Semigroup Well-posedness of A Linearized, Compressible Fluid with An Elastic Boundary

We address semigroup well-posedness of the fluid-structure interaction of a linearized compressible, viscous fluid and an elastic plate (in the absence of rotational inertia). Unlike existing work in the literature, we linearize the compressible Navier-Stokes equations about an arbitrary state (assuming the fluid is barotropic), and so the fluid PDE component of the interaction will generally include a nontrivial ambient flow profile $ \mathbf{U}$. The appearance of this term introduces new challenges at the level of the stationary problem. In addition, the boundary of the fluid domain is unavoidably Lipschitz, and so the well-posedness argument takes into account the technical issues associated with obtaining necessary boundary trace and elliptic regularity estimates. Much of the previous work on flow-plate models was done via Galerkin-type constructions after obtaining good a priori estimates on solutions (specifically \cite {Chu2013-comp}---the work most pertinent to ours here); in contrast, we adopt here a Lumer-Phillips approach, with a view of associating solutions of the fluid-structure dynamics with a $C_{0}$-semigroup $\left\{ e^{ \mathcal{A}t}\right\} _{t\geq 0}$ on the natural finite energy space of initial data. So, given this approach, the major challenge in our work becomes establishing of the maximality of the operator $\mathcal{A}$ which models the fluid-structure dynamics. In sum: our main result is semigroup well-posedness for the fully coupled fluid-structure dynamics, under the assumption that the ambient flow field $ \mathbf{U}\in \mathbf{H}^{3}(\mathcal{O})$ has zero normal component trace on the boundary (a standard assumption with respect to the literature). In the final sections we address well-posedness of the system in the presence of the von Karman plate nonlinearity, as well as the stationary problem associated with the dynamics.


Introduction
In this work, we consider a linearized fluid-structure model with respect to some reference state, including an arbitrary spatial vector field. The coupled system here describes the interaction between a linear plate and a flow of compressible barotropic, viscous fluid. Such interactive dynamics are crucially considered in the design of many engineering systems (e.g., aircraft, engines, and bridges). The study of compressible flows (gas dynamics) itself has implications to high-speed aircraft, jet engines, rocket motors, hyperloops, high-speed entry into a planetary atmosphere, gas pipelines, commercial applications (such as abrasive blasting), and many other fields (see [10,11,30], for instance). In these applications, the density of a gas may change significantly along a streamline. Compressibility -i.e., the fractional change in volume of the fluid element per unit change in pressure -becomes important, for instance, in flows for which In the study of incompressible flows, the associated analysis typically involves only two unknowns: pressure and velocity. These are usually found by solving two equations that describe conservation of mass and linear momentum, with the fluid density presumed to be constant. By contrast, in compressible flow, the gas density and temperature are variables. Consequently, the solution of compressible flow problems will require two more equations; namely, an equation of state for the gas, and a conservation of energy equation. 1 Moreover, the imposition of external forces to the governing equations may not immediately result in a uniform flow throughout the system. In particular, the fluid may compress in the vicinity of the applied force; that is to say, the density may increase locally in response to the given force.
