Hadamard well-posedness for a class of nonlinear shallow shell problems

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Hadamard well-posedness for a class of nonlinear

shallow shell problems

John Cagnol, Irena Lasiecka, Catherine Lebiedzik, Richard Marchand

To cite this version:

www.elsevier.com/locate/na

Hadamard well-posedness for a class of nonlinear shallow

shell problems

John Cagnol

, Irena Lasiecka

, Catherine Lebiedzik

, Richard Marchand

a_{Pˆole Universitaire L´eonard de Vinci, ESILV, DER CS, 92916 Paris La D´efense, France} b_{University of Virginia, Department of Mathematics, Charlottesville, VA 22904, USA}

c_{Wayne State University, Department of Mathematics, Detroit, MI 48201, USA} d_{Slippery Rock University, Department of Mathematics, Slippery Rock, PA 16057, USA}

Received 23 December 2005; accepted 7 September 2006

Abstract

This paper is concerned with the nonlinear shallow shell model introduced in 1966 by W.T. Koiter in [On the nonlinear theory of thin elastic shells. III, Nederl. Akad. Wetensch. Proc. Ser. B 69 (1966) 33–54, Section 11] and later studied in [M. Bernadou, J.T. Oden, An existence theorem for a class of nonlinear shallow shell problems, J. Math. Pures Appl. (9) 60(3) (1981) 285–308]. We consider a version of this model which is based upon the intrinsic shell modeling techniques introduced by Michel Delfour and Jean-Paul Zol´esio. We show existence and uniqueness of both regular and weak solutions to the dynamical model and that the solutions are continuous with respect to the initial data. While existence and uniqueness of regular solutions to nonlinear dynamic shell equations has been known, full Hadamard well-posedness of weak solutions, as shown in this paper, is a new result which solves an old open problem in the field.

c

2006 Elsevier Ltd. All rights reserved.

MSC:35Q99; 35L75; 74K25

Keywords:Uniqueness; Continuous dependence; Hyperbolic partial differential equation; Koiter nonlinear shell model; Intrinsic geometric shell modeling

1. Introduction

In [21], Koiter classifies shell problems according to the order of magnitude of the deflection. He arranges the classes in an order of increasing specialization, as follows: Large Deflections, characterized by the absence of restriction as to the magnitude of the displacement; Moderate Deflections, characterized by small values of all displacement gradients with respect to unity, although nothing is assumed a priori about their relative orders of magnitude; Small Finite Deflections, characterized by small displacement gradients and by rotations whose squares do not exceed the middle surface strain in order of magnitude; Infinitesimal Deflections, characterized by small displacement gradients none of which exceed the middle surface strains in order of magnitude.

∗_{Corresponding author. Tel.: +33 1 41 16 71 88; fax: +33 1 41 16 71 71.}

E-mail address:john.cagnol@devinci.fr(J. Cagnol).

0362-546X/$ - see front matter c 2006 Elsevier Ltd. All rights reserved.

The infinitesimal deflections hypothesis leads to the well-known linear theory. Instead, in this article we shall consider the small finite deflections case, which is studied in [21, Section 11]. This corresponds to the “medium bending case” in the sense of Naghdi (see [28]) and to the “approximation of small strains and moderately small rotations” in the sense of Lyell and Sanders (see [29]). This case leads to the geometrically nonlinear model discussed in [3] and summarized in [2, Section 7.3]. In [3], Bernadou and Oden have proven that a solution to the static model exists and that it is unique provided the load is sufficiently small. Philippe G. Ciarlet proposed in [8] a model where the exact change of curvature tensor is modified and Liliana Gratie generalized this latter model to the case of non-constant thickness in [17]. See [10] for the linear and nonlinear theories of shells.

This paper is concerned with the dynamical model. Dynamical systems of nonlinear elasticity have attracted much interest over the years [32,9,22,14,18,13]. Numerous physical phenomena can be described by these models, which mathematically take the form of a set of coupled partial differential equations. The shell model considered here is a nonlinear, strongly coupled system of hyperbolic equations with clamped (Dirichlet) boundary conditions. Its counterpart for a flat geometry is the full von K´arm´an system [32,9], which couples a plate-like equation for the normal displacement variable with wave equations for the tangential (in-plane) displacements.

Existence of weak solutions for such models follows from standard Faedo–Galerkin method [27]. However, the issue of well-posedness (uniqueness and continuous dependence on initial data) is much more subtle. The nonlinear term in the equation is neither locally Lipschitz nor bounded in the finite energy space. In addition, the problem is not monotone and weak solutions do not exhibit any additional regularity properties (unlike parabolic-like problems). Thus typical or known methods used for proving well-posedness of weak solutions are no longer applicable.

It is the main goal set in this paper to provide an affirmative answer to full Hadamard well-posedness of weak (often referred to as finite energy) solutions. More specifically, the main contribution of this work is threefold:

• Existence and uniqueness of weak solutions (i.e. finite energy solutions).

• Continuous dependence of solutions with respect to initial data measured in finite energy norm. • Regularity of weak solutions.

We shall use the intrinsic geometry method of Michel Delfour and Jean-Paul Zol´esio, which relies on the oriented distance function to describe the geometry [12,11]. This technique takes advantage of the intrinsic geometric properties of the shell. Here, the shell is described in terms of tangential differential operators which are defined by means of the oriented boundary distance function in R3. Sobolev spaces, Green’s formula, and key inequalities such as Poincar´e and Korn’s inequalities are all well defined. Delfour and Zol´esio have constructed models under a variety of assumptions. A linear version of the model developed in this paper was introduced in [5,6].

In addition, Hadamard well-posedness is proved in [18] by application of a finite difference method which allows for the rigorous derivation of an appropriate energy identity for the system.

It is the aim of this work to further extend the techniques developed in the case of “flat” geometry [26,18] to the curved case required by a shell model. The final result is full Hadamard well-posedness of finite energy solutions governing nonlinear dynamic shells.

Throughout this paper we will denote by Sl the shell and by Γ its mid-surface. We shall make the following

hypothesis:

(i) The shell is assumed to be made of an isotropic and homogeneous material, so that the Lam´e coefficientsλ > 0 andµ > 0 are constant. The density of the material will be denoted as ρ > 0.

(ii) (shallowness assumption) The thickness l of the shell is small enough to accommodate the curvatures H and K , i.e.the product of the thickness and the curvatures is small as compared to 1. In addition, the shell is shallow in the sense that the second fundamental form (here given in terms of the oriented distance function as D2b) and its derivative (D3b) are small. This assumption allows us to neglect certain terms in the strain energy in comparison to the model in [5,6] and yields a model in which the coupling between the normal and tangential displacements is of the first (rather than third) order. For a detailed justification of these assumptions we refer the reader to Koiter [20,21].

(iii) (Kirchhoff hypothesis) In the classical thin plate theory named after Kirchhoff, the displacement vectors of the shell Sland of the mid-surface Γ are related by the hypothesis that the filaments of the plate initially perpendicular

to the middle surface remain straight and perpendicular to the deformed surface, and undergo neither contraction nor extension. We generalize this hypothesis to the case of a shell using the intrinsic geometry in [5].

For the sake of self-containment, we provide background material on intrinsic modeling in Section2. In Section3.3, we state the main findings of this paper. Section5is concerned with establishing preliminary results that will be used in the subsequent sections. Section4is concerned with describing the model [2, Section 7.3] and subsequently justifying the investigation of the equations presented in(12). We prove the existence result in Section6, the uniqueness result in Section7, and the continuous dependence with respect to the initial data in Section8.

2. Intrinsic modeling

In this section we present a brief overview of the oriented distance function and the intrinsic tangential calculus that forms the basis of our shell model.

2.1. The oriented distance function and the intrinsic geometry

In order to improve readability we here include a brief discussion of the oriented distance function and the intrinsic geometric methods of Delfour and Zol´esio. Since by necessity this overview will lack detail, the reader is referred to [12,11] for a definitive exposition on this topic.

Consider a domain O ⊂ R3whose non-empty boundary∂O is a C1two-dimensional submanifold of R3. Define the oriented (or signed) distance function to O as

b(x) = dO(x) − dR3_\O(x) (1)

where d is the Euclidean distance from the point x to the domain O. In other words, b(x) is simply the positive or negative distance to the boundary∂O, depending on whether we are outside or inside the domain O. It can be shown that for every x ∈∂O, there exists a neighborhood where the function ∇b = ν, the unit outward external normal to ∂O [11].

Consider a subset Γ ⊆ ∂O which will eventually become the mid-surface of our shell. We define the projection p(x) of a point x onto Γ as p(x) = x − b(x)∇b(x). Then, we define a shell Slof thickness l as (seeFig. 1)

Sl(Γ ) ≡

n

x ∈ R3: p(x) ∈ Γ , |b(x)| < l/2o . (2) A natural curvilinear coordinate system(X, z) is thus induced on the shell Sl, where the coordinate vector X gives the

Fig. 1. The shell.

Using this notation, we also define the “flow mapping” Tz(X) as

Tz(X) = X + z∇b(X) (3)

for all X and z in Sl. This allows us to reconstruct the action at a given height z of the shell, once we know the action

of the mid-surface Γ . Define as Γzthe surface Tz(Γ ) at the ‘altitude’ z. Then, one can also describe the shell Slas

Sl = l/2

[

z=−l/2

Γz.

The curvatures of the shell will be denoted as H and K . These can be reconstructed from the boundary distance function b(x) by noting that at any point (X, z), the matrix D2bhas eigenvalues 0, λ1, λ2. The curvatures are then

given by tr(D2b) = 2H = λ1+λ2and K =λ1λ2.

