tool to show maximal regularity. For the details, see Sect. 3. c) We note that in the physically relevant situations n = 2 and n = 3, the case p = 2 is included. This might be of importance when considering the singular limit γ → 0 for vanishing damping of the plate. d) We formulated the result in the form of existence for fixed time and small data. By similar methods, one can also show short time existence for arbitrarily large data. This is more intricate, since then while estimating nonlinearities one has to carefully track the dependence of the constants on related smallness parameters. But we think that the known strategies, as elaborated, e.g., in [22], can be adapted.
The proof of Theorem 1.1 is based on several ingredients: First, we transform the system to a fixed domain and consider the linearization of the transformed system. By an application of the Newton polygon approach (see, e.g., [8] and [9]), we obtain maximal regularity for the linearized system. To deal with the nonlinearities, we employ embedding results on anisotropic Sobolev spaces given in [15].
The half-space model problem considered here can also be regarded as a first step towards an analysis on domains of more general geometry. By applying a suitable localization procedure, similar results are expected to hold, e.g., on bounded domains. On bounded domains, even global solvability for small data might be available.
b) An L p -L q -theory with p = q might be available as well. For the linear theory, in particular concerning the Newton polygon approach, the use of [9] then has to be replaced by the generalized approach developed in [8], see the proof of Lemma 3.2. Concerning the nonlinear system, so far there is no L p -L q analogon of the results on multiplication in [15] available in the existing literature. For this purpose, the corresponding estimates of the nonlinearities then had to be derived by more direct methods.

The transformed system
We start with a short discussion of the boundary conditions, where the Eulerian approach for the fluid has to be coupled with the Lagrangian description for the plate (see also [17] and [10]). Let Γ be given as in (1.2) and assume that η is sufficiently smooth. Following the Kirchhoff plate model, in-plate motions are ignored, and the velocity of the plate at the point (x , η(t, x )) τ is parallel to the vertical direction and given by (0, ∂ t η(t, x )) τ = ∂ t η(t, x )e n . As the fluid is assumed to adhere to the plate, we have no-slip boundary conditions for the fluid, and the equality of the velocities yields the first boundary condition The exterior normal at the point (x , η(t, x )) of the boundary Γ(t) is given by We define the transform of variables As it was discussed in [17], Section 1.2, the force F exerted by the fluid on the boundary is given by the evaluation of the stress tensor at the deformed boundary in the direction of the inner normal −ν(t, x ). More precisely, we obtain ( [17], Eq. (1.4)) x ) · e n , the equality of the forces gives the second boundary condition Conditions (2.1) and (2.2) are the precise formulation of the boundary conditions in (1.1).
To solve the problem (1.1), we first note that by a re-scaling argument we may assume that ρ = μ = 1 for the density ρ and viscosity μ from now on. Next, we transform the problem (1.1) to a problem on the fixed half-space R n + , using the above transformation θ. To this end, we set J := (0, T ) and write With the corresponding meaning, we write v , ∇ , etc. The pull-back is then defined as and correspondingly the push-forward as We Applying the transform of variables to (1.1) leads to the following quasilinear system for (v, p, η): The nonlinear right-hand sides are given as

The linearized system
The aim of this section is to derive maximal regularity for the linearized system In the sequel for k ∈ N 0 , 1 < p < ∞, a domain Ω ⊂ R n , and a Banach space X, and 0 W s p (J, X) accordingly. Observe that then we have 0 H 1 p (J, X) = {u ∈ H 1 p (J, X); u(0) = 0}. As references for vector-valued scales of Sobolev spaces, we mention [2], Chapter VII, and [13], Chapter 2. We will consider system (3.1) in spaces with exponential weight with respect to the time variable. Let ρ ∈ R and X be a Banach space. For u ∈ L p (R + , X), we define Ψ ρ as the multiplication operator with e −ρt , i.e., Ψ ρ u(t) := e −ρt u(t), t ∈ R + . The spaces with exponential weights are defined by R. Denk and J. Saal ZAMP 0 W s p , respectively. For mapping properties and interpolation results under the condition that X is a UMD space, we refer, e.g., to [9], Lemma 2.2. We also make use of homogeneous spaces, e.g., for Ω ⊂ R n we setḢ The corresponding dual spaces are defined aṡ see [22] Section 7.2. The homogeneous Sobolev-Slobodeckii spacesẆ s p (R n ) contain all functions u : R n → R such that where [s] = max{k ∈ N 0 ; k < s}, see [26]. Note that we havė W s p (R n ) =Ḃ s pp (R n ) for 1 < p < ∞, n ∈ N, and s ∈ R\Z, where the latter one denotes the homogeneous Besov space.
We refer to the pertinent monographs [1,5,24] for the scalar case and [2,13] for the X-valued case for properties, characterizations, and relations of the just introduced spaces.
In the following, we denote the time trace u → ∂ k t u| t=0 by γ t k and the trace to the boundary u → ∂ k n u| R n−1 by γ k . We set J = (0, T ) for T > 0. The solution (v, p, η) of (3.1) will belong to the spaces The function spaces for the right-hand side of (3.1) are given by By trace results with respect to the time trace, the spaces for the initial values are given by see the proof of Theorem 3.1 below (necessity part). Note also that in this section we have T = ∞ and that we skipped indicating the ρ dependence in E v , E p , etc., since we only deal with weighted time-dependent spaces for the rest of this section. We will also need the following compatibility conditions: There exists an η * ∈ E η with η * | t=0 = η 0 , ∂ t η * | t=0 = η 1 and (g, ∂ t η * ) ∈ H 1 p,ρ (J;Ḣ −1 p,0 (R n + )). Here, we define . Additionally, we have (g| t=0 , η 1 ) = (g| t=0 , v n 0 | R n−1 ) inḢ −1 p,0 (R n + ). We remark that only (3.2) is an additional condition, as it was shown in [9], Theorem 4.5, that for every η 0 ∈ γ t 0 E η and η 1 ∈ γ t 1 E η there exists an η * ∈ E η with η * | t=0 = η 0 and ∂ t η * | t=0 = η 1 . The main result of this section is the following maximal regularity result. The proof of this theorem will be done in several steps and follows from Sects. 3.1-3.4.