The effects due to compressibility and viscosity upon an (uncoupled) fluid dynamics will have to be taken in account when subsequently considering the mathematical properties of PDE's describing interactions of said fluid dynamics with some given structure. In aeroelasticity, the compressible gas is often assumed to be inviscid-i.e., viscosity-free-and the flow irrotational (potential flow). These assumptions are often invoked in practice, as they reduce the flow dynamics to a wave equation [24,30] (see Section 4 below). However, there are situations where viscous effects cannot be neglected, e.g., in the transonic region [30]. The mathematical literature-especially in the last 20 years-on fluidstructure interactions across each of these fluid regimes is quite vast. We will certainly not attempt here a general overview of this literature, but in Section 4 we will provide an in-depth discussion of those key modern references which pertain to the our present work. At this point, we do mention the primary motivating reference [16]: in this work, the author considers the dynamics of a (nonlinear) plate, located on a flat portion of the boundary of a three dimensional cavity, as it interacts with a compressible, barotropic (linearized) fluid which fills the cavity. In the present work, we will analyze a comparable setup, but with a purely linear plate model; the focus here will be on establishing and describing the essential semigroup dynamics which drives the coupled PDE model. Remark 1.1. The accommodation of physically relevant nonlinearities -i.e., those seen in [16,25,22] -can be readily made subsequent to the present analysis, which develops a "good" linear theory. Linearities which are amenable to such treatment include those of Berger, Kirchhoff, or von Karman type, inasmuch as they are locally Lipschitz [21,39] on the plate's natural energy space. In many cases, it is the addition of structural nonlinearity which ultimately leads to global-in-time boundedness of corresponding trajectories [34,40]. Accordingly, nonlinear dynamics will be considered in a forthcoming work.
We also consider a Lipschitz geometry, as opposed to the common assumption [16,29] that the domain is smooth. Given the transition between the elastic and inelastic components of the boundary, a Lipschitz boundary is surely more natural and physically relevant-see Figure 1 below-and also more amenable to pertinent generalizations, e.g., tubular domains (finite or infinite) [20,26]. Further distinguishing our work from [16], we take the linearization of the compressible Navier-Stokes equations about a rest state which has a nonzero ambient flow component. Since this linearization process produces some additional terms which depend on the ambient flow (which are not all compact perturbations of principal spatial operators), previous techniques to obtain the well-posedness result cannot be directly applied. 2 Instead, to obtain the primary result we invoke a semigroup approach, invoking the well known Lumer-Phillips theorem [39, p.13].
We believe that the present treatment fits nicely within the context of the recent work of I. Chueshov, where the interactive dynamics between fluid and a plate (or shell) are considered from various points of view [16,17,20,25,26,27]. Moreover, one can draw comparisons and contrasts between the well-posedness analysis here and that in [5] and [3], which deal with incompressible fluidstructure interactions: in [5] and [3], there is also a two dimensional elastic structure existing on the boundary of a three dimensional domain, in which a fluid evolves. However, the earlier well-posedness work [5] requires an appropriate mixed variational, Babuska-Brezzi formulation, which is nonstandard and instrinsic to the particular dynamics under consideration (see e.g., Theorem 3.1.5 of [33]); whereas the present effort combines the Lax-Milgram Theorem with a critical well-posedness result for (static versions of) the pressure PDE component of the fluid-structure system (see the first equation in (2.4) and Theorem 7.1 of the Appendix.) For fluid-structure well-posedness studies that involve a three dimensional solid immersed in a three dimensional fluid, and which utilize semigroup techniques, see [8], [9], [6].) Eventually, we are interested in learning if and how the presence of the dissipating fluid dynamics affects the stability of the structure (as in, e.g., [23,16,27,26], [4]). In particular, for the linear compressible fluid-structure interaction PDE model, we are interested in strong/uniform stability properties of the associated C 0 -semigroup. On the other hand, if one inserts nonlinearity into the structural component of the interaction, the existence and nature of global attractors become the primary objects of interest for the associated PDE dynamical system. Qualitative properties of fluid-structure models (such as well-posedness and stability of solutions, and the existence of compact global attractors) have been intensely investigated by many authors over the past 30 years. For the PDE model under consideration, (2.4)-(2.6) below, issues of long-time behavior of solutions will be addressed in forthcoming work [7].