2.2. Tangential differential calculus

Next, we mention briefly some useful aspects of the tangential differential calculus. Given f ∈ C1(Γ ), we define the tangential gradient ∇Γ of the scalar function f by means of the projection as

∇_Γ f ≡ ∇( f ◦ p)(x)|_Γ. (4a)

This notion of the tangential gradient is equivalent to the classical definition using an extension F of f in the neighborhood of Γ , i.e. ∇Γ f = ∇ F |Γ −∂ F_∂νν [11]. Following the same idea we can define the tangential Jacobian

matrix of a vector functionv ∈ C1(Γ )3as

DΓv ≡ D(v ◦ p)|Γ or (DΓv)i j =(∇Γvi)j, (4b)

the tangential divergence as

divΓv ≡ div(v ◦ p)|Γ, (4c)

the Hessian D2_Γf of f ∈ C2(Γ ) as

the Laplace–Beltrami operator of f ∈ C2(Γ ) as

1Γ f ≡divΓ (∇Γ f) = 1( f ◦ p)|Γ, (4e)

the tangential linear strain tensor of elasticity as

εΓ(v) ≡

1

2 DΓv +

∗

DΓv
= ε (v ◦ p)|Γ (4f)

and the tangential vectorial divergence of a second-order tensor A as

div_Γ A ≡ div(A ◦ p)|_Γ =div_Γ Ai. (4g)

Using the definitions given in(4)one can derive Green’s formula in the tangential calculus [11]: Z Γ fdivΓv dΓ + Z Γ h∇_Γf, vi dΓ = Z Υ hfv, νi dΥ + 2 Z Γ f H hv, ∇bi dΓ (5)

whereν is the outward unit normal to the curve Υ. We use the definition of εΓ(u) and of the tangential vectorial

divergence of a second-order tensor A as

divΓ A ≡ div(A ◦ p)|Γ =divΓ Ai (6)

to derive the integration formula Z Γ tr(ε_Γ(u)A) dΓ = Z ∂Γ hu, Aνi d(∂Γ ) − Z Γ hu, div_ΓAidΓ + Z Γ 2H hu, A∇bi dΓ (7) for A a symmetric matrix. Finally, from [11,12] we have that

h∇_Γw, ∇bi = 0, DΓv∇b = 0 (8)

by definition for any scalar w and vector v. In addition, if we consider a purely tangent vector v = vΓ, i.e.

hv_Γ, ∇bi = 0, we can take the tangential gradient of both sides of this expression and derive the following useful formula

D2bvΓ+ ∗DΓvΓ∇b =0. (9)

In the case of tangentialv_Γ, the last term of Eq.(5)is also zero. We conclude this section with some remarks about notation. We shall adopt the following notation:

|w|_s,Γ ≡ |w|_Hs_{(Γ )}; (u, v)_Γ ≡

Z

Γ

uv dΓ

and use h·, ·i to denote the scalar product of two vectors. Throughout this paper the conventions of [16] concerning tensors are used. For instance, we will make no distinction between a second-order tensor and a matrix, nor will we make a distinction between a first-order tensor and a vector. Consequently we will not distinguish simple contraction and multiplication. Finally, the notation ∂tϕ is used to denote the partial derivative of ϕ with respect to the time

variable, and the notation A · · B denotes the double contraction of two matrices — i.e. A · · B = tr(AB). 2.3. Some tangential operators and spaces

Let e be a vector function; we denote by eΓ andw the tangential component of e and its algebraic magnitude on

the normal part respectively, i.e.

w = he, ∇bi (10a)

e_Γ =e −w∇b. (10b)

Definition 1. We define G_Γw = 1 2((∇b ⊗ ∇Γw)D 2_{b + D}2_b(∇ Γw ⊗ ∇b)) V_Γe_Γ = 1 2((D 2_{b e} Γ) ⊗ ∇b + ∇b ⊗ (D2b eΓ)) SΓw = 1 2(D 2 Γw + ∗ D_Γ2w).

Operator G_Γ is a first-order tangential operator, V_Γ is a zero-order tangential operator, and S_Γ is the symmetrization of the Hessian (which is not we note symmetric in the tangential calculus [11]).

Definition 2. Let us define

BM(e) = εΓ(eΓ) + VΓeΓ +wD2b +

1

2∇Γw ⊗ ∇Γw BF(e) = SΓw + GΓw.

Operators BM and BFwill correspond to the membrane and bending (flexural) strains respectively. In the rest of this

paper the following functional space will be useful.

Definition 3. For m ∈ N, define the spaces Vm as Vm(Γ ) =  e ∈ Hm(Γ )2×Hm+1(Γ ) | e_Γ =0, w = 0, ∂ ∂νw = 0 on ∂Γ  . (11)

3. Statement of the main results

3.1. Equation to be studied

Letγ = l2/12, and for a matrix A define operator C by C(A) = λ(tr A)I + 2µA.

We consider the following equation in the variable e = [e_Γ, w], where e_Γ andw are respectively the tangential and vertical displacements and are defined above in Eqs.(10):

                           ρ∂t tw − ργ 1Γ∂t tw + (λ + 2µ)γ 12_Γw + ργ 2divΓ(D 2_b∂ t teΓ) + tr(C(εΓeΓ +VΓeΓ)D2b) +tr  C  wD2_{b +}1 2∇Γw ⊗ ∇Γw  D2b  −divΓ(C(εΓeΓ+VΓeΓ)∇Γw) −divΓ  C  wD2_{b +}1 2∇Γw ⊗ ∇Γw  ∇_Γw  +2µγ divΓ(K ∇Γw) + 2µγ divΓ((D2b)2∇w) = 0 ρ(I + γ (D2_b)2_)∂ t teΓ −ργ 2D 2_b∂ t tw − divΓ(C(εΓeΓ+VΓeΓ)) −divΓ  C  wD2_{b +}1 2∇Γw ⊗ ∇Γw  =0 (12)

where the derivatives are to be understood in the distributional sense.

Let k =λH2+µ(H2−2K) be a positive real number and m(e, ˆe), a(e, ˆe), n(e, ˆe) be defined by m(e, ˆe) = ρ[2(e_Γ, ˆe_Γ)_Γ +2γ ((D2b)e_Γ, (D2b) ˆe_Γ)_Γ −γ (∇_Γw, (D2b) ˆe_Γ)_Γ

−γ ((D2b)eΓ, ∇Γw)ˆ Γ+2(w, ˆw)Γ +2γ (∇Γw, ∇Γw)ˆ Γ] (13)

a(e, ˆe) = 8λ(kw, k ˆw)_Γ +4λ(Hdiv_ΓeΓ, ˆw)Γ+4λ(Hw, divΓeˆΓ)Γ

+2λ(div_ΓeΓ, divΓeˆΓ)Γ +4µ

Z

Γ

+4µ Z Γwtr((εΓ(ˆeΓ) + VΓ ˆ e_Γ) D2b) + 4µ Z Γ ˆ wtr((εΓ(eΓ) + VΓeΓ) D2b) +2γ λ(1Γw, 1Γw)ˆ Γ+4γ µ Z Γ tr((SΓw + GΓw)(SΓw + Gˆ Γw))ˆ (14)

−n(e, ˆe) = 2λ(Hk∇_Γwk2, ˆw)_Γ +λ(k∇_Γwk2, div_Γeˆ_Γ)_Γ +4λ(Hw∇Γw, ∇Γw)ˆ Γ+2λ(divΓeΓ∇Γw, ∇Γw)ˆ Γ +(λ + 2µ)(k∇_Γwk2∇_Γw, ∇_Γw)ˆ _Γ+2µ Z Γ tr(εΓ(ˆeΓ)(∇Γw ⊗ ∇Γw)) +2µ Z Γ ˆ wtr(D2_b(∇ Γw ⊗ ∇Γw)) + 4µ Z Γ whD2_b∇ Γw, ∇Γwiˆ +4µ Z Γ tr((∇_Γw ⊗ ∇_Γw)(εˆ _Γ(e_Γ) + V_Γe_Γ)). (15) The variational problem associated with(12)is, for each t ∈ [0, τ[ and all test functions ˆe ∈ V1(Γ ),

m(∂t te, ˆe) + a(e, ˆe) − n(e, ˆe) = 0 (16)

where e(0) = e0and∂te(0) = e1.

Definition 4 (Weak Solutions). We say that e ∈ C0([0, τ[, V1(Γ ))∩C1([0, τ[, V0(Γ )) is a weak solution to the small finite deflections shell problem if and only if(16)is satisfied.

Definition 5 (Regular Solutions). We say that e is a regular solution to the small finite deflections shell problem if and only if it is a weak solution (i.e. satisfies Eq.(16)) and e ∈ C0([0, T [, V2(Γ ))∩C1([0, τ[, V1(Γ ))∩C2([0, τ[, V0(Γ )). Remark 1. In the special case of a plate, the curvature tensor D2_{bvanishes so all the terms including it, like G}

Γ, VΓ,

K, vanish as well; we obtain:        ρ∂t tw − ργ 1Γ∂t tw + (λ + 2µ)γ 12_Γw − divΓ  C  εΓeΓ+ 1 2∇Γw ⊗ ∇Γw  ∇_Γw  =0 ρ∂t teΓ−divΓ  C(εΓeΓ) + 1 2∇Γw ⊗ ∇Γw  =0

which is the full von K´arm´an system; see [24].

3.2. Operators

Definition 6. Let M, A be the linear operators defined by (Me, ˆe) = m(e, ˆe); (Ae, ˆe) = a(e, ˆe),

let

H = D(A1/2_{) × D(M}1/2_{) = V}1_{(Γ ) × V}0_{(Γ ) = [H}1_{(Γ )]}2_×H2_{(Γ ) × [(L}2_{)(Γ )]}2_×H1_{(Γ ) (17)}

and let A : H → H be defined by A =  0 −I M−1A 0  (18)

D(A) = {(e1, e2) ∈ D(A1/2) × D(A1/2), Ae1∈ [D(M1/2)]0}.

Remark 2. A more compact but less explicit expression of the weak formulation can be given by 1

2(M∂t te, ˆe)Γ + Z

Γ

(C BM(e)) · · BM0 (e, ˆe) + γ

Z

Γ

where

B_M0 (e, ˆe) = ∂ BM(e + θ ˆe) ∂θ    _θ=0 =ε_ΓeˆΓ+VΓeˆΓ + ˆwD2b + 1 2∇Γw ⊗ ∇Γw +ˆ 1 2∇Γw ⊗ ∇ˆ Γw.

