An L 2 approach to viscous flow in the half space with free elastic surface

We consider a linearized fluid-structure interaction problem, namely the flow of an incompressible viscous fluid in the half space ℝ n + such that on the lower boundary a function h satisfying an undamped Kirchhoff-type plate equation is coupled to the flow field. Originally, h describes the height of an underlying nonlinear free surface problem. Since the plate equation contains no damping term, this article uses L 2 -theory yielding the existence of strong solutions on finite time intervals in the framework of homogeneous Bessel potential spaces. The proof is based on L 2 -Fou-rier analysis and also deals with inhomogeneous boundary data of the velocity field on the “free boundary” x n = 0 .

where u denotes the velocity of an incompressible, viscous fluid in ℝ n + and p its pressure coupled with a boundary function h satisfying an undamped Kirchhoff-type plate equation. This problem is a linearised version of the following more abstract system of equations in a time-depending domain Ω t = {x = (x 1 , ..., x n ) ∈ ℝ n ∶ x n > h(x � , t)} , t ∈ (0, T) , with lower free boundary, Γ t = {x = (x � , x n ) ∶ x n = h(x � , t) . Here, D(u) = 1 2 (∇u + (∇u) ⊤ ) is the deformation tensor, and denotes the force exerted by the incompressible fluid in Ω t onto the free boundary Γ t with exterior normal n t pointing downward. Moreover, at t = 0, Additionally, a body force g 2 exerts a force from the outside onto the boundary. Hence, the boundary is moving and its motion is tracked by the scalar function h describing the height of the boundary. On the boundary the fluid velocity, u, coincides with the velocity of the boundary, i.e., we assume the no-slip condition for its tangential part whereas its normal component coincides with t h (kinematic boundary condition); furthermore, for mathematical generality, we include an additional force g 1 satisfying the compatibility condition g 1 | t=0 = 0 , which also models the phenomenon of leaking. In the domain Ω t , the motion of the fluid is described by the Navier-Stokes equations with constant viscosity > 0. The linearised version with fixed boundary is obtained after transforming the equations to a problem in ℝ n + and ignoring all nonlinear terms appearing in this way as well as the convective term (u ⋅ ∇)u . The first part of the term F vanishes due to divu = 0 and u � = (u 1 , … , u n−1 ) = 0 on ℝ n + so that also u n ∕ x n = 0 (1.1) t u − Δu + ∇p = f in ℝ n + × (0, T) divu = 0 in ℝ n + × (0, T) u| x n =0 = (0, ..., 0, t h + g 1 ) on ℝ n−1 × (0, T) 2 t h + Δ 2 h = −p + g 2 on ℝ n−1 × (0, T) u| t=0 = u 0 in ℝ n in Γ 0 F ∶= ⟨(2 D(u) − pI)n t , −e n ⟩, n t = (∇ � h, −1) √ 1 + �∇ � h� 2 , Ω 0 ∶= {x ∈ ℝ n ∶ x n > h 0 (x � )}, Γ 0 ∶= {x ∈ ℝ n ∶ x n = h 0 (x � )}.

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An L 2 approach to viscous flow in the half space with free elas… on ℝ n + , and only the pressure term remains, see (1.1) 4 . For more details on this transformation see [8,Chapter 2], [6] and [11].
There is a lot of literature concerning fluids interacting with an elastic plate. Often an additional damping term like − t Δh is introduced. This term allows the equation to have maximal L p −regularity as considered by Denk and Saal, see [8], and [5,Chapter 4] in a space-time periodic setting. Weak solutions in L 2 -spaces are constructed by Chambolle et al. [6] when adding a viscous damping term of either the form Δ 2 t h or −Δ tt h in the plate equation. Similar results have been shown by Grandmont [11] in the case of a three-dimensional cavity with one part of the boundary being elastic and the other one being rigid; weak solutions are shown to exist until a possible time where intersections of the two boundary parts occur. However, the focus in [11] is on the behavior of solutions in the limit of a vanishing damping term and on a uniform positive lower bound of the time of existence.
Fluid structure interaction problems with a classical nonlinear von Kármán shallow shell allowing for both transversal and lateral displacements are considered by Chueshov and Ryzhkova [7]. Lengeler and Růžička [13] discussed the case of a linearly elastic Koiter shell instead of a flat plate and hence replaced the operator Δ 2 by an operator better suited for non-flat boundaries.
In the case of a bounded domain and no damping we still have the existence of a contraction semigroup, see Badra and Takahashi [1,Proposition 3.4]. In [4] Casanova, Grandmont and Hillairet construct weak solutions in a 2D periodic layer-type domain. For local-in-time strong solutions in this latter setting we refer to Beiraõ da Veiga [3]. Moreover, strong solutions are found by Badra and Takahashi [2] using a non-analytic semigroup of Gevrey class. Finally, in [10] the present authors construct weak solutions to the fluid-structure interaction problem of a viscous fluid coupled with a damped elastic plate under the nonlinear Coulomb boundary friction condition.
However, in our case of an unbounded domain even the existence of an L 2 -semigroup is doubtful, which is why we will make more basic considerations to solve the equation in the case of non-vanishing initial data. The methods used here admit a solution in the case p = 2 only since the corresponding multipliers are bounded but they are not Fourier multipliers for p ≠ 2 . Also note that although we show the existence of solutions on any finite time interval, we must exclude the case T = ∞ since the Fourier transform of a solution of the undamped plate equation, 2 t h + Δ 2 h = 0 , is given by terms involving cosine and sine functions which are not L 2 −integrable on (0, ∞) . Adding a damping term as in [8], this issue is solved and allows the use of solution spaces with exponential weights. Additionally, in [8] the damping term also guarantees that the solution belongs to an inhomogeneous Sobolev space, which is not possible in the present undamped case.
This work is structured as follows: First we introduce the relevant solution spaces. Secondly, we show existence of solutions in the case of vanishing initial data using partial Fourier transforms. Finally, we reconstruct the initial data. Here we do not use abstract semigroup theory, as usually done, but the specific form of the undamped plate equation.