The paper is organized as follows: In Section 2 we describe the PDE model and discuss our standing hypotheses. In Section 3 we provide a discussion of the principal dynamics operator (on the natural 2 In fact, the late author of [16]-to whom this work is dedicated-suggested in personal correspondence the precise model (2.4)-(2.6); [18]. In this communication, he remarked that the approaches utilized in [16] with U ≡ 0 were not amenable to the problem studied with U = 0. He noted that a semigroup approach might be fruitful, and this comment provided an impetus for the present work. space of finite energy), as well its domain; we then formally state the semigroup generation result which immediately yields well-posedness of fluid-structure model, in the sense of Hadamard. We also include in this section the notion of energy balance for semigroup solutions. Section 4 provides an in-depth discussion of the key pertinent references, and their relationship to the result presented here. Section 5 gives the proof of the main result via the Lumer-Phillips theorem: Namely, we establish dissipativity and maximality of a certain bounded perturbation of the modeling fluid-structure operator. Section 6 gives a brief description of stationary solutions to the dynamics at hand. Lastly, the Appendix provides a proof of a key technical lemma on the well-posedness of solutions to the stationary version of the decoupled pressure PDE component in (2.4)-(2.6) below.

Notation
For the remainder of the text we write x for ( )} , as dictated by context. For a given domain D, its associated L 2 (D) will be denoted as || · || D (or simply || · || when the context is clear). The symbols n and τ will be used to denote, respectively, the unit external normal and tangent vectors to O. Inner products in L 2 (O) or L 2 (O) are written (·, ·) O (or simply (·, ·) when the context is clear), while inner products L 2 (∂O) are written ·, · . We will also denote pertinent duality pairings as ·, · X×X , for a given Hilbert space X. The space H s (D) will denote the Sobolev space of order s, defined on a domain D, and H s 0 (D) denotes the closure of C ∞ 0 (D) in the H s (D)-norm · H s (D) or · s,D . We make use of the standard notation for the boundary trace of functions defined on O, which are sufficently smooth: i.e., for a scalar function φ ∈ H s (O), 1 2 < s < 3 2 , γ(φ) = φ ∂O , a well-defined and surjective mapping on this range of s, owing to the Sobolev Trace Theorem on Lipschitz domains (see e.g., [38], or Theorem 3.38 of [36]).

PDE Model
Let O ⊂ R 3 be a bounded and convex fluid domain (and so has Lipschitz boundary ∂O; see e.g., Corollary 1.2.2.3 of [31]). The boundary decomposes into two pieces S and Ω where ∂O = S ∪ Ω, with S ∩ Ω = ∅. We consider S to be the solid boundary, with no interactive dynamics, and Ω to be the equilibrium position of the elastic domain, upon which the interactive dynamics takes place. We also assume that the active component Ω is flat and embedded in the x 1 − x 2 plane, with the inactive component S lying below the x 1 − x 2 plane. This is to say, Letting n(x) denote the unit outward normal vector to ∂O, we have n| Ω = (0, 0, 1). (See Figure 1.) We consider the compressible Navier-Stokes system [15], assuming the fluid is barotropic, and linearize the system with respect to some reference rest state of the form {p * , U, * }. The pressure and density components p * , * are assumed to be scalar constants, and the arbitrary ambient flow field U : O → R 3 is given by: (2.3) Figure 1: The Fluid-Structure Geometry Deleting non-critical lower order terms (see Remark 2.1 below), and setting the pressure and density reference constants equal to unity, we obtain the following perturbation equations: Here, p(t) : R 3 → R and u(t) : R 3 → R 3 (pointwise in time) are given as the pressure and the fluid velocity field, respectively. The quantity η > 0 represents a drag force of the domain on the viscous fluid. In addition, the quantity τ in (2.4) is in the space T H 1/2 (∂O) of tangential vector fields of Sobolev index 1/2; that is, With respect to the "ambient flow" field U appearing in (2.4), we define the space and subsequently impose the standard assumption that (see the analogous-and actually slightly stronger-specifications made on ambient fields on p.529 of [29] and pp.102-103 of [42]). Given the Lamé Coefficients λ ≥ 0 and ν > 0, the stress tensor σ of the fluid is defined as where the strain tensor is given by (see [33, p.129]). With this notation it is easy to see that div σ(µ) = ν∆µ + (ν + λ)∇div(µ), where λ and ν are the non-negative viscosity coefficients. The boundary conditions that are invoked in (2.4) for the fluid PDE component are the so-called impermeability-slip conditions [11,15]. Their physical interpretation is that no fluid passes through the boundary (the normal component of the fluid field u on the active boundary portion Ω matches the plate velocity w t ), and that there is no stress in the tangential direction τ .