This compact expression(19)will be used in the proof of the existence of regular solutions (Section6), while the more detailed expression will be used in the proof of the uniqueness (Section7).

Definition 7. Let N be the nonlinear operator defined by(N(e), ˆe) = n(e, ˆe) and NΓ, Nn be respectively the vector

function and scalar function defined by

NΓ(w) = −λ∇Γ(k∇ΓwΓk2) + 2µdivΓ(∇Γw ⊗ ∇Γw)

Nn(eΓ, w) = −2λHk∇Γwk2+4λdivΓ(Hw∇Γw) + 4µdivΓ(w D2b∇Γw)

+(λ + 2µ)div_Γ(k∇_Γwk2∇_Γw) − 2µhD2b∇Γw, ∇Γwi

−2λdiv_Γ(div_ΓeΓ∇Γw) − 4µdivΓ((εΓeΓ +VΓeΓ)∇Γw)

so we have N(e) = NΓ(w) + Nn(eΓ, w)∇b.

It follows, from(16), that: ∂t  e ∂te  + A e_∂ te  =  0 M−1_N(e)  . (20) 3.3. Main results

Our main results are:

Theorem 1 (Regular Solutions). We consider Eq.(16). Assume thate(0) ∈ V2(Γ ) and ∂te(0) ∈ V1(Γ ). Then, there

exists a unique regular global solution

e ∈ C([0, τ[; V2(Γ )) ∩ C1([0, τ[; V1(Γ )) ∩ C2([0, τ[; V0(Γ )) (21) whereτ > 0 is arbitrary.

Theorem 2 (Existence and Uniqueness of Weak Solutions). We consider Eq.(16). Assume thate(0) ∈ V1(Γ ) and ∂te(0) ∈ V0(Γ ). Then there exists a weak solution

e ∈ C([0, τ[; V1(Γ )) ∩ C1([0, τ[; V0(Γ )) (22) which is unique, whereτ > 0 is arbitrary.

Theorem 3 (Continuous Dependence w.r.t. Initial Data). Weak solutions to Eq.(16)depend continuously on the initial data in the finite energy norm. That is to say, for allτ > 0 and all sequences of initial data such that

en(t) → e0 in V1(Γ ) ∂ten(t) → e1 in V0(Γ )

the corresponding solutionsen(t) ∈ C([0, τ[; V1(Γ )) and e(t) ∈ C([0, τ[; V1(Γ )) satisfy: en→e in C([0, τ[; V1(Γ ))

∂ten→∂te in C([0, τ[; V0(Γ )).

As a consequence we obtain an energy identity for weak solutions.

The result presented inTheorem 1is standard for an equation of the type of(16), though no references seem to be available for the proof this particular result.Theorems 2and3are new and solve an old problem in the field.

4. Derivation of the model

This section is devoted to deriving the model that leads to Eq. (12), which justifies the interest raised by this equation in the first place. The model will be obtained using Hamilton’s principle. The next two subsections are devoted to computing the energies. In Section4.3Hamilton’s principle is used to get the weak formulation, and in Section4.4the PDE formulation is derived. Finally in Section4.5, the total energy of the system is computed and the system is proved to be conservative (Proposition 1).

The reader solely interested in the Hadamard well-posedness of(12)may skip this section as notation introduced herein is explicitly referenced in the next sections andProposition 1is fairly standard.

Since the deflections are not infinitesimal, the strain–displacement relationship is not assumed to be purely linear, and instead is of the form given in Eq.(27)(see [9,22]). The displacement of the shell T and the displacement of the mid-surface e are related by the Kirchhoff hypothesis, that is that the filaments of the plate initially perpendicular to the middle surface remain straight and perpendicular to the deformed surface, and undergo neither contraction nor extension. Following [5], we have

T =e ◦ p − b(∗DΓe ∇b) ◦ p (23)

which leads to

T = eΓ◦ p +(w∇b) ◦ p + b (D2b eΓ − ∇Γw) ◦ p. (24)

4.1. The kinetic energy

The kinetic energy is given by

Ek(e) =ρ

2 Z

Sl

|∂_tT |2. (25)

Following [5], the kinetic energy of the system at any time t is given by

Ek(e) = ρl 2 Z Γ |∂_teΓ|2+ |∂tw|2+ ρlγ 2 Z Γ |D2b∂teΓ|2+ |∇Γ∂tw|2− hD2b∂teΓ, ∇Γ∂twi. (26)

4.2. The elastic energy

The key hypothesis, and main difference from the model studied in [5], is the strain–displacement relations ε(T ) = 1 2( ∗ DT + DT) +1 2 ∗_{DT DT. (27)}

Following [21, Section 11], several parts of the nonlinear term ∗DT DT will be neglected; this will in particular be the case for all the terms involving ∗DΓeΓ that appear in ∗DT DT. This is consistent with the shallowness

assumption (Hypothesis (ii), Section1), as these nonlinear terms appear in combination with higher order derivatives of the oriented distance function b. Eqs.(23)and(27)yield

ε(T ) =εΓ(eΓ) + wD2b + VΓeΓ+ 1 2∇Γw ⊗ ∇Γw + 1 2k∇Γwk 2_{∇b ⊗ ∇b}  ◦p − b(SΓw + GΓw) ◦ p. (28)

From now on, and when no confusion is possible, we will writeε instead of ε(T ).

Assuming isotropy and homogeneity of the material, one can apply Hooke’s law to give the elastic energy:

At this point in the computation of the elastic energy, it is customary to impose the hypothesis of plane stresses: σ · · (∇b ⊗ ∇b) = 0 (which in local coordinates is denoted as σ33 = 0). In order to satisfy this requirement, we

changeε(T ) to ε(T ) =εΓ(eΓ) + wD2b + VΓeΓ + 1 2∇Γw ⊗ ∇Γw  ◦ p − b(S_Γw + G_Γw) ◦ p. (30) With the notation ofDefinition 2we have

ε(T ) = (BM(e)) ◦ p + b (BF(e)) ◦ p.

This new form ofε(T ) is in line with [2, page 38, Section 7.3]. Eq.(29)yields Ep(e) = 1 2 Z Sl [C(BM(e)) · · BM(e)] ◦ p + 1 2 Z Sl

b[C(BM(e)) · · BF(e) + C(BF(e)) · · BM(e)] ◦ p

+1 2 Z Sl b2[C(BF(e)) · · BF(e)] ◦ p. (31) We define EM p (e) = 1 2 Z Sl [C(BM(e)) · · BM(e)] ◦ p (32) EF p(e) = 1 2 Z Sl b2[C_(BF(e)) · · BF(e)] ◦ p (33)

as the membrane and bending (flexural) energy, respectively. Since the second integral of the right hand side of(31)

vanishes, we have

Ep(e) = EpM(e) + EpF(e).

We compute the explicit form of the bending energy:

E_pF(e) = 1 2 Z Sl b2[C(BF(e)) · · BF(e)] ◦ p = λ 2 Z Sl b2(tr(BF(ε)))2◦ p +µ Z Sl b2tr((BFε)2) ◦ p = λ 2 Z l/2 −l/2 z2 Z Γz(tr(BF(ε))) 2_+µZ l/2 −l/2 z2 Z Γz tr((BFε)2) = λl 3 24 Z Γ(tr(BF(ε))) 2₊µl3 12 Z Γ tr((BFε)2).

Using tr G_Γw = 0 and tr S_Γw = 1_Γw we get E_pF(e) = λl 3 24k1Γwk 2 L2(Γ )+ µl3 12 Z Γ tr((SΓw + GΓw)2). (34)

Analogous computations can be carried out with EM_p (e). We use that D2b∇b = 0 and that for any vector u we have tr VΓu = 0 and tr(VΓu D2b) = 0 for any vector u. We also use that tr(∇Γw ⊗ ∇Γw) = k∇Γwk2and that

1b = tr D2_{b =2H . We obtain that the elastic energy of the system is given by}

+λl 3 24 |1Γw| 2 0,Γ+ µl3 12 Z Γ trh(S_Γw + G_Γw)2i . (35) 4.3. Weak formulation of the model

Let us consider a final timeτ > 0. Among all kinematically admissible displacements, the actual motion of the shell will make stationary the Lagrangian

L(e) = Z τ 0 Ek(e) − Ep(e) = Z τ 0 Ek(e) − EMp(e) − E F p(e).

We consider the Gˆateaux derivative in a direction ˆe = ˆeΓ + ˆw∇b; we have

∀ˆe , ∂L(e + θ ˆe) ∂θ    _θ=0 =0. (36)

Note that S symmetric implies S · · M = S · ·(∗M + M) so the two last terms of B_M0 (e, ˆe) will often be replaced by twice one of them.

