Smooth attractors of finite dimension for von Karman evolutions with nonlinear frictional dampinglocalized in a boundary layer.

Smooth attractors of finite dimension for von Karman evolutions with nonlinear frictional damping
localized in a boundary layer.

Pelin G. Geredeli Irena Lasiecka Justin T. Webster Hacettepe University University of Virginia University of Virginia Ankara, Turkey Charlottesville, VA Charlotesville, VA pguven@hacettepe.edu.tr il2v@virginia.edu jtw3k@virginia.edu
Abstract

In this paper dynamic von Karman equations with localized interior damping supported in a boundary collar are considered. Hadamard well-posedness for von Karman plates with various types of nonlinear damping are well-known, and the long-time behavior of nonlinear plates has been a topic of recent interest. Since the von Karman plate system is of ”hyperbolic type” with critical nonlinearity (noncompact with respect to the phase space), this latter topic is particularly challenging in the case of geometrically constrained and nonlinear damping. In this paper we first show the existence of a compact global attractor for finite-energy solutions, and we then prove that the attractor is both smooth and finite dimensional. Thus, the hyperbolic-like flow is stabilized asymptotically to a smooth and finite dimensional set. Key terms: dynamical systems, long-time behavior, global attractors, nonlinear plates, nonlinear damping, localized damping

1 Introduction

We consider the evolution of a nonlinear von Karman plate subject to nonlinear frictional damping with essential support in a boundary collar. Our aim is to consider the long-time behavior of the corresponding evolution. This includes studying (a) existence of global attractor which captures long-time behavior of the dynamics, and (b) properties of this attractor, such as smoothness and finite dimensionality.

In short, our goal is to show that the original infinite dimensional and non-smooth dynamics of hyperbolic type can be reduced (asymptotically) to a finite dimensional and regular set, with respect to the topology of “finite energy”. The latter is associated with weak (or generalized) solutions of the underlying semigroup for the dynamics. This type of result then allows the implementation of tools from finite dimensional control theory in order to achieve a preassigned outcome for the dynamics.

1.1 Model and Energies

Let be a bounded domain with taken to be sufficiently smooth. We consider a plate model where the real-valued function models the out-of-plane displacement of a plate with negligible thickness. Then the von Karman model [16, 34] requires that satisfies

(1.1)

The von Karman nonlinearity

(1.2)

is given in terms of (a) the Airy Stress function , satisfying

(1.3)

and (b) the von Karman bracket given by

(1.4)

The internal force , , and external force play an essential role in shaping the nontrivial stationary solutions. (In this paper denotes the Sobolev space of order on domain .) In the absence of these forces, the stationary solution of the corresponding nonlinear boundary value problem becomes trivial and simply reduces to zero.

In this treatment we focus on the stabilizing properties of the damping term . In particular, we take to be a monotone increasing function, with and further boundedness and smoothness assumptions to be imposed later; additionally, is a nonnegative localizing function which restricts the damping term to a particular subset . This is to say or for . Initially we will take to be a general set , but more specifically, we are interested in taking to be an open collar of the boundary . This type of damping represents localized, viscous damping near the boundary .

The boundary conditions we consider for the plate are:

  1. Clamped, denoted (C)

    (1.5)
  2. Hinged (simply-supported), which we denote by (H)

    (1.6)
  3. Free-type, denoted by (F)

    (1.7)

    where we have partitioned the boundary (with possibly empty). For simplicity we assume that . Otherwise the regularity theory for elliptic problems with mixed boundary conditions must be invoked. The boundary operators and are given by [34]:

    where is the outer normal to , is the unit tangent vector along . The parameters and are nonnegative, the constant has the meaning of the Poisson modulus.

Notation: Note, when referencing the plate equation above in (1.1) we will write (1.1)(C), (1.1)(H), or (1.1)(F) to indicate which boundary conditions we are taking. We write the norm in as and ; for simplicity (when the meaning is clear from context) norms and inner products written without subscript ( ,  ), are taken to be of the appropriate domain . Additionally, we employ the notation that gives the closure of in the norm.

The von Karman plate equation is well-known in nonlinear elasticity, and constitutes a basic model to describe the nonlinear oscillations of a thin plate with large displacements [34] (and references therein). In particular, we take the thickness of the plate to be negligible (as is usual in the modeling of thin structures [16]).

Remark 1.1.