Remark 2.1. The above fluid equations in (2.4) might be referred to as the Oseen equations for viscous compressible barotropic fluids. In the linearization procedure, without making additional assumptions on U, we obtain: for a prescribed scalar function F 1 and vector field F 2 . In our analysis we retain only the principal mathematical terms in (2.4)-(2.6), as the others may be viewed as zeroth order perturbations, and handled in a standard fashion.
Remark 2.2. Other possible physically relevant boundary conditions have appeared in the literature. We mention the Kutta-Joukowski type condition (2.11) [24], as well as the adherence condition (2.12).

Main Results
We are interested in Hadamard well-posedness of the linearized coupled system given in (2.4)-(2.6). Specifically, we will ascertain well-posedness of the PDE model (2.4)-(2.6) for arbitrary initial data in the natural space of finite energy. To accomplish this, we will adopt a semigroup approach; namely, we will pose and validate an explicit semigroup generator representation for the fluid-structure dynamics (2.4)-(2.6).
With respect to the coupled PDE system (2.4)-(2.6), the associated space of well-posedness will be H is a Hilbert space, topologized by the following inner-product: In what follows, we consider the linear operator A : D(A) ⊂ H → H, which expresses the compressible fluid-structure PDE system (2.4)-(2.6) as the abstract ODE: To wit, Here, the domain D(A) is given as on Ω (and so µ 0 | ∂O ∈ T H 1/2 (∂O)).
In the following theorem, we provide semigroup well-posedness for A : D(A) ∈ H → H, the proof of which is based on the well known Lumer-Phillips Theorem.
, w t (t)} defines a strongly continuous semigroup {e At } on the space H, and hence the system (2.4)-(2.6) is well-posed (in the sense of mild solutions-see Remark 3.2 below). In addition, the semigroup enjoys the following estimate: Remark 3.1. Given the existence of a semigroup {e At } for the fluid-structure generator A : In particular, the solution satisfies the condition (A.iv) in the definition of the generator. This means that one has that the tangential boundary condition satisfied in the sense of distributions. That is to say, ∀τ ∈ T H 1/2 (∂O) and ∀φ ∈ D(∂O), Remark 3.2 (Notions of Solution). We note that semigroup solutions, as arrived at in Theorem 3.1 for initial data y ∈ H, correspond to so called mild solutions (satisfying an integral form of (2.4)-(2.6)) in the sense of [39,Section 4.2]. Moreover, for initial data y ∈ D(A), we obtain so called strong solutions.
In [16], semigroup techniques are not used in demonstrating well-posedness. As such, the author takes care to define an appropriate notion of weak solution corresponding to a Galerkin construction (see [16, pp.653-654]). Such a notion of weak solution is relevant here, and can be obtained by making minor modifications that take into account the vector field U. Here we assert that mild solutions (obtained via our semigroup) are in fact weak solutions as in [16]. In this way, we recover the wellposedness result of [16] (in the fully linear case with O bounded) by simply letting U ≡ 0.
Finally, we describe the energy balance equation for semigroup solutions to (2.4)-(2.6). We introduce the notion of energy into the analysis. Semigroup solutions obtained on the finite-energy space H are measured in the finite energy norm, which provides us with the energy functional: for Let us also introduce the convenient notation: With mild solutions in hand, we may test (2.4)-(2.6) with p, u, and w t (repsectively) to obtain the energy balance. Equivalently, the semigroup solution admits, for y 0 ∈ H, p ∈ L 2 0, t; L 2 (O) , u ∈ L 2 0, t; L 2 (O) , and w t ∈ L 2 0, t; L 2 (Ω) , making them valid "multipliers". This yields the energy balance below.