∂EM p (e + θ ˆe) ∂θ     _θ=0 = Z Γ

(C BM(e)) · · B0M(e, ˆe) (37a)

∂EF p(e + θ ˆe) ∂θ     _θ=0 =γ Z Γ(C BF(e)) · · BF(ˆe) (37b)

and it follows that ∂Ep(e + θ ˆe) ∂θ    _θ=0 = Z Γ(C BM(e)) · · B 0 M(e, ˆe) + γ Z Γ(C BF(e)) · · BF(ˆe) =λl Z Γ  2Hw + divΓeΓ + 1 2k∇Γwk 2  (2H ˆw + divΓeˆΓ + h∇Γw, ∇Γwi)ˆ +2µl Z Γ tr  εΓeΓ +D2bw + VΓeΓ + 1 2∇Γw ⊗ ∇Γw  ×  εΓeˆΓ +D2b ˆw + VΓeˆΓ + 1 2∇Γw ⊗ ∇ˆ Γw + 1 2∇Γw ⊗ ∇Γwˆ  +λl 3 12 Z Γ 1Γw1Γw +ˆ µl 3 6 Z Γ tr((S_Γw + G_Γw)(S_Γw + G ˆw)).ˆ

Consequently, if e is an admissible displacement of the mid-surface of the shell which is sufficiently regular then the following expression vanishes for any ˆe = ˆeΓ+ ˆw∇b ∈ V1(Γ )

Z τ 0 Z Γ −2ρh∂_teΓ, ∂teˆΓi −2ρ∂tw∂tw + γˆ Z τ 0 Z Γ −2ρhD2b∂teΓ, D2b∂teˆΓi −2ρh∇Γ∂tw, ∇Γ∂twiˆ +ρhD2b∂teΓ, ∇Γ∂twi + ρhDˆ 2b∂teˆΓ, ∇Γ∂twi + Z τ 0 Z Γ 8λH2w ˆw + 4λH divΓeΓw + λHk∇ˆ Γwk2wˆ

+4λHw div_Γeˆ_Γ+2λdiv_Γe_Γdiv_Γeˆ_Γ+2λk∇_Γwk2div_Γeˆ_Γ +4λHwh∇Γw, ∇Γwi + 2λdivˆ ΓeΓh∇Γw, ∇Γwiˆ

+λh∇_Γw, ∇_Γwik∇ˆ _Γwk2+4µ tr((ε_Γ(e_Γ) + V_Γe_Γ)(ε_Γ(ˆe_Γ) + V_Γeˆ_Γ))

+4µw tr((ε_Γ(ˆe_Γ) + V_ΓeˆΓ) D2b) + 4µ ˆwtr((εΓ(eΓ) + VΓeΓ) D2b) + 8µ(H2−2K)w ˆw

+2µk∇Γwk2h∇Γw, ∇Γwi + 2µtr((εˆ Γ(ˆeΓ) + VΓeˆΓ)(∇Γw ⊗ ∇Γw)) + 2µ ˆwtr(D2b(∇Γw ⊗ ∇Γw))

+2µwtr((∇Γw ⊗ ∇ˆ Γw + ∇Γw ⊗ ∇Γw) Dˆ 2b) +γ Z τ 0 Z Γ 2λ(1Γw)(1Γw) + 4µ tr((Sˆ Γw + GΓw)(SΓw + Gˆ Γw)).ˆ

Therefore, for all ˆe_Γ ∈ H₀1(Γ ), we have Z τ 0 Z Γ −2ρh∂teΓ, ∂teˆΓi + γ Z τ 0 Z Γ −2ρhD2b∂teΓ, D2b∂teˆΓi +ρhD2b∂teˆΓ, ∇Γ∂twi

+4λHwdivΓeˆΓ +2λdivΓeΓdivΓeˆΓ +λk∇Γwk2divΓeˆΓ

+4µtr((ε_Γ(e_Γ) + V_Γe_Γ)(ε_Γ(ˆe_Γ) + V_Γeˆ_Γ)) + 4µwtr((ε_Γ(ˆe_Γ) + V_Γeˆ_Γ) D2b) +2µtr((εΓ(ˆeΓ) + VΓeˆΓ)(∇Γw ⊗ ∇Γw)) = 0

and for all ˆw ∈ H₀2(Γ ) ∩ H₀1(Γ ) we have Z τ 0 Z Γ −2ρ∂tw∂tw + γˆ Z τ 0 Z Γ −2ρh∇_Γ∂tw, ∇Γ∂twi + ρhDˆ 2b∂teΓ, ∇Γ∂twiˆ + Z τ 0 Z Γ 8λH2w ˆw + 4λHdivΓeΓw + 2λHk∇ˆ Γwk2wˆ +4λHwh∇_Γw, ∇_Γwi + 2λdivˆ _ΓeΓh∇Γw, ∇Γwiˆ +λh∇_Γw, ∇_Γwik∇ˆ _Γwk2+4µ ˆwtr((εΓ(eΓ) + VΓeΓ) D2b) +8µ(H2−2K)w ˆw + 2µk∇_Γwk2h∇_Γw, ∇_Γwi + 2µ ˆwtr(Dˆ 2b(∇_Γw ⊗ ∇_Γw)) +2µtr((∇Γw ⊗ ∇ˆ Γw + ∇Γw ⊗ ∇Γw)(εˆ Γ(eΓ) + VΓeΓ)) +2µwtr((∇_Γw ⊗ ∇ˆ _Γw + ∇_Γw ⊗ ∇_Γw) Dˆ 2b) +γ Z τ 0 Z Γ 2λ(1Γw)(1Γw) + 4µtr((Sˆ Γw + GΓw)(SΓw + Gˆ Γw)) = 0.ˆ

These equations are in line with [2, Section 7.3].

Lemma 1. The following identities hold true

tr(V_Γu∇Γv ⊗ ∇Γv) = 0 (38) Z Γ tr(ε_Γu ∇Γv ⊗ ∇Γv) = − Z Γ hdiv_Γ(∇v ⊗ ∇v), ui. (39)

Proof. The term tr(VΓu∇Γv ⊗ ∇Γv) can be written as

1 2tr(((D 2_{b u) ⊗ ∇b)(∇} Γv ⊗ ∇Γv) + (∇b ⊗ (D2b u))(∇Γv ⊗ ∇Γv)) =1 2tr(h∇b, ∇Γvi(D 2_{b u) ⊗ ∇} Γv + hD2b u, ∇Γvi(∇b ⊗ ∇Γv)) =1 2h∇b, ∇ΓvihD 2_{b u, ∇} Γvi + 1 2hD 2_{b u, ∇} Γvih∇b, ∇Γvi

and since h∇b, ∇Γvi = 0 we obtain the first identity. The second one can be obtained by writing

Z Γ tr(εΓu∇Γv ⊗ ∇Γv) = Z ΓεΓ u · ·(∇v ⊗ ∇v) = − Z Γ hdivΓ(∇v ⊗ ∇v), ui. 

4.4. PDE formulation of the model

In this subsection we are interested in showing why (12)governs the displacement of the shell. We go back to Eq.(19). We have

(M∂t te, ˆe)Γ = (2ρ(I + γ (D2b)2)∂t teΓ−ργ D2b∂t tw, ˆeΓ)Γ

+ [2ρ∂t tw − 2ργ 1Γ∂t tw + γρdivΓ(D2b∂t teΓ)] ˆw (40)

Z

Γ

C BM(e) · · B0M(e, ˆe) = (−divΓ(C BM(e)), ˆeΓ)Γ + [tr(C(BM(e))D2b) − divΓ(C(BM(e))∇Γw)] ˆw. (41)

Using [6, Lemma 2.3], we get Z

Γ

C BF(e) · · BF(ˆe) = [(λ + 2µ)12_Γw + 2µdivΓ(K ∇Γw) + 2µdivΓ((D2b)2∇Γw)] ˆw. (42)

Combining(40)–(42), we obtain        ρ∂t tw − ργ 1Γ∂t tw + (λ + 2µ)γ 12_Γw + ργ 2divΓ(D 2_b∂ t teΓ) + tr(C(BM(e))D2b)

−div_Γ(C(BM(e))∇Γw) + 2µγ divΓ(K ∇Γw) + 2µγ divΓ((D2b)2∇Γw) = 0

ρ(I + γ (D2_b)2_)∂ t teΓ −ρ γ 2D 2_b∂ t tw − divΓ(C BM(e)) = 0. (43)

The definition of BM (Definition 2) gives the desired result(12).

4.5. Total energy of the system

The total energy of the system is given by the sum of the kinetic and potential (elastic) energy:

E(e) = Ek(e) + Ep(e)

= ρl 2 Z Γ |∂_te_Γ|2+ |∂_tw|2+ρlγ 2 Z Γ |D2b∂teΓ|2+ |∇Γ∂tw|2+ |D2b∂teΓ− ∇Γ∂tw|2 +λl 2     2Hw + divΓeΓ + 1 2k∇Γwk 2     2 0,Γ +µl Z Γ tr "_ εΓ(eΓ) + D2bw + VΓeΓ + 1 2∇Γw ⊗ ∇Γw 2# +λl 3 24 |1Γw| 2 0,Γ+ µl3 12 Z Γ tr[(S_Γw + G_Γw)2]. (44) Note that E_{(e(t)) ≥ C(keΓ(t)k}2 H1_{(Γ )}+ k∂teΓ(t)k2_L2_{(Γ )}+ k∂tw(t)k2_H1_{(Γ )}+ kw(t)k2_H2_{(Γ )}). (45)

The sum of the kinetic energy with the bending (flexural) energy will be of some interest in Section8. It is defined as

EF(e) = Ek(e) + EpF(e) (46)

which can also be written as

EF_{(e) = E(e) −} l

2 Z

Γ

C(BM(e)) · · BM(e). (47)

Note that the nonlinear terms appear only in BM whose energy terms are in EpM. As a consequence the energy terms

Proposition 1. Let e be a regular solution to the small finite deflections shell problem; then

∀t ∈ [0, τ[, E(e(t)) = E(e(0)).

Proof. Since e ∈ C0([0, T [, V2(Γ )) ∩ C1([0, τ[, V1(Γ )) ∩ C2([0, τ[, V0(Γ )) one can replace ˆe by ∂te in(16); this

yields

m(∂t te, ∂te) + a(e, ∂te) − n(e, ∂te) = 0

and after some computations, it follows that

∂t[m(∂te, e)] + Z Γ C(BM(e)) · · B0M(e, ∂te) + γ Z Γ C(BF(e)) · · BF(∂te) = 0. Eqs.(37)yield ∂tEk(t) + ∂tEFp(t) + ∂tEpF(t) = 0

and therefore∂tE(t) = 0, which leads to the desired result. 

For the sake of simplicity, we shall not introduce new notation for the energy as a function of time and we shall use E(t) (resp. Ep(t), . . .) to refer to E(e(t)) (resp. Ep(e(t)), . . .).

5. Preliminary properties

The remainder of this paper is devoted to the proof ofTheorems 1–3. The properties established in this section will be used in the proofs of these theorems.

5.1. Properties of the operators

Lemma 2. We have hAX, XiH=0 for all X ∈ D(A).

