On a Quasilinear Parabolic-Hyperbolic System Arising in MEMS Modeling

A coupled system consisting of a quasilinear parabolic equation and a semilinear hyperbolic equation is considered. The problem arises as a small aspect ratio limit in the modeling of a MEMS device taking into account the gap width of the device and the gas pressure. The system is regarded as a special case of a more general setting for which local well-posedness of strong solutions is shown. The general result applies to different cases including a coupling of the parabolic equation to a semilinear wave equation of either second or fourth order, the latter featuring either clamped or pinned boundary conditions.


Introduction
Electrostatically actuated micro-electromechanical systems (MEMS) are ubiquitous in today's electronic devices.Idealized MEMS often consist of a fixed ground plate and an elastic membrane (or plate) that are close.Keeping the two components at different potential induces a Coulomb force deflecting the membrane.In the past two decades MEMS devices have been a highly active mathematical research focus, in particular due to their interesting qualitative behaviors with respect to pull-in instabilities (as a result from a possible touching of membrane and ground plate) and the inherent challenges related to local and global well-posedness of the corresponding models.We refer to [7] and the references therein for more details on MEMS models and their mathematical investigation in general.
In this paper, we consider a model introduced in [5,6] arising as a small aspect ratio limit of equations governing an electrostatically actuated MEMS, where the narrow gap separating the membrane and the ground plate is filled with a rarefied gas.More precisely, we consider ∂ t (wu) = div w 3 u∇u , t > 0 , x ∈ Ω , (1.1a) u(0, x) = u 0 (x) , w(0, x) = w 0 (x) , ∂ t w(0, x) = w ′ 0 (x) , x ∈ Ω , where w = w(t, x) denotes the varying width of the gap and u = u(t, x) is the local pressure of the gas.The (sufficiently) smooth bounded subset Ω of R n with n ∈ {1, 2} represents the shape of the membrane and the ground plate.The constants a, b, θ 1 , θ 2 > 0 and σ ≥ 0 in (1.1b) -(1.1c) as well as the initial data u 0 , w 0 , and w ′ 0 in (1.1d) are given.The degeneracy of (1.1a) and the singularity in (1.1b) occurring for a vanishing gap width w(t, x) = 0 capture the instabilities related to a touchdown of the membrane on the ground plate.A detailed account of the modeling aspects is given in [5,6] to which we refer.
In [5] the short-time existence of solutions to this MEMS model is established for the one-dimensional case n = 1.The approach chosen therein consists of solving first the hyperbolic equation for w (via a fixed point argument for a given, fixed u) and so reducing the coupled system (1.1) to a single fixed point equation for u which is then solved using parabolic semigroup theory.Instead of decoupling the system, we proceed differently and solve the mild formulation of (1.1) simultaneously for u and w, also relying on semigroup theory for semilinear hyperbolic and quasilinear parabolic equations described in [4,8] respectively [2,3].A key ingredient for this is the observation that mild solutions to the hyperbolic equation (1.1b) enjoy a priori Hölder continuity properties with respect to time (and values in spaces of sufficiently high spatial regularity) that guarantee an evolution operator for the quasilinear parabolic equation (1.1a) in the sense of [3] (see also Remark 3.2 and Proposition A.1 below).In this way we provide a considerably shorter proof for local existence including also the case n = 2: The initial values are compatible with the Dirichlet boundary conditions and the regularity of strong solutions at t = 0.In fact, the solution component u has even better regularity properties than stated in Theorem 1.1, see Remark 4.1.Moreover, the solution can be extended to a maximal solution on [0, T max ) existing as long as u(t) > 0 and w(t) > 0 in Ω as well as the C 1 -norm of (u, w) does not blow up.It is worth pointing out that a common feature in MEMS models [7] is the possible occurrence of finite time quenching inf x∈Ω w(t, x) → 0 as t → T max preventing global existence of solutions.
A similar result as Theorem 1.1 can also be shown for a related fourth-order equation when the Laplacian ∆ in (1.1b) is replaced by −∆ 2 + ∆ subject to pinned or clamped boundary conditions (see Theorem 5.1 below for details).This corresponds to a MEMS device involving an elastic plate instead of a membrane.
In fact, Theorem 1.1 (and Theorem 5.1) can be regarded as a special case of a more general setting including a quasilinear parabolic equation coupled to a semilinear wave equation of the form where A(u, w) are generators of analytic semigroups on a Banach space and −A is a generator of a cosine function on a Hilbert space (see Section 3 below for details).
In the next Section 2 we first identify (1.1) as a special case of (1.2) (see (2.11) below).The latter is treated in Section 3 in an abstract functional analytic framework that is not restricted to the particular setting of (1.1).The main result of this research is Theorem 3.1 on the local well-posedness of (1.2) that is established using semigroup theory and then implies Theorem 1.1 for the particular case (1.1) as shown in Section 4. In Section 5 we briefly show how to apply Theorem 3.1 for the case of the fourth-order problem (5.1) including a bi-Laplacian.

Functional Formulation of the Problem
We demonstrate how to express the system (1.1) in the abstract form of problem (1.2) and list relevant properties of the functions involved.
The previous considerations ensure that problem (2.11) (and thus problem (1.1)) fits into the more general framework of Theorem 3.1 of the next section.We shall then continue from here in Section 4 and finish off the proof of Theorem 1.1 by applying Theorem 3.1.