Remark 3.3. We note two features of the energy identity: first, when the field U ∈ V 0 is also divergence free, the energy identity remains the same as in the case where U ≡ 0 (like [16]). Secondly, the dissipation integral ∞ 0 a O (u, u)dτ depends on the quantity div U as well: again, with div U ≡ 0, we see that

Discussion of Main Results in Relation to the Literature
The model under consideration describes the case of a (possibly viscous) compressible gas/fluid flow and was recently studied in [16] in the case with zero speed (U = 0) of the unperturbed flow. Beginning with compressible Navier-Stokes, one can obtain several fluid-plate cases which are important from an applied point of view: • Incompressible Fluid, i.e., div u = 0 and density constant: In the viscous case, the standard linearized Navier-Stokes equations arise; fluid-plate interactions in this case were studied in [27,25,26,28]. Results on well-posedness and attractors for different elastic descriptions and domains were obtained. In this case, we also mention the work [5,3,4] which addresses semigroup wellposedness of a related linear fluid-plate model, and decay rates via frequency domain techniques.
The inviscid case was studied in [17] in the same context.
• Compressible Fluid: In the inviscid case we can obtain wave-type dynamics for the (perturbed) velocity potential (u = ∇φ, potential flow) of the form (see also [10,11,30]): In these variables, the pressure/density of the fluid has the form p = (∂ t + U · ∇)φ. Due to the impermeability assumption, in the case of the perfect fluid, we have only one Neumann-type boundary condition given above via the operator L. The (semigroup) well-posedness [22,40] and stability properties [21,23,34] of this model have been intensively studied.
The viscous case was studied in [16], and is the motivation of the current work.
In all the papers cited above, the interactive dynamics between fluid and a plate (or shell) are considered. These analyses are distinguished from those for other fluid-structure interactive PDE models in that the elastic structure is two dimensional, and evolves on the boundary of the three dimensional fluid domain. One of the key issues for the present configuration-and indeed, one of the main points in the bulk of the literature above-is the determination of how, and to what extent, the fluid (de)stabilizes the structure. In [16,27,26], after obtaining well-posedness of the models (with structural nonlinearity), the existence of compact global attractors for the dynamics is shown; in some cases the existence of this invariant set is due strictly to the presence of the fluid, rather than some underlying structural phenomenon. 5 In addition, it is sometimes possible (perhaps under additional assumptions) to show strong stabilization to equilibrium for the fluid-structure dynamics (e.g., [34,21]). In all cases where the ambient flow field U = 0, the stability properties of the model depend greatly on the structure and magnitude of the flow field U [1]. This will certainly be the case for the dynamics considered here, as one can see from (3.7).
Remark 4.1. The survey papers [28,24] provides a nice overview of the modeling, well-posedness, and long-time behavior results for the family of dynamics described above.
We emphasize that in any study involving compressible fluids, the enforced compressibility produces additional density/pressure variables, and, as a result, well-posedness cannot be obtained in a straightforward way. In fact, the primary difficulty lies in showing the maximality (range) condition of the generator, since one has to address this density/pressure component. This variable cannot be readily eliminated, and therefore accounts for an elliptic equation which must be solved. To overcome this, we develop a methodology based on the application of a static well-posedness result given in the Appendix of [29] (see also [35]). That paper, as well as [42], deals with the stationary compressible Navier-Stokes equations. Their principal results (obtained independently, through different methodologies) is a small data well-posedness for the fully nonlinear fluid problem. However, both approaches first necessarily provide a framework for the linearized problem; in particular, [29] provides a strategy for our analysis of the stationary compressible fluid-structure PDE which is associated with maximality of the generator.