Proof. We note that the inner product above is on the space H which is defined in Eq.(17). Let X = [e1e2_{]. Thus we}

have hAX, X iD(A1/2_)×D(M1/2₎ = −he2, e1i_D(A1/2₎+ hM−1Ae1, e2i_D(M1/2₎ = −hA1/2_e2_{, A}1/2_e1_i [L2(Γ )]3+ hM 1/2_M−1_Ae1_{, M}1/2_e2_i [L2(Γ )]3 = −he2, Ae1i [L2(Γ )]3+ hM 1/2_M1/2_M−1_Ae1_{, e}2_i [L2(Γ )]3 = −he2, Ae1i_[L 2(Γ )]3+ hAe 1_{, e}2_i [L2(Γ )]3 =0. 

Lemma 3. The bilinear form a(e, ˆe) defined in(14)is elliptic on V1(Γ ), that is, there exists a constant C > 0 such that

C kek2

V1_{(Γ )}≤a(e, e) ∀e ∈ V

1_{(Γ ). (48)}

In addition the bilinear form m(e, ˆe) is elliptic on the space V0(Γ ). This gives coercivity of the related linear operators A, M.

5.2. Norm estimates

Lemma 4. Letw ∈ H2_{(Γ ); then}

k∇_Γw ⊗ ∇_Γwk_L2_{(Γ )}≤C kwk_H2_{(Γ )}kwk_H1_{(Γ )}. (49)

Proof. We have

k∇_Γw ⊗ ∇_Γwk_L2_{(Γ )}≤ k∇_Γwk2 L4_{(Γ )}.

Using Sobolev’s embedding H1/2(Γ ) ⊂ L4(Γ ), we obtain k∇_Γw ⊗ ∇_Γwk_L2_{(Γ )}≤C k∇_Γwk2

H1/2_{(Γ )}

and the classical interpolation inequality k∇_Γwk2

H1/2(Γ )≤ k∇ΓwkH1(Γ )k∇ΓwkL2(Γ )yields(49). 

Let us denote by {ϕi}the family of the eigenvectors of the Laplace operator andλithe eigenvalues corresponding

toϕi (the eigenvaluesλnshould not to be confused with the Lam´e coefficientλ). Let Pnbe the orthogonal projection

on the subspace spanned by the n eigenvectorsϕ0, . . . , ϕn−1. If f ∈ L2and g ∈ H1, then the estimate resulting from

Sobolev’s embedding and the Holder inequality gives [30]: k(P_nf)gk_L2 ≤C p ln(1 + λn)k f kL2kgk_H1 (50a) k(P_ng) f k_L2 ≤C p ln(1 + λn)k f kL2kgk_H1 (50b) kPngkL∞≤C p ln(1 + λn)kgkH1. (50c) We define Qn=I − Pn.

Remark 3. The sequence(λn) is increasing and lim λn= +∞.

Lemma 5. Let > 0, α > 0 and f ∈ H+α; then kQnf k2H ≤ 1 λαkQnf k2_H+α. (51) Proof. We have Qnf = +∞ X i =n hf, ϕiiϕi

and, forη > 0, we have kQnf k2Hη=

+∞

X

i =n

hf, ϕii2ληi. (52)

Applying(52)withη =  yields kQnf k2H = +∞ X i =n hf, ϕii2λi ≤ sup i ≥n 1 λα i !_+∞ X i =n hf, ϕii2λ+αi .

Using that(λi) is an increasing sequence, we obtain

kQnf k2H ≤ 1 λα n +∞ X i =n hf, ϕii2λ+α_i .

Applying(52)withη =  + α yields kQnf k2H ≤

1 λα

n

Lemma 6. Let > 0, α > 0, f ∈ H+αand g ∈ H1; then k(Q_nf)gk2_L2 ≤ 1 λα n kf k2_H+αkgk2_H1. (53) Proof. We have k(Q_nf)gk2 L2 ≤ kQnf k2Hkgk2H1. Lemma 5gives k(Q_nf)gk2_L2 ≤ 1 λα n kQnf k2_H+αkgk2_H1. Using kQnf k2_H+α ≤ kf k2H+αwe obtain(53). 

Proposition 2. Let > 0, α > 0, f ∈ H+αand g ∈ H1; then kf gk2 L2 ≤C  ln(1 + λn)k f k2_L2 + 1 λα n kf k2_H+α  kgk2 H1. (54) Proof. We have kf gk2 L2 = k(Pnf)g + (Qnf)gk2_L2 ≤ k(Pnf)gk2_L2+ k(Qnf)gk2_L2

applying(50a)to k(Pnf)gkL2and(53)to k(Qnf)gkL2 gives(54). 

Corollary 1. Let > 0, α > 0, v1∈(H+α)3andv2∈(H1)3; then

khv₁, v₂ik2 L2 ≤C  ln(1 + λn)kv1k2_L2+ 1 λα n kv₁k2 H+α  kv₂k2 H1 (55) kv₁⊗v₂k2 L2 ≤C  ln(1 + λn)kv1k2_L2+ 1 λα n kv₁k2 H+α  kv₂k2 H1. (56)

Proof. These inequalities follow fromProposition 2for each term of hv1, v2ior each component ofv1⊗v2. 

6. Proof ofTheorem 1(existence of regular solutions)

In this section we proveTheorem 1, the existence and uniqueness of regular solutions to the problem(16). We shall use the nonlinear Galerkin method, essentially following the proof in [25] for the full von K´arm´an plate equation.

6.1. Semidiscrete approximation

Let h be a non-negative real number and V_h2(Γ ) be a finite dimensional subspace dense in V2(Γ ) as defined in(11). We consider the following semidiscrete approximation of (19). Given e0,h = (eh,0

Γ , wh,0) ∈ Vh2(Γ ) and eh,1=(eh,1 Γ , w 1,h_{) ∈ V}1 h(Γ ), find eh =(e h Γ, wh) ∈ V 2 h such that (M∂t teh, ˆe)Γ + Z Γ (C BM(eh)) · · B0M(eh, ˆe) + γ Z Γ

(C BF(eh)) · · BF(ˆe) = 0, ∀ ˆe ∈ Vh2(Γ ) (57a)

eh(0) = eh,0 (57b)

∂teh(0) = eh,1. (57c)

Let us define Eh(t) = E(eh) where E is defined in Eq. (44). Let Eh(t) denote the energy computed along the

semidiscrete solution eh. Since eh(t) belongs to V_h2(Γ ) it is a regular solution, soProposition 1applies.

Global existence and uniqueness of semidiscrete solutions follows from the fact that the nonlinear terms contained in BM are locally Lipschitz on V_h2(Γ ) together with the standard a priori bound(58). It follows that

k∂_tw_h(t)k2 H1+ k∂teΓ(t)k2_L2 + kwh(t)k2_H2 + k∂teΓ(t)k2_H1 ≤ Eh(t) ≤ Eh(0) (59) k∂_tw_hk L2 ≤ p Eh(0) (60)

for allτ > 0 such that the solution eh_{(t) ∈ C}∞_{([0, τ[, V}2 h).

6.2. Stability in higher norms

In the following equations, we shall use ¯u =∂tuh. Let us differentiate(57a)with respect to t ; then the following

equation holds true for all tests ˆe ∈ V_h2

(M∂t t¯e, ˆe) + Z Γ(C B 0 M(eh, ¯e)) · · B 0 M(eh, ˆe) + γ Z Γ(C B F(¯e)) · · BF(ˆe) = 0.

Replacing the test function by∂te inˆ (57a)yields a simplification in the equation above; it results in the following

equation holding true for all tests ˆe ∈ V_h2

(M∂t t¯e, ˆe) + Z Γ C B0 M(eh, ¯e) · · B 0 M(eh, ˆe) + Z Γ C BM(eh) · · (∇Γw ⊗ ∇ˆ Γw) + γ¯ Z Γ (C BF(¯e)) · · BF(ˆe) = 0.

We note that the differentiation of the nonlinear quantities results in extra terms in the equation. In order to see them we will write explicitly:

Z Γ C B0_M(eh, ¯e) · · B0_M(eh, ˆe) + Z Γ C BM(eh) · · (∇Γw ⊗ ∇ˆ Γw)¯ = Z Γ tr  C  εΓe¯Γ +VΓe¯Γ + ¯wD2b + 1 2∇Γw h_{⊗ ∇} Γw +¯ 1 2∇Γw ⊗ ∇¯ Γw h ×  εΓeˆΓ +VΓeˆΓ+ ˆwD2b + 1 2∇Γw h_{⊗ ∇} Γw +ˆ 1 2∇Γw ⊗ ∇ˆ Γw h  + Z Γ tr  C  εΓ(eΓ)h+VΓe_Γh +whD2b + 1 2∇Γw h_{⊗ ∇} Γwh  ∇_Γw ⊗ ∇ˆ _Γw
¯  . (61)

Taking ˆe =∂t¯e =∂t teh, we obtain

(M∂t t¯e, ∂te¯) + γ Z Γ trC S_γw + G¯ _Γw
(S¯ _Γ∂_tw + G¯ _Γ∂_tw)¯ + Z Γ tr  C  εΓe¯Γ +VΓe¯Γ + ¯wD2b + 1 2∇Γw h_{⊗ ∇} Γw +¯ 1 2∇Γw ⊗ ∇¯ Γw h  ×  εΓ∂te¯Γ+VΓ∂te¯Γ +∂twD¯ 2b + 1 2∇Γw h_{⊗ ∇} Γ∂tw +¯ 1 2∇Γ∂tw ⊗ ∇¯ Γw h  + Z Γ tr  C  εΓ(eΓ)h+VΓe_Γh +whD2b + 1 2∇Γw h_{⊗ ∇} Γwh  (∇Γ∂tw ⊗ ∇¯ Γw)¯  =0. (62) Since ∂tBM(eh) = ∂t  εΓ(eΓ)h+VΓeh_Γ +whD2b + 1 2∇Γw h_{⊗ ∇} Γwh  =ε_Γe¯Γ+VΓe¯Γ + ¯wD2b + 1 2∇Γw h_{⊗ ∇} Γw +¯ 1 2∇Γw ⊗ ∇¯ Γw h_=B0 M(¯e, eh) (63)