It is worth noting that the von Karman plate model can accomodate plates with non-negligible thickness - the equation in (1.1) then gives the vertical displacement of the central plane of the plate. This is tantamount to adding the term , to the LHS of (1.1). This term corresponds to rotational inertia in the filaments of the plate, and (a) is regularizing from the energetic point of view and (b) forces the dynamics of the plate to be hyperbolic. In this treatment we take , since it constitutes the most challenging problem mathematically, however, a future manuscript will address the case and the limiting problem (convergence of solutions and attractors) as .

The energies associated to the above equation are given by (in the case of clamped (C) or hinged (H) boundary conditions)

where

(1.8)

The above (linear) energy dictates our state space , which depends on boundary conditions. In the case of clamped boundary conditions (C) we have . For hinged boundary conditions (H) we have .

Lastly, for free boundary conditions (F) we have (where is the Sobolev space with clamped conditions on ); the potential energy in this case is given by the bilinear form

(1.9)

where

(1.10)

Then the energy becomes

The total energy becomes

Remark 1.2.

We note that this last form of the energy described by the bilinear form can also be applied to clamped or hinged boundary conditions. Indeed, in this latter case the bilinear form collapses just to .

It will be convenient to introduce an elliptic operator given by , where incorporates the corresponding boundary conditions (clamped, hinged, or free). It is useful to note that by elliptic regularity

It is important to note the total potential energy may not be positive, or even not bounded from below. This is due to the presence of internal force which may drive the energy to . However, the presence of the von Karman bracket in the model, along with appropriate regularity properties imposed on , assures that the energy is bounded from below. This can be seen from the following lemma [12, 13]:

Lemma 1.1.

Let , and , . Then, there exists such that in the clamped and hinged case

and in the free case with ,

As a consequence we have the following bounds from below for the energy:

There exist positive constants such that

(1.11)
(1.12)

where denotes a continuous function.

1.2 Motivation and Literature

Well-posedness for von Karman’s plate equation with interior and/or boundary dissipation has been known for some time for smooth solutions in the case of homogeneous [8] or inhomogeneous nonlinear boundary conditions [12, 21] and references therein. The issue of well-posedness for ‘weak’ (finite-energy) solutions is more recent [12, 21]. In this paper, we are interested in homogeneous type boundary conditions and we will be considering generalized nonlinear semigroup solutions [4, 43] which also can be shown to be weak variational solutions. For a detailed and complete discussion regarding the wellposedness and regularity of von Karman solutions the reader is referred to [12, 28]. In the context of this paper we will need the following well-posedness result, which is contingent upon the recently shown sharp regularity of the Airy Stress function in (1.3) [21, 12]:

Theorem 1.2.

With reference to problem 1.1(C) with initial data , or 1.1(H) with initial data , or 1.1(F) with intial data , there exists a unique global solution of finite-energy (i.e. for resp., for any ). Additionally, depends continuously on .

Thus, for any initial data in the finite energy space , there exists a well defined semiflow (nonlinear semigroup) which varies continuously with respect to the initial data in . The domain of the corresponding generator is given by . For initial data taken in , the corresponding solutions are regular and remain invariant in [4, 39, 43]. With an additional assumption that is bounded polynomially at infinity, one has . Equipped with the regularity of the domain , one derives the energy identity for all regular solutions. Due to the density of the embedding , monotonicity of the damping, and sharp regularity of the Airy stress function (see Lemma 3.4) the same energy equality remains valid for all generalized solutions corresponding to any boundary conditions under consideration. Thus we have the energy identity for boundary conditions (C), (H), or (F) satisfied for all generalized (semigroup) solutions (complete details of this argument are given in [12]).

This equality reads: for all , strong and generalized solutions to (1.1) satisfy

(1.13)