The pioneering work [16], which we cite as the primary motivating reference, considers the model presented here with U ≡ 0. In this paper, solvability and dynamical properties of the model are considered in the case of a general (possibly unbounded) smooth domain. Along with the well-posedness result, the existence of a finite dimensional compact global attractor is proved when the domain is bounded. The techniques used are consistent with those in [27,25,17]; namely, Galerkin-type procedures are implemented, along with good a priori estimates, in order to produce solutions. As with many fluid-structure interactions, the critical issue in [16] is the appearance of ill-defined traces at the interface. In the incompressible case, one can recover negative Sobolev trace regularity of the pressure p at the interface via properties of the Stokes' operator. However, in the viscous compressible case this is no longer true. Our semigroup approach does not require the use of approximate solutions. Indeed, we overcome the key difficulty of trace regularity issues by exploiting cancellations at the level of solutions with data in the generator. In this way we do not have to work component-wise on the dynamic equations, though we must work carefully (and component-wise) on the static problem associated with maximality of the generator. We remark that, despite these trace regularity issues, when U ≡ 0, uniform decay of finite energy solutions is obtained in [16] through a clever Lyapunov approach that makes use of a Neumann lifting map with associated estimates; this construction is fundamentally obstructed by the addition of the U · ∇p term in the pressure equation. See the forthcoming work [7] on the decay properties of the model considered here.

The Proof of Theorem 3.1
Our proof of well-posedness hinges on showing that the matrix A : D(A) ⊂ H → H generates a C 0semigroup. At this point, we should note that due to the existence of the generally nonzero ambient vector field U in the model, we have a lack of dissipativity of the operator A. Accordingly, we introduce the following bounded perturbation A of our generator A: Therewith, the proof of Theorem 3.1 is geared towards establishing the maximal dissipativity of the linear operator A; subsequently, an application of the Lumer-Phillips Theorem will yield that A generates a C 0 semigroup of contractions on H. In turn, applying the standard perturbation result [32] (given, for instance, in [39, Theorem 1.1, p.76]) yields semigroup generation for the original modeling fluid-structure operator A of (3.4), via (5.1).

Dissipativity
Considering the inner-product for the state space H given in (3.2), for any y = [p 0 , u 0 , w 1 , Applying Green's Theorems to right hand side, we subsequently have is then a cancellation of boundary terms so as to have Moreover, via Green's Theorem, as well as the assumption that U ∈ V 0 (as defined in (2.8)), we Applying these relations to the right hand of (5.4), we then have which establishes the dissipativity of A : D(A) ⊂ H → H.

Maximality
In this section we show the maximality property of the operator A on the space H. To this end, we will need to establish the range condition, at least for parameter ξ > 0 sufficiently large. Namely, we must show Range(ξI − A) = H, for some ξ > 0.
This necessity is equivalent to finding [p, v, w 1 , w 2 ] ∈ D(A) which satisfies, for given [p * , v * , w * 1 , w * 2 ] ∈ H, the abstract equation Given the definition of A in (3.4), then in PDE terms, solving the abstract equation (5.7) is equivalent to proving that the following system of equations, with given data [p * , v * , w * 1 , w * 2 ] ∈ H, has a (unique) solution [p, v, w 1 , w 2 ] ∈ D(A): (5.10) We will give our proof of maximality in two steps. In the first step, we will show the existence and uniqueness of "uncoupled" versions of the compressible fluid-structure PDE system (5.9) which is satisfied by variables {p, v}. To this end, the key ingredient will be the well-posedness result Theorem 7.1, which is applicable to (uncoupled) equations of the type satisfied by the pressure variable. (See also [29] and [35].) Subsequently, we proceed to establish the range condition (5.8), by sequentially proving the existence of the pressure-fluid-structure components {p, v, w 1 , w 2 } which solve the coupled system (5.9)-(5.10). This work for pressure-fluid-structure static well-posedness involves appropriate uses of the Lax-Milgram Theorem.