(M∂t te¯, ∂te¯) + Z Γ(C B 0 M(eh, ¯e)) · · ∂tB0M(eh, ¯e) + γ Z Γ(C BF(¯e)) · · ∂t BF(¯e) = Z Γ C B0_M(eh, ¯e) · · (∇Γw ⊗ ∇¯ Γw) −¯ Z Γ C BM(eh) · · (∇Γ∂tw ⊗ ∇¯ Γw).¯ (64) Let us define ¯ E(t) = kM∂tek¯ L2_{(Γ )}+ Z Γ C B_M0 (eh, ¯e) · · B_M0 (eh, ¯e) + γ Z Γ C BF(¯e) · · BF(¯e). (65) We have kM∂te¯(t)kL2_{(Γ )}+ k ¯w(t)k_H2+ k ¯e_Γ(t)k_H1 ≤ ¯E(t). (66)

It is straightforward to verify that ¯E(t) is bounded above and below by a constant depending on k¯ek_V1_{(Γ )} and

k∂_tek¯ _V0_{(Γ )}. This fact will be used frequently and without further mention. Eq.(64)can be rewritten as

1 2∂t ¯ E(t) = Z Γ C B0 M(eh, ¯e) · · (∇Γw ⊗ ∇¯ Γw) −¯ Z Γ C BM(eh) · · (∇Γ∂tw ⊗ ∇¯ Γw)¯ = Z Γ tr  C  εΓe¯Γ +VΓe¯Γ + ˆwD2b + 1 2∇Γw h_{⊗ ∇} Γw +¯ 1 2∇Γw ⊗ ∇¯ Γw h  (∇Γw ⊗ ∇¯ Γw)¯  − Z Γ tr  C  εΓehΓ+VΓehΓ+w h_D2_{b +}1 2∇Γw h_⊗ 1 2∇Γw h_(∇ Γ∂tw ⊗ ∇¯ Γw)¯  . (67) Integrating this equation between 0 and t yields

1 2( ¯E(t) − ¯E(0)) = Z t 0 Z Γ C B0 M(eh, ¯e) · · (∇Γw ⊗ ∇¯ Γw) −¯ 1 2 Z t 0 Z Γ C BM(eh) · · ∂t(∇Γw ⊗ ∇¯ Γw).¯ (68)

Integrating by parts the second term of the right hand side of(68)gives

¯ E(t) = ¯E(0) + 3 Z t 0 Z Γ C B0_M(eh, ¯e) · · (∇Γw ⊗ ∇¯ Γw) +¯ Z Γ(C B M(eh)) · · (∇Γw ⊗ ∇¯ Γw)¯ t 0 . (69) Therefore ¯ E(t) ≤ ¯E(0) + 3 2 Z t 0  kC B0_M(eh, ¯e)k2_L2_{(Γ )}+ k∇Γw ⊗ ∇¯ Γwk¯ 2_L2_{(Γ )}  ds + 1 4kC BM(eh)k2_L2_{(Γ )}+k∇Γw ⊗ ∇¯ Γwk¯ 2 L2_{(Γ )} t 0 (70)

where is an arbitrarily small non-negative number.

FromLemma 4we have k∇Γw ⊗ ∇¯ Γwk¯ L2_{(Γ )}≤C k ¯wk_H2_{(Γ )}k ¯wk_H1_{(Γ )}. Using(45)with(58)gives k ¯wk_H1_{(Γ )} ≤

Eh(0), and then using(66), we obtain

k∇ ¯w ⊗ ∇ ¯wk2

L2_{(Γ )}≤C Eh(0)k ¯wk2_H2_{(Γ )}≤C Eh(0) ¯E(t) (71)

where C is a constant. In addition, the following inequalities follow from the fact that tensor C is coercive and the a prioribound on the energies:

Let us now set = 1/(2C Eh(0)); then ¯ E(t) ≤ ¯E(0) + C(1 + Eh(0)) Z t 0 ¯ E(s) ds +1 2(C Eh(0)) 2₊1 2 ¯ E(t) (74) and therefore ¯ E(t) ≤ C( ¯E(0) + (Eh(0))2) + C(1 + Eh(0)) Z t 0 ¯ E(s) ds. (75)

Gronwall’s lemma yields

∀t ∈ [0, τ[, E¯(t) ≤ C( ¯E(0) + [Eh(0)]2) exp(C(1 + Eh(0))τ) (76)

where C does not depend on h.

6.3. A priori bounds for the discrete initial data

In order to provide an effective estimate (independent of h) of the right hand side of Eq.(76)we need to estimate Eh(0) and ¯E(0). This is the point of the next two sections.

6.3.1. A priori bounds for Eh(0)

We consider e0,h and e1,h “good” approximations of the initial data, that is lim h→0ke 0_−eh,0_k V2_{(Γ )}=0 and lim h→0ke 1_−eh,1_k V1_{(Γ )}=0. (77)

By the stability result of the estimates resulting from (77) and the regularity of the initial condition, we obtain Eh(0) ≤ C2.

6.3.2. A priori bounds for ¯E(0)

Using H1(Γ ) ⊂ L6(Γ ) ⊂ L4(Γ ) as well as k ˆwk_H1_{(Γ )}≤ kˆek_V0_{(Γ )}and k ˆe_Γk_L2_{(Γ )}≤ kˆek_V0_{(Γ )}we get

(∂te¯(0), ˆe)V0_{(Γ )} ≤ C(keh,0k_V0_{(Γ )}+ keh,0k_V1_{(Γ )}+ keh,0k_V2_{(Γ )})kˆek_V0_{(Γ )} (81a)

≤ C keh,0k_V2_{(Γ )}kˆek_V0_{(Γ )}. (81b)

For h small enough, keh,0k_V2_{(Γ )}≤2ke0k_V2_{(Γ )}, and it follows that

(∂te¯(0), ˆe)V0_{(Γ )}≤C ke0k_V2_{(Γ )}kˆek_V0_{(Γ )} (82)

where C is a constant independent of h; therefore

k∂_te¯(0)k_V0_{(Γ )}≤C ke0k_V2_{(Γ )}. (83)

Using k¯e(0)k_V2_{(Γ )}≤C and(83)we conclude that ¯E(0) is bounded by a constant independent of h. Eq.(76)yields,

for all t< τ, ¯

E(t) ≤ C. (84)

Hence the a priori bounds

kehk_V1_{(Γ )}+ k∂tehkV1_{(Γ )}+ k∂t tehkV0_{(Γ )}≤C (85)

hold uniformly in h, for all t< τ where τ is arbitrary. 6.4. Passage to the limit

We can now extract convergent subsequences, denoted by the same symbol, such that the following weak∗ convergence holds:

eh→e in L∞(0, τ; V1(Γ )) (86a)

∂teh→∂te in L∞(0, τ; V1(Γ )) (86b)

∂t teh→∂t te in L∞(0, τ; V0(Γ )). (86c)

Now, we can pass to the limit on the original semidiscrete form of Eq.(57a). Note that this is possible due to the weak continuity of ∇_Γw ⊗ ∇_Γw. The clamped boundary condition also passes to the limit thanks to the strong convergence of the boundary traces. Therefore, we can conclude that e satisfies the weak form of the original equation

(16). Moreover, it has the following regularity

e ∈ C(0, τ, V1(Γ )), ∂te ∈ C(0, τ, V1(Γ )), ∂t te ∈ C(0, τ, V0(Γ )). (87)

In order to obtain existence of weak solutions, we use the a priori bound for the original energy function which implies weak convergence of

eh→e in L∞(0, τ; V1(Γ )) (88)

∂teh→∂te in L∞(0, τ; V0(Γ )). (89)

The above weak convergence and weak continuity of nonlinear terms allows passing to the limit. This part of the argument is straightforward. The limit equation defines a weak solution.

6.5. The improved spatial regularity

The regularity established in Section6.4gives the following regularities:

∂t teΓ ∈ [L2(Γ )]2, ∂t teΓ ∈ [L2(Γ )]2, divΓ(C(wD2b)) ∈ [H1(Γ )]2.

In addition ∇_Γw ⊗ ∇_Γw is in [H1−(Γ )]2as the product of two functions in [H1(Γ )]2. It follows that div_Γ(C(∇_Γw ⊗ ∇_Γw)) ∈ [H−(Γ )]2

and consequently divΓ(C(εΓeΓ +VΓeΓ)) ∈ [H−(Γ )]2so

eΓ ∈C(0, τ; [H2−(Γ )]2). (91)

Forw, the first equation of(12)gives (λ + 2µ)γ 12 Γw = −ρ∂t tw + ργ 1Γ∂t tw − ρ γ 2divΓ(D 2_b∂ t teΓ) − tr(C(εΓeΓ +VΓeΓ)D2b) −tr  C  wD2_{b +}1 2∇Γw ⊗ ∇Γw  D2b  +div_Γ(C(ε_Γe_Γ +V_Γe_Γ)∇_Γw) +div_Γ  C  wD2_{b +}1 2∇Γw ⊗ ∇Γw  ∇_Γw  −2µγ div_Γ(K ∇_Γw) − 2µγ div_Γ((D2b)2∇w). (92) We now inspect the regularity of each term of(92). This regularity follows from Section6.4for all but the fifth term:

∂t tw ∈ H1(Γ ), 1Γ∂t tw ∈ H−1(Γ ), divΓ(D2b∂t teΓ) ∈ H−1(Γ ) tr(C(εΓeΓ +VΓeΓ)D2b) ∈ L2(Γ ), tr  C  wD2_{b +}1 2∇Γw ⊗ ∇Γw  D2b  ∈ H1−(Γ ) divΓ  C  wD2_{b +}1 2∇Γw ⊗ ∇Γw  ∇_Γw  ∈H−(Γ ), divΓ(K ∇Γw) ∈ L2(Γ ), divΓ((D2b)2∇w) ∈ L2(Γ )

where we have used that ∇w ∈ [H1(Γ )]2implies ∇Γw ⊗ ∇Γw ∈ [H1−(Γ )]2. The regularity of the sixth term of

(92)follows from(91). We have C(εΓeΓ +VΓeΓ) ∈ H1−(Γ ) and ∇Γw ∈ [H1(Γ )]2so the product is in H1−(Γ ),

div_Γ(C(ε_Γe_Γ +V_Γe_Γ)∇_Γw) ∈ [H−(Γ )]2 and consequently12_Γw belongs to H−1(Γ ) so

w ∈ C(0, τ; H3_{(Γ )). (93)}

Going back to(90)we further improve the regularity with, obtaining eΓ ∈C(0, τ; [H2(Γ )]2). It follows that

e ∈ C(0, τ, V2(Γ )) (94)

as desired for the proof of the existence of regular solutions.