With the well-posedness of the semiflow established in Theorem 1.1, it is natural to investigate long time behavior of the dynamical system generated by (1.1). It is clear from (1.13) that the essential mechanism for dissipating the energy is the damping term . In the simplest possible scenario when the energy function is equivalent topologically to the norm of the phase space . Since is nonincreasing on the trajectories, it becomes a Lyapunov function for the corresponding nonlinear dynamical system, whose only equilibrium is the zero point. If one assumes that in , it is well known that becomes a strict Lyapunov function and zero equilibrium is strongly stable. However, the above condition imposed on is not sufficient to guarantee uniform convergence to the equilibrium (this is also the case for linear dynamics without the von Karman term). In order to secure uniform convergence or, more generally, convergence to a compact attractor, a stronger form of the damping is necessary. For example, and , provides a classical model for which uniform convergence to zero in the absence of external/internal forcing (or more generally to an attractor) can be shown [7, 8, 12, 34]. The goal in this paper is to consider nonlinear damping of a reduced essential support whereby the inequality will be enforced only in a small set , while the dynamics will be forced by nontrivial sources . Existence of a compact and possibly smooth finite dimensional attracting set for the dynamics generated by (1.1) with boundary conditions (C), (H), or (F) and geometrically constrained dissipation is of great physical interest. Such a result is tantamount to asserting that the infinite dimensional, non-smooth dynamics are asymptotically reduced to a smooth and finite dimensional set. While such a reduction is expected for dynamical systems that exhibit some smoothing effects (e.g. parabolic-like) [44, 37, 17, 41, 2, 38, 26], it is a much less evident phenomena in the case of hyperbolic-like dynamics, where the ‘taking-off’ of the dynamics produces no smoothing effect. The role of the frictional damping in such a system is instrumental; in fact, it is the induced friction that creates a stabilizing and asymptotically regularizing effect on the evolution, ultimately reducing it to a compact set. On the other hand it is well known that the hyperbolic-like dynamics can not be stabilized by a compact feedback operator [30] (and references therein). This is due to the fact that instabilities in the system are inherently infinite dimensional and the essential part of the spectrum can not be dislodged by a compact perturbation. Thus, any effective damping cannot be compact (with respect to the phase space). The above feature combined with (a) nonlinearity of the damping and (b) lack of compactness of the nonlinear von Karman source makes the analysis of long-time behavior for this class of systems challenging. In fact, critical exponent nonlinearities and nonlinear dissipation are known to constitute endemic difficulties in the study of hyperbolic-like systems [22].

To orient the reader and to provide some perspective for the problem studied, we shall briefly describe some of the principal contributions to this area of research. A detailed account is given in [12].

In the discussion of global attractors for von Karman evolution equations, we must distinguish between two types of dynamics for the problem: (a) the rotational case (as addressed above) when the term is added to the LHS of (1.1) and (b) nonrotational (). In case (a), we note that the von Karman nonlinearity (in the finite energy topology) is compact, which considerably simplifies the analysis of long-time dynamics. In the latter case (b) (which we consider here), a very different type of analysis is needed. Here, we shall focus on part (b) only. In fact, the very first contribution to this problem is a pioneering paper [8] where the existence of weak attractors with a linear, fully supported damping was demonstrated. Later on, owing to new results on the regularity of Airy’s stress function [21, 12], weak attractors were improved to strong attractors, and the restriction of linear damping was removed in order to allow nonlinear, monotone damping [11]. In order to incorporate fully nonlinear interior damping, [11] assumes that the dissipation parameter is sufficiently large. This restriction was later removed in [27], whose paper introduces a very clever way of bypassing a lack of compactness and replacing it with an “iterated convergence” trick. Further studies of the attractor (including properties such as dimensionality and smoothness) in the fully nonlinear setup, without “size” restrictions imposed on the parameters, are presented in [14] and in monograph form in [10, 12].

It should be noted that the results described above pertain to the interior and fully supported dissipation. The situation is much more delicate when the dissipation is geometrically constrained, where the essential support of the damping is localized to a subset of the spatial domain . In that case, the issue of propagating the damping from one area to another becomes the critical one. While this sort of problems has been previously studied in the context of stabilization to equilibria [34, 24, 25], the estimates needed for attractors are much more demanding. Previous methods developed in the context of stabilization no longer apply. Some long-time behavior results with boundary damping are presented in [13, 14], wherein nonlinear dissipation on the boundary acting via free boundary conditions is considered. These works, however, impose the rather stringent geometric restrictions of the entire boundary being star-shaped. Such restrictions are removed in [12], where dissipation via hinged boundary conditions is considered; however this is done at the expense of limiting the class of dissipation to those of linearly bounded type. This restriction is needed since the elimination of the geometric condition is achieved via microlocal estimates [32], which in turn force velocity dependent nonlinear terms to be linearly bounded.