Main Theorem
As just pointed out above, Theorem 1.1 is a special case of a more general setting: Consider where A(u, w) ∈ L(E 1 , E 0 ) for some continuously and densely injected Banach couple A(u 0 , w 0 ) with domain E 1 generates an analytic semigroup on E 0 ), and We formulate (3.1) as a coupled system of two first order equations relying on results for cosine functions [4, Section 5.5 & Section 5.6], see also Appendix A. To this end note that (3.2) ensures that the powers A z for z ∈ C are well-defined closed operators (bounded for z ≤ 0).Consequently, the matrix operator generates a strongly continuous semigroup (e tA ) t≥0 on the Hilbert space (in fact, it generates a group (e tA ) t∈R ).Using the notion w = (w, w ′ ) and setting we can write (3.1b) as a semilinear hyperbolic Cauchy problem ∂ t w = Aw + F(u, w) , t > 0 , w(0) = w 0 := (w 0 , w ′ 0 ) , in H.In fact, for greater flexibility (and to cope with the particular case (2.1)) we shift this problem to the interpolation space (for some α ∈ [0, 1)) where we recall (due to the Fourier series representation of A α or [9, Theorem 1.15.3]) that Then, the H α -realization A α of A, given by generates a strongly continuous semigroup (e tAα ) t≥0 on H α according to [3, Chapter V].
We shall then consider (3.1) in the equivalent form In the following, let (•, •) θ be arbitrary admissible interpolation functors [3, I.Section 2.11] and set and suppose (3.2).Moreover, assume that (a) There is a unique mild solution to the Cauchy problem (3.4) on some interval [0, T ]. ) is a strict solution to (3.1a).In this case, if ).We emphasize that one may rely on the regularity properties (3.7) and (3.11) when checking (3.12) or (3.13).
Given T ∈ (0, 1) (to be specified later) we then introduce where z 0 = (u 0 , w 0 ) and the notation w = (w, w ′ ) is used throughout.Then V T is a complete metric space when equipped with the metric Then, for z = (u, w) ∈ V T , we have by interpolation (see 1 The notation A ∈ H E1, E0; κ, ω means that ω − A ∈ Lis(E1, E0) and and it thus follows from (3.15) and ρ < 1 − µ that for some constant r(u 0 , w 0 ) > 0 (independent of z ∈ V T ) and from (3.14) that Now, [3, II.Corollary 4.4.2] and (3.18) imply that for each z = (u, w) ∈ V T , the operator A(u, w) generates a parabolic evolution operator on E 0 with regularity subspace E 1 and that we may apply the results of [3, II.Section 5].
Remark 3.2.It is worth emphasizing that one of the key ingredients of the proof of Theorem 3.1 is the observation that the first component w of a mild solution w = (w, w ′ ) to the hyperbolic equation (3.4b) enjoys a Hölder continuity property with respect to time and values in spaces of sufficiently high spatial regularity (see (3.17)) as stated in Proposition A.1.In fact, this ensures the Hölder continuity of the operator t → A(u(t), w(t)) and thus that the associated evolution operator is well-defined according to [3, II.Corollary 4.4.2].

Proof of Theorem 1.1
We can now complete the proof of Theorem 1.1.From Section 2 we know that problem (1.1) is equivalent to (2.1) (recalling that (u, w) is identified with (u − θ 1 , w − θ 2 )) which, in turn, is a special case of (3.1) (see (2.11)).

A Fourth-Order Problem
As pointed out in the introduction, Theorem 3.1 also applies to certain fourth-order wave equations.Indeed, consider ∂ t (wu) = div w 3 u∇u , t > 0 , x ∈ Ω , (5.1a) where δ ∈ {0, 1}.Equations (5.1) govern the gap width w and the gas pressure u for a MEMS device involving an elastic plate (instead of a membrane) of shape Ω, where Ω ⊂ R n with n ∈ {1, 2} is assumed to be a (sufficiently) smooth bounded set.The elastic plate is either clamped at its boundary (corresponding to δ = 0) or is hinged along its boundary so that it is free to rotate (corresponding to δ = 1).We assume that D 1 > 0 and D 2 ≥ 0 and that a, b, θ 1 , θ 2 > 0 and σ ≥ 0. This MEMS model was introduced in [6] and the short-time existence of solutions was established for the pinned case δ = 1 (for both cases n = 1, 2).We derive the result for pinned and clamped boundary conditions simultaneously as a consequence of Theorem 3.1 (the assumptions on the initial values are compatible with the regularity of the solution): Proof.The proof is very much the same as for Theorem 1.1 and we thus only sketch it and point out the new aspects.Arguing as in Section 2 by shifting u and w we may focus on ∂ t u = A(u, w)u + g(u, w, ∂ t w) , t > 0 , u(0) = u 0 , (5.2a) ∂ 2 t w + σ∂ t w = −Aw + f (u, w) , t > 0 , (w(0), ∂ t w(0)) = (w 0 , w ′ 0 ) , (5.2b) with A defined in (2.5) and g and f in (2.8) respectively (2.10).The only difference now is that we consider the fourth-order operator densely defined, self-adjoint, positive operator with bounded and compact inverse (3.2) on a Hilbert space H with scalar product (•|•).Here, positive operator means that (Ax|x) ≥ 0 for x ∈ D(A).Let σ ≥ 0.