7. Proof ofTheorem 2(uniqueness property)

In order to proveTheorem 2we use a method developed by V.I. Sedenko for Marguerre–Vlasov equations and then utilized for the von K´arm´an equations in [25]. Let e1and e2be two weak solutions to the small finite deflections shell problem, with the same initial condition. Let ˜e = e2−e1. The goal of this section is to prove that ˜e = 0.

7.1. First step

Proof. Eq.(20)yields ∂tA−1  _˜ e ∂te˜  + AA−1  _˜ e ∂te˜  = A−1  0 M−1[N(e2) − N(e1)]  (96) and consequently  ∂tA−1  ˜ e ∂te˜  , A−1  ˜ e ∂te˜  H +  AA−1  ˜ e ∂te˜  , A−1  ˜ e ∂te˜  H =  A−1  ₀ M−1 γ [N(e2) − N(e1)]  , A−1  _˜ e ∂te˜  H . Within this proof, we shall define

X(s) = A−1  _˜ e(s) ∂te˜(s)  and F(s) = A−1  0 M−1[N(e2) − N(e1)]  . UsingLemma 2on(96)yields

(∂tX(s), X (s))H=(X (s), F(s))H and therefore 1 2 Z t 0 ∂ skX(s)k2_Hds = Z t 0 (X (s), F(s))H ds

whence it follows that 1 2kX(t)k 2 H≤ 1 2kX(0)k 2 H+ Z t 0 kX(s)kHkF(s)kHds.

Since ˜e(0) = ∂te(0) = 0 and A−1is linear, we have X(0) = 0; it follows that

1 2kX(t)k 2 H≤ sup s∈[0,t] kX(s)kH ! Z t 0 kF(s)kHds 

and thus, for all ¯t ≤ t , we have

               1 2kX(¯t)k 2 H≤ sup s∈[0,¯t] kX(s)kH ! Z t¯ 0 kF(s)kHds ! sup s∈[0,¯t] kX(s)k2_H≤ sup s∈[0,t] kX(s)k2_H Z t¯ 0 kF(s)kHds ≤ Z t 0 kF(s)kHds

whence it follows that

and using kX(t)kH≤sups∈[0,t]kX(s)kH, we get kX(t)kH≤ Z t 0 kF(s)kHds, that is(95). _

Lemma 8. The following inequality holds true

kM1_γ/2ek˜ _L2 ≤ A −1  _˜ e ∂te˜  _H. (97) Proof. We have A−1 e_∂ t˜e  =A −1_M γ∂te −˜e  and therefore A −1  _˜ e ∂te  2 H = kA−1/2M_γ∂tek˜ 2_L2 + kM1/2γ ek˜ 2_L2. Inequality(97)follows. 

Proposition 3. The following inequality holds true

kM1/2_γ ek˜ _L2 ≤ Z t 0 kA−1/2[N(e2(s)) − N(e1)(s)]k_L2ds. (98) Proof. We have A−1  ₀ M−1 γ [N(e2) − N(e1)]  =A −1_[N(e2_{) − N(e}1_)] 0  and therefore A −1  ₀ M−1_γ [N(e2) − N(e1)]  H = kA−1/2[N(e2) − N(e1)]k_L2.

FromLemma 7, we get A −1 e ∂te  _H ≤ Z t 0 kA−1/2[N(e2) − N(e1)]k_L2ds. Lemma 8yields kM1_γ/2ek˜ _L2 ≤ Z t 0 kA−1/2[N(e2) − N(e1)]k_L2ds which proves(98). _

Corollary 2. FollowingProposition3, we obtain

k ˜eΓk2_L2+ k ˜wk2_H1 ≤t

Z t

0

(kNΓ(w2) − NΓ(w1)k2_H−1+ kNn(e_Γ2, w2) − Nn(e1_Γ, w1)k2_H−2)ds. (99)

7.2. Second step Lemma 9. We have NΓ(w2) − NΓ(w1) = λ∇Γ(h∇Γw, ∇˜ Γ(w2+w1)i) +2µdiv_Γ(∇_Γw ⊗ ∇˜ _Γw2+ ∇_Γw1⊗ ∇_Γw)˜ (100) Nn(e2_Γ, w2) − Nn(e1_Γ, w1) = −2Hλh∇Γw, ∇˜ Γ(w2+w1)i +4λdiv_Γ(Hw2∇_Γw + H ˜w∇˜ _Γw1) + 4µdiv_Γ(D2b(w2∇_Γw + ˜w∇˜ _Γw1)) +(λ + 2µ)div_Γ(k∇_Γw2k2∇_Γw + ∇˜ _Γw1h∇_Γw, ∇˜ _Γ(w2+w1)i)

−2µhD2b∇Γw, ∇˜ Γw2i −2µhD2b∇Γw1, ∇Γwi − 2λdiv˜ Γ(divΓe˜Γ∇Γw2+divΓe_Γ1∇Γw)˜

−4µdivΓ((εΓeΓ2 +VΓe2Γ)∇Γw + (ε˜ Γe˜Γ+VΓe˜Γ)∇Γw1). (101)

Proposition 4. Let > 0 and α > 0 be such that  + α ≤ 1. There exists a real C depending only on λ, µ, w1and w2_{such that} kNΓ(w2) − NΓ(w1)k2_H−1 ≤Cln(1 + λn)k ˜wk2_H1+ C λα n . (102)

Proof. FollowingDefinition 7gives

kNΓ(w2) − NΓ(w1)k2_H−1 ≤ λk∇Γh∇Γw, ∇˜ Γ(w2+w1)ik2_H−1

+2µkdiv_Γ(∇_Γw ⊗ ∇˜ _Γw2+ ∇_Γw1⊗ ∇_Γw)k˜ 2

H−1. (103)

Since divΓ is a bounded operator on H−1, there exists a constant C such that

kN_Γ(w2) − N_Γ(w1)k2

H−1 ≤ λ 2_{C kh∇}

Γw, ∇˜ Γ(w2+w1)ik2_L2

+4µ2C k∇Γw ⊗ ∇˜ Γw2+ ∇Γw1⊗ ∇Γwk˜ 2_L2. (104)

Applying(55)withv1= ∇Γw and v˜ 2= ∇Γ(w2+w1) yields

kh∇_Γw, ∇˜ _Γ(w2+w1)ik2 L2 ≤C  ln(1 + λn)k ˜wk2_H1+ 1 λα n kw2−w1k2 H1++α  kw2+w1k2 H2. (105)

Applying(56)withv1= ∇Γw and v˜ 2= ∇Γw2yields

k∇_Γw ⊗ ∇˜ _Γw2k2 L2 ≤C  ln(1 + λn)k ˜wk2_H1 + 1 λα n kw2−w1k2 H1++α  kw2k2 H2. (106)

Applying(56)withv1= ∇Γw and v˜ 2= ∇Γw1yields

k∇_Γw1⊗ ∇_Γwk˜ _L2 ≤C  ln(1 + λn)k ˜wk2_H1 + 1 λα n kw2−w1k2 H1++α  kw1k2 H2. (107)

These inequalities, andw1andw2belonging to H2, give(102). _

Proposition 5. Let > 0 and α > 0 be such that  + α ≤ 1. There exists a real C depending only on λ, µ, w1_,w2_,

e_Γ1, e2_Γ, and b, such that

kNn(e_Γ2, w2) − Nn(e_Γ1, w1)k2_H−2 ≤Cln(1 + λn)k ˜wk2_H1+Cln(1 + λn)k˜eΓk2_L2+

C λα n

Proof. Let us define

X1( ˜w, w1, w2) = −2λHh∇_Γw, ∇˜ _Γ(w2+w1)i + 4λdiv_Γ(Hw2∇_Γw + H ˜w∇˜ _Γw1) +4µdivΓ(D2b(w2∇Γw + ˜w∇˜ Γw1))

−2µhD2b∇Γw, ∇˜ Γw2i −2µhD2b∇Γw1, ∇Γwi˜

+(λ + 2µ)div_Γ(∇_Γwk∇˜ _Γw2k2+ ∇_Γw1h∇_Γw, ∇˜ _Γ(w2+w1)i) X2( ˜w, ˜eΓ, w1, eΓ1, w2, e2Γ) = −2λdivΓ(divΓe˜Γ∇Γw2+divΓe1Γ∇Γw)˜

−4µdiv_Γ((ε_Γe_Γ2 +VΓe2Γ)∇Γw + (ε˜ Γe˜Γ+VΓe˜Γ)∇Γw1)

so we have

Nn(e2_Γ, w2) − Nn(e_Γ1, w1) = X1( ˜w, w1, w2) + X2( ˜w, ˜eΓ, w1, eΓ1, w2, e2Γ) (109)

and therefore

kNn(e2_Γ, w2) − Nn(e_Γ1, w1)kH−2 ≤ kX1( ˜w, w1, w2)k_H−2+ kX2( ˜w, ˜e_Γ, w1, e1_Γ, w2, e_Γ2)k_H−2. (110)

Estimation of the first term of the right hand side of (110)

kX1( ˜w, w1, w2)k_H−2 ≤C(kh∇_Γw, ∇˜ _Γ(w2+w1)ik_H−2+ kw2∇_Γwk˜ _H−1

+ k ˜w∇_Γw1k_H−1+ khD2b∇_Γw, ∇˜ _Γw2ik_H−2 + khD2b∇_Γw1, ∇_Γwik˜ _H−2

+ kdiv_Γ((ε_Γe2_Γ +VΓe2Γ)∇Γw + (ε˜ Γe˜Γ +VΓe˜Γ)∇Γw1)kH−2).