This brings us to the main contribution of the present manuscript. Our goal is to show that the fully nonlinear damping with essential support in an arbitrarily small layer near the boundary provides not only the existence of compact attractors but also desirable properties such as smoothness and finite dimensionality. Thus the original hyperbolic-like non-smooth flow is asymptotically reduced to smooth and finite dimensional dynamics. The result is valid for all types of boundary conditions with geometrically constrained dissipation, which can be nonlinear of any polynomial growth at infinity and with no restriction on the size of the damping parameter.

We obtain this result by proving that the dynamics are quasi-stable - a concept introduced in [10] and [12]. The ability to show quasi-stability is dependent upon: (a) a new method of localization of multipliers that allows smooth propagation of the damping from the boundary collar into the interior (even in the presence of boundary conditions (free) that do not comply with the Lopatinski conditions [42]) and (b) “backward” smoothness of trajectories from the attractor - a method used also in [14] and in [15, 5] - the latter in the context of geometrically constrained dissipation for wave dynamics.

Lastly, we would like to note that while some of the methods developed for boundary dissipation [12, 14] can also be used in the case of partially localized dissipation and Dirichlet - clamped boundary conditions, this is not the case with Neumann type (free) boundary conditions which violate strong Lopatinski [42] condition. In this latter case, propagation of the damping from the boundary layer via boundary damping estimates is obstructed by the well known lack of sufficient regularity (the absence of so called “hidden” regularity [31]) of boundary traces corresponding to the linear model [32]. Our aim in this paper is to develop a method which is effective for all kind of boundary conditions and does not depend on hidden regularity, where the latter restricts the analysis to Lopatinski type of models. The key element for this are suitably localized multipliers estimates.

1.3 Statement of Results

Equipped with well-posedness of finite energy and regular solutions corresponding to (1.1) under one of the boundary conditions (C), (H), or (F), we are now ready to state our main results pertaining to long time behavior of solutions. In order to do this, we shall introduce the following unique continuation condition, denoted . We say that the system satisfies the property iff the following implication is valid for any weak solution to (1.1): There exists such that

It is clear that the property holds if .

Remark 1.3.

It is worth noting that due to the non-local character of von Karman nonlinearity, the unique continuation property for the von Karman plate is not fully understood. A now classical set of tools developed for plate equations and based on Carleman estimates [18, 1, 20, 19] do not apply. The non-locality of the von Karman bracket prevents propagation across the entire domain of weak damping localized to a small set. Therefore, we have the question: if the damping in the equation (represented by ) is zero in an open set of positive measure inside of , does this imply that the solution must also be 0 in ?; it remains open. In relation to our analysis here, if the general unique continuation property holds for the von Karman plate, then it immediately strengthens our result by allowing to vanish away from an open collar of the boundary. However, at present, the best we can state is a sufficient condition, namely that a.e. in order to satisfy the property.

In addition, we shall assume validity of an asymptotic growth condition from below imposed on . Such condition is typical [34] and necessary in order to obtain uniform decay rates of solutions in hyperbolic-like dynamics. It allows control of the kinetic energy for large frequencies.

Assumption 1.

There exist positive constants and a constant such that

We now state the primary result in this treatment.

Theorem 1.3.

Take Assumption 1 to be in force. Let and in , where is any full collar near the boundary . Then for all generalized solutions corresponding to solutions with initial data , there exist compact attractor . If, in addition, the UC property holds, then said attractor is global, i.e for all .

Remark 1.4.

The property is needed in order to construct a strict Lyapunov function for the plate. In the absence of this Lyapunov function, the obtained results are of local character - as in the first part of Theorem 1.3. The property allows us to conclude that local attractors coincide with a global one.

Theorem 1.4.

In addition to Assumption 1 and the UC property, assume that there exists , and such that , for all . Then,

(a) the attractor is regular, which is to say is a bounded set in that topology.

(b) The fractal dimension of is finite.

Remark 1.5.

If we consider , then one can show that satisfies the above condition.

There are three main difficulties/novelties pertaining to the proof of the results stated above:

(a) The nonlinear source is of critical exponent (lack of compactness).

(b) The essential damping is geometrically constrained to a small subset .

(c) The damping is genuinely non-linear (any polynomial growth at the infinity is allowed).

These three difficulties are well-recognized in the context of studying long time behavior of hyperbolic-like systems where there is no inherent smoothing mechanism present in the model. In order to provide some perspective, it helps to add that geometrically constrained damping forces to use higher order multipliers which become supercritical when dealing with energy terms and nonlinear critical terms. Thus, any successful approach must rely on suitable cancellations, which must be uncovered for the specific dynamics in question.