Necessity
be a solution of (3.1). By standard continuity and trace results, the righthand sides f v , and g as well as the time trace v 0 belong to the spaces above. Noting that div : , and as for all , we obtain the compatibility condition (C1) for all p > 1 (see also [22], Theorem 7.2.1).

Reductions
We can reduce some part of the right-hand side of (3.1) to zero by applying known results on the Stokes system. For this, let (v (1) , p (1) ) ∈ E v × E p be the unique solution of the Stokes problem in the half space The unique solvability of (3.3) follows from [22], Theorem 7.2.1. To show that this theorem can be applied, we remark in particular that the compatibility condition (e) in [22, p. 324] holds because of (C4). Moreover, by the embedding E η ⊂ H ) and the compatibility condition (C3), we see that also the compatibility condition (d0) in [22, p. 324] holds.

Solution operators for the reduced linearized problem
In the following, we show solvability for the reduced problem (3.4), omitting the tilde again. An application of the Laplace transform formally leads to the resolvent problem We observe that the second and the third line of (3.5) imply that Applying partial Fourier transform in x ∈ R n−1 , we obtain the following system of ordinary differential equations in x n for the transformed functionsv,p andη: Here, we have set ω := ω(λ, ξ ) := λ + |ξ | 2 and Multiplying the first equation with (iξ , ∂ n ) and combining it with the second one yields (−|ξ | 2 + ∂ 2 n )p = 0 for x n > 0. The only stable solution of this equation is given bŷ Putting the pressure term on the right-hand side, v formally solves a vector-valued heat equation. Hence, to solve the above system we employ the ansatẑ (3.8) with the Green functions subject to Dirichlet resp. Neumann conditions k ± (λ, ξ, x n , s) := 1 2ω e −ω|xn−s| ± e −ω(xn+s) .
Formula (3.14) defines the solution operator for η as a function of f η on the level of its Fourier-Laplace transform. The following result is based on the Newton polygon approach and shows that the solution operator is continuous on the related Sobolev spaces. In the following, we consider (−Δ ) 1/2 as an unbounded operator in L p ρ (R + ; L p (R n−1 ) and define N L (∂ t , (−Δ ) 1/2 ) by the joint H ∞ -calculus of ∂ t and (−Δ ) 1/2 (for details, we refer to, e.g., [9], Corollary 2.9). We will apply the Newton polygon approach on the Bessel potential scale H s p with respect to time and on the Besov scale B r pp with respect to space.
Due to the last result, we obtain the existence of a solution (v, p, η) of (3.4). In fact, for η, φ n , and p 0 defined as in Lemma 3.2(b), we can define p and v by (the Laplace and Fourier inverse transform of) (3.6) and (3.7)-(3.8), respectively. Here, φ is given by (3.11). As we know that φ n and p 0 belong to the canonical spaces by Lemma 3.2(b), we get v ∈ E v and p ∈ E p by standard results on the Stokes equation (see, e.g., [12], Section 2.6, and [22], Section 7.2). By construction, (v, p, η) is a solution of (3.4).

Uniqueness of the solution
To show that the solution of (3.1) is unique, let (v, p, η) be a solution with zero right-hand side and zero initial data. Then, the Laplace transform in t and partial Fourier transform in x is well-defined, and the calculations above show, in particular, that for almost all ξ ∈ R n−1 . Therefore, η = 0 which implies that (v, p) is the solution of the Dirichlet Stokes system with zero data. Therefore, v = 0 and p = 0. This finishes the proof of Theorem 3.1.