An inviscid free boundary fluid-wave model

We consider the local existence and uniqueness of solutions for a system consisting of an inviscid fluid with a free boundary, modeled by the Euler equations, in a domain enclosed by an elastic boundary, which evolves according to the wave equation. We derive a priori estimates for the local existence of solutions and also conclude the uniqueness. Both, existence and uniqueness are obtained under the assumption that the Euler data belongs to Hr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{r}$$\end{document}, where r>2.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r>2.5$$\end{document}, which is known to be the borderline exponent for the Euler equations. Unlike the setting of the Euler equations with vacuum, the membrane is shown to stabilize the system in the sense that the Rayleigh–Taylor condition does not need to be assumed.


Introduction
In this paper, we derive a priori estimates for solutions to an inviscid flow-structure interaction model. The system consists of the incompressible Euler equations defined in a channel with a free boundary at the top of the channel. The free boundary is elastic and deforms according to a scalar second-order linear equation satisfied by the transversal displacement. For simplicity, we consider the wave equation as an approximation for the dynamics of elasticity.
We consider the problem with periodic boundary conditions in the horizontal directions, and we reformulate the problem on a fixed domain using the Arbitrary Lagrangian Eulerian (ALE) variable, which is a change of coordinate involving the harmonic extension of the interface graph function. The interaction between the flow and the structure is captured mathematically through the pressure acting as a forcing term in the wave equation, while the normal velocity of the flow is matched with the normal component of the velocity of the interface through the kinematic condition.
The a priori estimates combine the energy estimate for the elastic displacement with the vorticity and pressure estimates. A characteristic feature of the system is that no stability condition is required for the local-in-time existence of solutions. The Rayleigh-Taylor stability condition is well-known to be necessary for problems involving the free boundary Euler equation with constant pressure at the free interface. However, in the setting we consider, the elasticity of the interface acts as a stabilizing and a regularizing agent, even though the wave equation does not incorporate any smoothing effects. Compared to the case of the plate equation considered in [21], the system considered here features an additional stabilization mechanism. Namely, the pressure term is more regular, and separate tangential estimates on the Euler equation that exploit the cancellation of the boundary term with energy estimates on the structural variable are unnecessary. Instead, the pressure forcing term in the wave equation can be estimated directly using the regularity property of the Robin-to-Dirichlet map, with the pressure term as the solution to a boundary value elliptic problem with Robin boundary data. The normal component of the fluid velocity can then be controlled directly using energy estimates on the wave equation. The full regularity of the fluid velocity can then be recovered using div-curl type estimates. We emphasize that the regularity of solutions considered in this paper are at the sharp level (H 2.5+δ where δ > 0 is arbitrarily small) expected for the Euler equation, leading to diffeomorphic flow maps of the changing domain.
Physically speaking, it is notable that second-order hyperbolic dynamics used to model the interface are also used to model the torsion or twist motions of an airfoil subject to airflow, though the model involved is slightly different since it involves the torsion angle as a variable instead of elastic deformation. Moreover, the incompressibility condition imposed on the flow is appropriate for flows at low Mach numbers. In our model, we choose the simple configuration where the flow happens along a horizontal channel with periodic boundary conditions, with the moving elastic interface at the top and the fixed wall at the bottom. However, this type of simplification is not essential and can be extended to the case of a general domain using flattening of the boundary and a partition of unity ( [14]).
The well-posedness of free boundary models involving viscous fluid flow interacting with a lower-dimensional structure has also been studied extensively. The first instance of such treatments is due to Desjardins et al. [7,13] who obtained weak solutions to a system involving the incompressible Navier-Stokes equations coupled with a lower-dimensional interface modeled by a strongly damped linear plate equation.
Models involving second-order equations for the lower-dimensional interface have been studied by Lequeurre [25,26]. More recent works on the interaction between viscous flow and linear plate structural dynamics with and without damping have been considered in [3,16,17]. Other intricate systems involving the interaction between viscous flow and a linear or nonlinear Koiter shell have also been studied in [23,24,[28][29][30][31][35][36][37].
Our main result in this paper is the derivation of a priori bounds for the existence of local-in-time smooth solutions to the system, as well as uniqueness of solutions. We note that solutions to the system can be constructed using the scheme introduced in [21], whereby we solve the Euler system with variable coefficients and non-homogeneous boundary and divergence data. In [21], the inviscid flow interacts with a fourth-order plate equation, while here, the elastic body evolves according to a second-order wave equation. We present the a priori estimates and the uniqueness; the construction is omitted as it can be obtained by following [21].
The paper is organized as follows. In Sect. 2, we introduce the model and state the main results, Theorems 2.1 on existence and 2.3 on uniqueness. The a priori bounds for the existence are provided in Sect. 3. They couple the energy estimates for the wave equation with the tangential estimates for the Euler equations. An important feature of the estimates is that they are performed at a lower regularity level for the velocity. The velocity is then reconstructed using the pressure bounds and the estimates for the ALE vorticity. Finally, the proof of uniqueness is provided at the end of the section.