STEP 1 consists of proving the following (driving) lemma for the existence and uniqueness of the solution {p, v} of (5.11)-(5.15).  = 0 for all τ ∈ T H 1/2 (∂O), (5.18) and so the boundary condition (5.13) is satisfied in the sense of distributions; see (3.6) of Remark 3.1 (iv) The fluid and pressure solution components (p, v) satisfies the following estimates, for ξ = ξ(U) large enough: . (5.20) Proof of Lemma 5.1. We give the proof in two parts. Our beginning point is to resolve the pressure term; this will be accomplished by applying Theorem 7.1 of the Appendix. To this end: If we initially consider the equation With the well-posedness above, in order to find the existence and uniqueness of the fluid component v, we now turn our attention to (5.11)-(5.15). In view of the well-posedness of (5.21), we decompose the fluid term v and pressure term p as follows: with estimates-for ξ = ξ(U) large enough, see (5.23) and (5.26) - where again, fluid variable u will be sought after. Now, the rest part of the proof relies on the application of Lax-Milgram theorem, by way of solving for u in (5.24). For this reason, we firstly define the operator A ∈ L(V 0 , [V 0 ] ) to be, for all ψ ∈ V 0 , where again, p[u] solves (5.27). So, with a view of finding the solution pair (p, v), via the expressions (5.24) and (5.25), we are led to consider the following variational problem: Find u ∈ V 0 which solves, for every ψ ∈ V 0 , where the forcing term F ∈ [V 0 ] is given by for given ψ ∈ V 0 . After considering the definition of V 0 and using the divergence theorem we note that In particular, if ψ ∈ [D(O)] 3 , we then have (v * , ψ) O = (ξv + U · ∇v  (5.20). This concludes the proof of Lemma 5.1, and so STEP 1 of the maximality argument.

Stationary Problem
Since the stationary problem associated with a dissipative dynamical system is of interest when studying long-time behavior of solutions [21], we briefly discuss the stationary problem associated with (2.4)-(2.6). (Such a discussion is in line with [16, p.658].) Formally, we introduce the problem: As in [16], this problem is to be interpreted variationally. This is to say, we define a weak solution to (6.1)-(6.2) to be a triple (p, u, w) ∈ L 2 (O) × V 0 × H 2 0 (Ω), which must satisfy: for all q ∈ H 1 (O) and all ψ ∈ V Ω ≡ {v ∈ H 1 (O) : v · n = 0 on S} with ψ · n Ω = β ∈ H 2 0 (Ω). We note that this is a natural definition of a weak (variational) solution, in line with weak solutions [16] to the dynamic equations (2.4)-(2.6).
Thus, for any ψ as above, we utilize the latter two identities to see that a stationary solution must satisfy: We may also take ψ = u, as well as test the pressure equation with p, to obtain: As discussed above, it is clear that the long-time behavior (stability) properties of solutions to the dynamic (2.4)-(2.6) depend on the structure of the flow field U. Thus, it is not an unwelcome assumption (see [7]) to consider a divergence-free flow field U. If div U ≡ 0, then we see from (6.7) that u ≡ 0 (in the sense of H 1 (O)), since η > 0. Moreover, testing with any ψ such that (ψ) 3 Ω = β = 0, we see from (6.6) that p = const. Thus, in the case where U is divergence free, the stationary problem reduces to a simple problem for the clamped biharmonic operator, i.e., find w ∈ H 2 0 (Ω) such that (∆w, ∆v) Ω = (c, v) Ω , ∀ v ∈ H 2 0 (Ω). (6.8) In this situation, a stationary solution to (2.4)-(2.6) (or equivalently, a solution to (6.1)-(6.2)) will be a triple of the form (c, 0, w c ) ∈ L 2 (O) × V 0 × H 2 0 (Ω), with c = const, and where w c solves the corresponding biharmonic problem on H 2 0 (Ω).