Similar issues appear when dealing with nonlinear damping. The damping term must be critical (in hyperbolic dynamics) in order to be effective (we recall that the essential spectrum of an operator can not be altered by a compact perturbation). The property of monotonicity of the problem does help when dealing with a single solution at the energy level. However, when dealing with long-time behavior, the protagonist is not a single solution but the difference of two solutions. In the study of the corresponding dynamics at the non-energetic levels (resulting from multipliers), monotonicity is destroyed. There is a “spillover” of the noncompact (in fact, supercritical) damping that must be absorbed. For this issue, different mechanisms need to be discovered (e.g. backward smoothness of trajectories, compensated compactness, etc).

While recent developments in the field provide tools enabling us to handle a combination of any two of the difficulties listed above, the inclusion of the third prevents us from utilizing existing mathematical technology. The principal contribution of this treatment is to develop method which is capable of dealing with all three aforementioned difficulties simultaneously. The main ingredients of this new approach are (i) a localization method which allows us to show propagation of the damping without any requiring that the Lopatinski condition be satisfied, and (ii) backward smoothness of trajectories from the attractor with geometrically localized dissipation.

We conclude this section by listing few problems that are of interest to pursue and still open. (1) smoothness of attractors in the presence of nonlinear damping

Regarding the first item, smoothness of an attractor can be proved under certain restrictions on nonlinear damping by methods developed in [12] and also in an influential paper [23]. The treatment of a fully nonlinear and monotone dissipation is still not fully understood. (2) Damping restricted to a portion of an open collar

Secondly, dissipation localized to part of the collar could be considered by assuming certain geometric conditions imposed on the uncontrolled part of the collar. Certain ideas presented in [15, 5] should prove useful. (3) The property for a larger class of dampings

Lastly, the third item is an wide open and important problem. Other mild forms of dissipation - such as viscoelastic weak damping - are also valid alternatives, however, a full understanding of this problem would require the development of a substitute for Carleman estimates, which are not applicable due to localized structure of the von Karman bracket. The associated problem is related to controlling low frequencies - an endemic problem principally associated with strong stability [30, 29]. It is worth noting that this problem is non-existent in the case of other nonlinear plates, such as semilinear plates with local nonlinearities or even Berger’s model [6], where some forms of Carleman’s estimates do apply [1, 18].

2 Long-time Behavior of Dynamical Systems

In this manuscript we will make ample use dynamical systems terminology (see [2, 38, 41, 7, 12]); let be a dynamical system with the set of its stationary points.

We say that a dynamical system is asymptotically compact if there exists a compact set which is uniformly attracting: for any bounded set we have that

(2.1)

in the sense of the Hausdorff semidistance.

is said to be asymptotically smooth if for any bounded, forward invariant set there exists a compact set such that (2.1) holds. An asymptotically smooth dynamical system should be thought of as one which possesses local attractors, i.e. for a given ball of radius in the space there exists a compact attracting set in the closure of , however, this set need not be uniform with respect to or .

A global attractor is a closed, bounded set in which is invariant (i.e. for all ) and uniformly attracting (as defined above).

A strict Lyapunov function for is a functional on such that (a) the map is nonincreasing for all , and (b) for all and implies that is a stationary point of . If the dynamical system has a strict Lyapunov function, then we say that is gradient.

In the context of this paper we will use a few keys theorems (which we now formally state) to prove the existence of the attractor. (For proofs and references, see [12] and references therein.) First, we address attractors for gradient systems and characterize the attracting set:

Theorem 2.1.

Suppose that is a gradient, asymptotically smooth dynamical system. Suppose its Lyapunov function is bounded from above on any bounded subset of and the set is bounded for every . If the set of stationary points for is bounded, then possesses a compact global attractor which coincides with the unstable manifold, i.e.

Secondly, we state a useful criterion (inspired by [27]) which reduces asymptotic smoothness to finding a suitable functional on the state space with a compensated compactness condition:

Theorem 2.2.

Let be a dynamical system, Banach with norm . Assume that for any bounded positively invariant set and for all there exists a such that

with a functional defined on depending on and such that

for every sequence . Then is an asymptotically smooth dynamical system.