The model and the main results
Our goal is to address a flow-structure problem in which an incompressible fluid, modeled by the Euler equations in an open bounded domain interacts with a membrane on a top boundary, which evolves according to the wave equation For simplicity (the general situation can be addressed using straightening of the boundary and a partition of unity, as in [14]), we assume the periodic boundary conditions on the side, i.e., the initial domain is represented by while the upper boundary of the domain (t) evolves according to The initial condition for the displacement w reads noting that the general nonzero w(0) can be considered using identical approach. On the bottom of the domain, we impose the impermeability condition for the velocity where N denotes the outward unit normal. The same analysis and theorems also apply if we have a second-order nonlinear elastic equation (2.3) replaced by the equation of the form where σ (w) is a sufficiently smooth nonlinear tensor satisfying the coercivity condition

ALE change of variable
It is essential to switch to the ALE variables as follows. Let ψ : → R be the harmonic extension of 1 + w to the domain = (0), which means that ψ is a solution of With ψ as above, we define the ALE variable η : The inverse of the derivative of ∇η is computed as is the Jacobian. Note that we have the Piola identity where, unless indicated otherwise (usually with a summation symbol), we use the summation convention on repeated indices. With denoting the ALE velocity and pressure, the Euler equations (2.1) take the form in , with the initial condition v| t=0 = v 0 (2.10) and the boundary condition v 3 = 0 on 0 . (2.11) Note that, using the form of the third column of (2.7), the first equation in (2.9) may also be expressed as The kinematic condition (2.4) takes the form with the pressure normalized by The next statement on a priori bound for the local existence is our main result.

16)
where M 0 ≥ 1 and r > 2.5. Then v, w, and q satisfy where K is a constant depending on M 0 and T ∈ (0, T 0 ] is a sufficiently small time depending on M 0 . Here and in the sequel C ≥ 2 denotes a sufficiently large generic constant. Also, if the domain of the Sobolev space is not specified, as in (2.17), it is understood to be .

Remark 2.2.
Here we point out differences in the membrane case treated here compared to the plate setting from [21]. While in both cases we achieve the sharp level of regularity for the Euler equations (H 2.5+δ , where δ > 0 is arbitrary), the regularity for the membrane displacement w and its velocity w t is lower here. However, the different form of the critical pressure quantity (3.16) below, which drives the energy for the wave part, is directly controllable. This allows for a simplification of the approach since the tangential estimates for the velocity are not needed. In addition, the fact that the pressure term is controllable allows for a simpler construction; in particular, the smoothing term in the wave equation is no longer needed. Due to this and the new treatment of the pressure, the uniqueness and construction also hold here for the data in H 2.5+δ . In addition to the simplification induced by the higher regularity of the pressure energy term, we also simplify the vorticity estimate, which no longer depends on the Sobolev extension of the Jacobian.
Throughout the paper, we keep r > 2.5 fixed, allowing all the constants to depend on this parameter without mention. Next, we assert the uniqueness of solutions.
Both theorems are proven in the next section. Finally, we state the existence theorem for solutions in the class asserted in the a priori estimates.  solution (v, q, w, w t ) to the Euler-plate system (2.9)-(2.15) with the initial data The proof of this existence theorem follows the parallel statement in [21]. Note however that the regularity properties of v, q, and w in Theorem 2.1 allow us to conclude the boundedness of the critical term t 0 1 q 2(r −0.5) w t in (3.15) below, which may be rewritten as t 0 1 r −0.5 q r −0.5 w t , and thus the regularization step with the dissipative term involving ν in [21] is not necessary, leading to a great simplification of the construction. To avoid repetition, we omit further details.

Proof of the a priori bounds
Here, we provide a priori estimates for the system, thus proving Theorem 2.1. With C 0 and K to be determined, we have (2.17) on some interval [0, T ], and we intend to provide a lower bound on T .
We start with the bounds on the inverse matrix a and the Jacobian J .