Appendix
For the reader's convenience, we provide an explicit proof for the well-posedness of the (uncoupled) pressure equation (5.21).
Theorem 7.1. (See [29] and [35]) Consider the linear equation (as before, O ⊂ R 3 is a bounded, convex domain). Also, the parameter k > 0 and forcing term G ∈ L 2 (O). Moreover, the fixed vector field v in (7.1) is in H 3 (O) and further satisfies:

2)
where C S > 0 is a constant which gives rise to the Sobolev embedding inequality, Then for given G ∈ L 2 (O), there exists a unique λ ∈ L 2 (O) which is a weak solution of (7.1). Moreover, the solution satisfies the bound By the L 2 (O)-function λ being a weak solution of (7.1), we mean that it satisfies the following variational relation: Proof of Theorem 7.1: In large part, the present proof is taken from that of Appendix I of [29] (on p.541)-see also [35]-which however was undertaken on the assumption that geometry O is smooth. Accordingly, adjustments are made here for general convex domain O, as well as for the perturbation λ 2 div(v). Given > 0, we denote λ to be the solution of the following regularized boundary value problem: − ∆λ + kλ + v · ∇λ + 1 2 λ div(v) = G in O, λ | ∂O = 0. (7.6) Assume initially that data G ∈ H 1 0 (O). We note that the Lax-Milgram Theorem insures the existence of the H 1 0 (O)-function λ which solves (7.6): Indeed, multiplying the left hand side of the equation in (7.6) by solution variable λ , integrating, and then integrating by parts, we have Thus we infer H 1 0 (O)-ellipticity for the bilinear form associated with the PDE (7.6). Subsequently, since the bounded domain O is convex, we can invoke Theorem 3.2.1.2, p.147, of [31] to have that solution λ ∈ H 2 (O) ∩ H 1 0 (O). This extra regularity allows for a multiplication of both sides of (7.6) by term ∆λ , and a subsequent integration and integration by parts, so as to yield the relation To handle the third term on left hand side: Using classic vector field identities for the wave equationsee e.g., [37], [14], or [41]  (In stating this relation, we are using assumption (i) of (7.2).) With respect to the first term on right hand side of (7.9): using Proposition 4, p.702 of [12], and the fact that λ | ∂O = 0, we have on ∂O ∇λ | ∂O = ∇ ∂O ( λ | ∂O ) + n ∂λ ∂n = n ∂λ ∂n .
(Above, ∇ ∂O ( λ | ∂O ) ∈ L 2 (∂O) denotes the tangential gradient of λ | ∂O ; see [38] or p.701 of [12].) Applying this relation to (7.9), and considering v · n| ∂O = 0, we then have For the fourth term on the left hand side of (7.8): Using Green's Theorem, we have A further integration by parts then yields Applying the relations (7.10) and (7.12) to (7.8), we then have Now, concerning the fourth term on left hand side: and subsequently applying Hölder's inequality with conjugates p = 3/2 and p * = 3, we have then where positive constant C S is that in (7.3).
With the convergences above in hand, we multiply the (7.6) by test function ϕ ∈ H 1 (O), and integrate. (Recall that each λ is a strong solution of (7.6).) This gives the relation GϕdO for every ϕ ∈ H 1 (O).
(7.18) Passing to the limit on left hand side, and invoking (7.17) we have that strong L 2 -limit λ satisfies (7.5).
Moreover, from the relations (7.6) and (7.7) for the regularized problem, and the strong limit posted in (7.17), we have the estimate k λ O ≤ G O . (7.19) In sum: we have justified the existence of a operator L 0 , say, which satisfies, for given G ∈ H 1 0 (O), L 0 (G) = λ, where λ ∈ L 2 (O) solves (7.5), and which yields the estimate after using (7.19). An extension by continuity now yields the solvability of equation (7.1) for given L 2 -data G, and this solvability is unique because of the inherent dissipativity in this equation. Lastly, the estimate (7.4) is just (7.20).