In order to establish both smoothness of the attractor and finite dimensionality a stronger estimate on the difference of two flows is needed.

Theorem 2.3.

Let where is a forward invariant set for the flow . Assume that the following inequality holds for all with positive constants

(2.2)

where is compactly embedded. Then the attractor associated with the flow posesses the following properties:
(a) The fractal dimension of is finite.
(b) For any one has .

Remark 2.1.

The estimate in (2.2) is often referred to as a “quasistability” estimate. It reflects the fact that the flow can be stabilized exponentially to a compact set. Alternatively, we might say that the flow is exponentially stable, modulo a compact perturbation (lower order terms). We assert that the lower order terms being quadratic is important for the validity of Theorem 2.3.

The proof of Theorem 2.3 employs the idea of “piecewise” trajectories introduced in [35, 40]. This allows to generalize previous criteria for finite-dimensionality [2, 44, 41, 17] by reducing the problem to validity of quasistable estimate.

2.1 Approach and Outline of the Paper

To show our main result on the existence of the global attractor for (1.1) with boundary conditions (C), (H), or (F) we make use of the theorems above. First, we note that in the case of any boundary conditions, the von Karman system in (1.1) is gradient with Lyapunov function (under the assumption that the property is satisfied, e.g. in the case that a.e. in ). We refer to [12] for the details. Moreover, the set of stationary points for the dynamical system generated by (1.1)(C), (1.1)(H), or (1.1)(F) is bounded. This latter fact follows from (1.11) (see [12]). Hence we are in a position to use Theorem 2.1 if we can obtain an inequality of the form in Theorem 2.2 to show asymptotic smoothness of the system. We will analyze , taken to be the difference of strong solutions, and make use of the linear energy (then, via a standard limiting procedure obtain our estimate for generalized solutions as well); this estimation will produce our functional in Theorem 2.2. Our main tool in estimating will be the use of two multipliers: and , where will be a suitably chosen vector field and are appropriate localization functions.

First, we perform multiplier analysis as generally as possible, without imposing boundary conditions. Later on, we shall use boundary conditions (either clamped, or hinged or free) in order to obtain the smoothness inequality in Theorem 2.2.

After establishing the existence of the attractor, we proceed to show that it has additional regularity than that of the state space, and also that it has finite fractal dimension. The ultimate goal is to prove a “quasistability” estimate for the difference of general trajectories and apply abstract Theorem 2.3, however doing so directly in this case is difficult. Instead, we will prove a modified quasistability estimate (via similar methods in the asymptotic smoothness calculation) which applies to the difference of two terms along the same trajectory (i.e. the difference of ). This term is substantially easier to analyze, since we have continuity in in and both and converge to the same point of equilibrium for and . Our modified quasistability estimate hinges upon the bounding terms being quadratic, so upon division by and taking we can show additional regularity of elements from the attractor for sufficiently negative times . Proving this will depend upon a trajectory being ‘close’ to a point of equilibrium, and hence yielding ‘smallness’ of the velocity of the solution. Propagating this regularity forward via the dynamical systems property will then allow us to show the additional regularity of the attractor. We will then proceed in a standard fashion to show that this additional regularity of the attractor yields the true and sought after quasistability estimate in Theorem 2.3, which will produce the finite fractal dimension of the attractor.

3 Asymptotic Smoothness

In this section we prove that the dynamical system generated by (1.1) is asymptotically smooth. We will refrain from imposing boundary conditions until absolutely necessary in the hope of unifying the treatment of (C), (H), and (F).

Note that the new variable , where , are solutions to (1.1) with initial data taken in bounded set in . On the strength of Lemma 1.1 and (1.11) we may assume that there exists such that

(3.1)

The difference of two trajectories solves the following PDE:

(3.2)

where

The above evolution is equipped with appropriate boundary conditions (C), (H), or (F) which will be specified later.

3.1 Multipliers

Ultimately, we will need a pointwise bound (in time) on the functional as defined above. To achieve this bound, we will employ multiplier methods based on specially chosen cut-off functions and . These functions are taken to be . Later, we will choose the supports of the derivatives of and to be contained in the damping region , where the damping is effectively localized; the cut-off functions will be chosen in this way so as to reconstruct the full energy via the multipliers, bounded in terms of the damping. However, for now, we can consider to be arbitrary.