1)
where M ≥ 1 is as in the statement of Theorem 2.1. Then we have

3)
and C is a sufficiently large constant depending on C 0 .
First, note that, using the definitions of ψ and η, we have Also, we have and For the matrix a, one may easily check that and provided J ≥ 1/2. As usual, A B stands for A ≤ C B for a sufficiently large constant C ≥ 1.  .3) and C is sufficiently large. In particular, if > 0 is sufficiently small, we have J ≥ 1/2 for t ∈ [0, T 0 ] with T 0 as in (3.3). Differentiating a∇η = I in time, we get a t = −a∇η t a, and then, using also a(0) = I , it follows that With T 0 as in (3.3), we obtain the bound for the first two norms in (3.2).
In the sequel, we do not distinguish between T 0 and T ; thus we assume that (3.1) is satisfied and that where = 1/C and C ≥ 2 is sufficiently large. Note that, in particular, T ≤ 1.
We are now ready to prove the main result.
Proof of Theorem 2.1. Assume that [0, T ], where T > 0 is as in (3.10), is an interval such that (v, q, w) is a C ∞ solution with the initial data v 0 satisfying (2.17)-(2.18). We intend to prove then that T can be chosen so it depends only on M 0 .
We start with the pressure estimates. The interior elliptic equation for the pressure is obtained by applying J a ji ∂ j to the first equation in (2.9). After some rewriting, as in [21,Remark 3.5], we obtain On the other hand, to obtain the boundary conditions on 0 and 1 , we test the Euler equations with a 3i and evaluate. On the bottom boundary, we have while for the top boundary, after some work, as in [21,Remark 3.5], we obtain a Robin-type boundary condition Using (3.8), and the algebra property of H r −1 , and elliptic regularity, we get on the interval [0, T ]; here and in the sequel, the symbol P denotes a generic nonnegative polynomial of its arguments. The control of the displacement is obtained by using a tangential estimate. With Using w(0) = 0, we obtain from the wave equation (2.14) Estimating the last term on the right-hand side as with a help of (3.14) in the last step. We obtain and since for t ∈ [0, T ]; when not specified, the time argument is understood to be t.
With the pressure and tangential estimates in (3.14) and (3.17) established, we use the ALE vorticity where ω = curlu, to get control of the velocity v. In the ALE variables, the vorticity reads ζ i = i jk a m j ∂ m v k and thus satisfies At this point, we use the Sobolev extension operator f →f , which is a continuous where k is not necessarily an integer. We require the extension to be such that suppf vanishes in a neighborhood of in 0 = T 2 × R, with the initial condition θ(0) =ζ (0). (Note that (3.19) is a simpler equation than the one considered in [21].) By the properties of the extension operator and since the equation for θ is of transport type, we have We now introduce the quantity Note that, by the continuity properties of the Sobolev extensions, we have To prove (3.21), we differentiate (3.20) and use (3.19) to obtain where I represents the first two terms, which are the ones with an additional derivative on θ , whileĪ stands for the rest. For I , we integrate by parts in x j , obtaining Estimating the expression for I in (3.22) and bounding all the terms inĪ directly, we obtain and thus (3.21) is established.
To relate the quantity X to the actually vorticity, we claim that for i = 1, 2, 3. This leads to the energy equality where the boundary terms vanish since the equation (3.25) can be readily checked using (2.7), (2.11), and (2.13). From (3.24) and σ (0) = 0 in , we obtain ζ = θ in , and thus (3.23) holds. Finally, we summarize the obtained estimates. To use the div-curl estimate from [5], we bound curlv in divv and v · N in appropriate Sobolev spaces. For curlv, we have using also (3.2). Therefore, with the help of (3.21) and (3.23), we have For the divergence, we simply use the divergence-free condition to write while for the bottom boundary term, we have Also, by (2.13), we get the difference. In this section, we allow all the implicit constants to depend on the norms of (v, q, w) and (ṽ,q,w) in (2.17)- (2.18).
To obtain an estimate on the difference of the pressures, we subtract (3.11)-(3.13) and the analogous equations forq. Applying the elliptic regularity estimate to the difference, we obtain Next, from the equation satisfied by W = w −w, we have the energy equality since W (0) = W t (0) = 0. We estimate the last term on the right-hand side as in (3.16) and then bound the norm of the pressure Q in H r −1 using (3.33) obtaining r −0.5 W 2 L 2 ( 1 ) + r −1.5 W t 2 L 2 ( 1 ) t 0 ( V H r −1 + W H r −0.5 ( 1 ) + W t H r −1.5 ( 1 ) ) 2 ds.
For the vorticity, we need to estimate the difference Z = ζ −ζ . For this purpose, we extend ζ andζ to θ andθ so they are defined on 0 = T 2 × R. For simplicity, we do not distinguish in notation between the functions defined on and their extensions defined on 0 . With this agreement, we get from (3.18) that the equation for = θ −θ reads where + k a mk ∂ mṽi +θ k A mk ∂ mṽi , i = 1, 2, 3.
Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/ by/4.0/.
Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.