Solution spaces
One drawback of working with an undamped plate equation is that -unlike in [8] we no longer obtain solutions u ∈ L 2 (0, T;H 2 (ℝ n + )) where H 2 (ℝ n + )) denotes the usual inhomogeneous Sobolev space of order 2 over L 2 ; rather, we have to work in homogeneous spaces, i.e., u ∈ L 2 (0, T;Ḣ 2 (ℝ n + )) . Since there are non-equivalent definitions of homogeneous Sobolev spaces and Bessel potential spaces, we choose a notion that fits well with our method of choice to obtain solutions, namely via the partial Fourier transform.

Definition 1.1
Given s ∈ ℝ we define the homogeneous Sobolev space Ḣ s (ℝ n ) as equipped with the norm where P(ℝ n ) denotes the space of all polynomials on ℝ n and S � (ℝ n ) denotes the set of Schwartz distributions. For the domain ℝ n + we define the homogeneous Sobolev space Ḣ s (ℝ n + ) via restriction: We have the following useful properties that we will frequently use.
Proposition 1.2 Let s ∈ ℝ and let Λ = (−Δ) 1∕2 denote the operator defined by For a proof of iii) see Theorem 2.1 in [12]. For i) and ii) as well as for more information about homogeneous spaces see [14,Chapter 5].
Concerning Bochner spaces with respect to time over a Banach space X we .

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An L 2 approach to viscous flow in the half space with free elas… cases, X will be a homogeneous Bessel potential space Ḣ s (Ω) where Ω = ℝ n , ℝ n + or Γ ∶= ℝ n−1 = ℝ n + . Since the Fourier transform is used only on ℝ n−1 , it will, for the sake of simplicity, be denoted by F rather than F ′ , its inverse by F −1 rather than (F � ) −1 . However, the phase variable still is ′ . For the one-dimensional Fourier transform with respect to time the phase variable will be called .

Main results
Now we can formulate the main result on existence of solutions. Recall that be given. Then the system admits a solution Furthermore, we have the estimate � .

This implies
We exploit this representation to solve (2.1) 2 and get that Now we determine two identities for ĥ by using (2.1) 4 and (2.1) 5 : Combining these identities we solve for the still unknown term and see that We define and use the identity
Proof We note that B = ReB + i 2 ReB with ReB > 0 and decompose N(A, ) −1 into its real and imaginary part: First we prove that AN(A, ) is bounded: The function 3∕2 N(A, ) is also bounded due to the previous considerations and Young's inequality To simplify the analysis of the multiplier functions, we reduce the question of their boundedness to a problem in ℝ n−1 rather than in ℝ n + by the following lemma.
(2.11) (i) u � ∈ L 2 (0, ∞;Ḣ 2 (ℝ n + )) : Concerning tangential derivatives k i u j , i, j, k ∈ {1, ..., n − 1} , we estimate as follows: For the normal derivatives 2 n u ′ we have: The second summand can be treated as above. Hence we only have to consider the first one: For mixed derivatives k n u j , j, k ∈ {1, ..., n − 1} we have that The first summand can be treated as in (2.13), the second one as in (2.12).
(ii) u n ∈ L 2 (0, ∞;Ḣ 2 (ℝ n + )) : Since div u = 0 and thus k n u n = − as shown in (i), it is left to consider k i u n for i, k ≠ n . Moreover, the first summand of v n can be treated in the same way as v ′ , see (2.11). Therefore, we only have to discuss the second one: (iii) p ∈ L 2 (0, ∞;Ḣ 1 (ℝ n + )) : Since np = −Ap , it suffices to consider the tangential derivatives ∇ � p where, since M is bounded, we get that (2.12) Although the following estimates are similar to the previous ones, we present some details for the convenience of the reader: (i) u � ∈ L 2 (0, ∞,Ḣ 2 (ℝ n + )) : For tangential derivatives k i u j , 1 ≤ i, j, k ≤ n − 1 , we use the interpolation inequality and that |B| ≤ | | | | 1∕2 + |A| , and calculate as follows: For the normal derivatives 2 n u ′ we compute: (2.14)

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An L 2 approach to viscous flow in the half space with free elas… The second summand can be treated as in (2.15). Hence we only have to consider the first one: For mixed derivatives k n u j , k = 1, ..., n − 1 , there holds The first summand can be treated as in (2.16), the second one as in (2.15).

The case of non-vanishing initial data
Next we want to recover non-vanishing initial data starting with h 0 ∈Ḣ 7∕2 (Γ) . Consider the classical plate equation Applying the partial Fourier transform F with respect to the spatial variable x � ∈ Γ = ℝ n−1 we get a solution Now the general idea is to replace h from Proposition 2.3 with h + 0 to satisfy the initial condition and to leave the plate equation unchanged. However this replacement alters the equation u n | x n =0 = t h which is why we need the following proposition on the Stokes system with inhomogeneous Dirichlet boundary data in the nth component.

Remark 2.7
Despite h ∈Ḣ 2 (0, T;Ḣ −1∕2 (Γ)) we did not show that h is very regular with respect to space, let alone h(t) ∈Ḣ 7∕2 (Γ) for almost all t ∈ (0, T) . This is due to the fact that while 3∕2 N(A, ) is bounded, see Lemma 2.1, the multiplier function A 3 N(A, ) is not uniformly bounded in A, .
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