We define the variables and . The use of the cut-off functions will produce commutators active in the regions of where the cut-off functions are non-constant. Lastly, we will make use of the following notational conventions. First, to describe (a) lower order terms:

where and (b) boundary terms:

Remark 3.1.

We note that the use of different notations for lower order terms is necessary in the handling of dissipation estimates. Specifically, we must treat the dissipation terms differently when dealing with asymptotic smoothness type estimates, and the estimates which will ultimately yield the quasistability estimate.

3.1.1 Multiplier

Let and be two differential operators. We will make use of the commutator symbol given by

We shall work with smooth solutions guaranteed by Theorem 1.2. Multiplying the PDE in (3.2) by we arrive at

Now, we employ the multiplier . This is an equipartition multiplier which allows us to reconstruct the difference between the potential and kinetic energies. The following Green’s identities are available [34] for sufficiently smooth functions and :

Using the first formula for clamped or hinged boundary conditions yields:

(3.3)

Making use of standard splitting and Sobolev embeddings, we arrive at

(3.4)

In the case of free boundary conditions, the equipartition of energy takes the form

(3.5)

We note for all boundary conditions (C), (H), the boundary terms . In the free case (F) we have where the latter term contributes a lower order term to the estimate.

To continue with our observability estimation, we must explicitly bound the commutator Purely algebraic calculations give

(3.6)

The calculation above implies that the commutator is a differential operator of order three. In order to exploit this in the calculations with the energy, we need to reduce the order of differential operator acting on a solution via integration by parts. This is done below.

This computation makes sole use of Green’s theorem. For the sake of exposition, we do not impose any boundary conditions:

(3.7)
(3.8)

Note that due to the fact that the support of is away from the boundary, all of the boundary terms in the above expressions (3.7) and (3.8) will vanish. Moreover,

(3.9)

Hence to conclude our multiplier estimate, we have the following technical lemma:

Lemma 3.1 (Preliminary Estimate).

Let , as defined above, where solves(3.2) with boundary conditions (C) or (H). Then, there exists such that

(3.10)

In the free case (F)

(3.11)
Proof.

Taking into account (3.9) in (3.4), we have

Taking into consideration boundary conditions (C) or (H) in (3.4), noting that and accounting for the fact that the boundary terms resulting from the commutators vanish leads to the first statement in the Lemma. Calculations in the free case are analogous, and result from (3.5) and , where the latter term contributes a lower order term to the estimate:

3.1.2 Multiplier 2:

For the first part of this section, we specify only that ; otherwise, we keep as general as possible, specifying it at the last possible moment. Additionally, define a set . Now, if we multiply (3.2) by and recall that , we obtain

where and , as before. We now make use of the multiplier , which we write as henceforth; there are various choices for the vector field , situationally dependent, however here we need only take in order to obtain control on the potential energy of the plate. Now, as in the previous section, we multiply the last equality by our multiplier and use Green’s Theorem to obtain

By explicitly writing out the commuator, and taking into account the support of , upon splitting we obtain:

(3.13)

Now, at this point we specify the specific structure of the supports for and (which up to now have been general). The following picture illustrates our choice for these supports and their relationship to the damping region :

We emphasize that (a) the set and (b) and overlap inside the damping region and that . Since , we have the following inequality:

(3.14)

3.2 Energy Recovery Estimate

We may now appeal to our calculation with the multiplier previously, to obtain our preliminary estimate:

Lemma 3.2 (Preliminary Estimate).

Let , as defined above, where solves (3.2) with any boundary conditions under considerations. Moreover, assume is bounded away from . Then, in the case of clamped (C) or hinged (H) boundary conditions we have

(3.15)

In the case of free boundary conditions (F)

(3.16)

We note that in the nonlinear boundary term associated with the operator vanishes due to the fact that the support of is away from the boundary.

We may now combine the estimates from Lemma 3.1 and Lemma 3.2 to obtain an estimate on the total energy (with either form of boundary conditions (C) or (H) or (F)):

(3.17)

By our choice of supports for and we note that the LHS of the above equation overestimates the total energy . On the RHS of the estimate we have the term , which we replace by since and on , , so we have that

Making the appropriate changes above in Lemma 3.1 and Lemma 3.2, we have the analogous result for the free boundary conditions (F). Hence we can conclude

Lemma 3.3 (Preliminary Energy Estimate).

For any boundary condition (C), (H), or (F) we have