Bounding kX1( ˜w, w1, w2)k_H−2is an easy task, because all the terms are of lower order. For instance,

h∇Γw, ∇˜ Γ(w 2_+w1_)i H−2 ≤ C kh∇Γw, ∇˜ Γ(w 2_+w1_)ik L2 ≤ C k∇Γwk˜ L2k∇_Γ(w2+w1)k_L2 ≤ C k ˜wk_H1kw2+w1k_H1 ≤ C k ˜wk_H1

where C depends only on w1andw2. An analogous argument is repeated for the four other terms. Eventually, we obtain that

kX1( ˜w, w1, w2)k_H−2 ≤C k ˜wk_H1 (111)

where C depends only onw1,w2, the Lam´e coefficients and b. Estimation of the second term of the right hand side of (110)

kX2( ˜w, ˜e_Γ, w1, e1_Γ, w2, e_Γ2)k_H−2 ≤ C(kdiv_Γ(div_Γe˜_Γ∇_Γw2+div_Γe_Γ1∇_Γw)k˜ _H−2

+ kdivΓ((εΓeΓ2 +VΓe2Γ)∇Γw + (ε˜ Γe˜Γ+VΓe˜Γ)∇Γw1)kH−2)

≤ C(kdiv_Γe˜Γ∇Γw2kH−1+ kdiv_Γe1_Γ∇_Γwk˜ _H−1

+ k(ε_Γe2_Γ+V_Γe_Γ2)∇_Γwk˜ _H−1 + k((ε_Γe˜_Γ+V_Γe˜_Γ)∇_Γw1)k_H−1).

Let us look first at the term kdivΓe˜Γ∇Γw2kH−1. Using Green’s formula for tangential derivatives(5)gives

kdivΓe˜Γ∇Γw2kH−1 = sup {ϕ, kϕk H 1₀=1} Z Γ(divΓ ˜ eΓ∇Γw2, ϕ)H−1_×H1 0 = sup {ϕ, kϕk H 1 0 =1} Z Γ −h ˜eΓ, ∇Γ(∇Γw2, ϕ)H−1_×H1 0 i +2 Z Γ H(∇Γw2, ϕ)H−1_×H1 0 h ˜eΓ, ∇bi + Z ∂Γ h∇_Γ(∇_Γw2, ϕ) H−1_×H1 0 ˜ eΓ, νi = − sup {ϕ, kϕk H 1₀=1} Z Γ

since the shell is clamped on the boundary and h ˜eΓ, ∇bi = 0. Using Cauchy–Schwarz and the inequality(56)we have     Z Γ h ˜e_Γ, D2_Γw2ϕi     =     Z Γ D2_Γw2 · ·(˜e_Γ ⊗ϕ)     ≤ kD2_Γw2k_L 2k ˜eΓ ⊗ϕkL2 ≤ C |w2|_H2  ln(1 + λn)k˜eΓkL2+ 1 λα n k ˜eΓkH+α  kϕk_H1

and the second term can be bounded as well:     Z Γ h ˜eΓ, ∇Γw2DΓϕi     =     Z Γ DΓϕ · · ˜eΓ⊗ ∇Γw2     ≤ kDΓϕkL2k ˜eΓ ⊗ ∇Γw 2_k L2 ≤ C kϕk_H1  ln(1 + λn)k˜eΓkL2+ 1 λα n k ˜e_Γk_H+α  kw2k H2 so that kdiv_Γe˜_Γ∇_Γw2k H−1 ≤C  ln(1 + λn)k˜eΓkL2+ 1 λα n k ˜e_Γk_H+α  kw2k H2. (112)

The second and third terms are more direct

kdivΓe1Γ∇Γwk˜ H−1 = sup {ϕ, kϕk H 1₀=1} Z Γ divΓe1Γh∇Γw, ϕi˜ ≤ sup {ϕ, kϕk H 1₀=1} kdivΓe1ΓkL2kh∇Γw, ϕik˜ L2 ≤ C  ln(1 + λn)k ˜wkH1+ 1 λα n k ˜wk_H1++α  ke1_Γk_H1 (113) and k(ε_Γ(e2_Γ) + V_Γe2_Γ)∇Γwk˜ H−1 = sup {ϕ, kϕk H 1₀=1} Z Γ(εΓ(e 2 Γ) + VΓeΓ2) · · (∇Γw ⊗ ϕ)˜ ≤ sup {ϕ, kϕk H 1₀=1} k(ε_Γ(e2_Γ) + V_Γe_Γ2)kL2k∇Γw ⊗ ϕk˜ L2 ≤ C  ln(1 + λn)k ˜wkH1 + 1 λα n k ˜wk_H1++α  ke_Γ2k_H1. (114)

so that we have     Z Γ(εΓ(˜eΓ) + VΓ(˜eΓ)) · · (∇Γw 1_⊗ϕ)     ≤ 1 2     Z Γ

heΓ, divΓ(∇Γw ⊗ ϕ) + divΓ(ϕ ⊗ ∇Γw)i

    +1 2   h2H ˜eΓ +D 2_{b ˜e} Γ, ∇Γw1ihϕ, ∇bi    ≤ 1 2 Z Γ

|h ˜e_Γ, D_Γ2w1ϕi + h˜e_Γ, ∇_Γw1div_Γϕi + h ˜e_Γ, D_Γϕ∇_Γw1i + h ˜e_Γ, 1_Γw1ϕi| +C Z Γ h ˜eΓ, ∇Γw1ihϕ, ∇bi ≤ C(kD2_Γw1k_L 2k ˜eΓ ⊗ϕkL2+ kdivΓϕkL2kh ˜eΓ, ∇Γw 1_ik L2 + kDΓϕkL2k ˜eΓ ⊗ ∇Γw 1_k L2+ k1Γw 1_k L2kh ˜eΓ, ϕikL2 + kh ˜eΓ, ∇Γw1ikL2khϕ, ∇bikL2)

and after application of Eqs.(55)and(56)we have

k(ε_Γe˜Γ) + (VΓe˜Γ)∇Γw1kH−1 ≤C  ln(1 + λn)k˜eΓkL2+ 1 λα n k ˜eΓkH+α  kw1k_H2. (115)

Finally, collecting equations(112)through(115)allows us to estimate the second term as kX2( ˜w, ˜eΓ, w1, eΓ1, w2, eΓ2)kH−2 ≤ Cln(1 + λn)k ˜wkH1+Cln(1 + λn)k˜eΓkL2 + C λα n k ˜wk_H1++α+ C λα n k ˜eΓkH+α (116)

where C depends only onw1,w2, the Lam´e coefficients and b. Conclusion:From(110),(111)and(116), we derive

kNn(e2_Γ, w2) − NΓ(eΓ1, w1)k2H−2 ≤Cln(1 + λn)k ˜wk2_H1+Cln(1 + λn)k˜eΓk2_L2+

C λα n

which is the announced result.  7.3. Third step

From now on we will considerα = 1 − . This choice of α complies with the requirements ofProposition 4and

Proposition 5.

Lemma 10. The following inequality holds true

k ˜e_Γk2

L2+ k ˜wk2_H1 ≤C t2λC t 2_−α

n (117)

where C depends only one1,e2, the Lam´e coefficients and the geometric information embedded in b.

Proof. Propositions 3–5give

Gronwall’s inequality yields k ˜e_Γk2 L2 + k ˜wk2_H1 ≤ C t2 λα n expC t2ln(1 + λn)  (120) and consequently k ˜eΓk2_L2 + k ˜wk2_H1 ≤ C t2 λα n (1 + λn)C t 2 (121) Eq.(117)follows. _ Corollary 3. We have ˜e = 0.

Proof. The sequence(λn) tends toward +∞. When t ∈

h

0,qα_Ch, the right hand side of(117)tends to 0. Consequently ˜

e = 0 onh0,q_Cαh. The bootstrap argument completes the proof of the corollary and the proof ofTheorem 2. _ Remark 4. The proof of uniqueness does not give the continuous dependence, which requires a separate argument. This is the point of the next section of this paper.

8. Proof ofTheorem 3(continuous dependence with respect to the initial data)

The main technical difficulty in the proof ofTheorem 3is the derivation of the energy identity for weak solutions. We denote as B([0, τ[; X) the space of X-valued functions which are bounded on the interval [0, τ[. This space is endowed with the norm |x|B([0,τ[;X)=supt ∈[0,τ[|x(t)|X.

8.1. Energy identity result

Lemma 11. We consider solutions to(16)with the following a priori regularity: e ∈ B([0, τ[; V1(Γ )), ∂te ∈ B([0, τ[; V0(Γ )).

Then, the following energy identity holds:

E(t) = E(s), 0 ≤ s ≤ t ≤ τ (122)

whereE is defined in(44).

Note thatProposition 1cannot be applied as is because of the lack of regularity of e. In order to prove the energy identity we shall use a finite difference approximation of time derivatives (the same approximation as was used in [1,

18] for the von K´arm´an plate).

Let h> 0 be a small parameter designed to go to zero. Let g ∈ B([0, τ[; X) where X is a Hilbert space. We extend g(t) to all t ∈ R by defining g(t) = g(0) for t ≤ 0 and g(t) = g(τ) for t ≥ τ. With these extensions we define three finite difference operators depending on the parameter h.

g_h+(t) ≡ g(t + h) − g(t) g_h−(t) ≡ g(t) − g(t − h) ∂hg(t) ≡ 1 2hg + h(t) + g − h(t) . (123)

Proposition 6. The following properties hold true: (i) Let g be weakly continuous with values in X . Then

(ii) Let g ∈ H1([0, τ[; X). Then the following limits are well defined in L2([0, τ[; X). lim h→0∂hg =∂tg; h→0lim 1 hg + h =∂tg; _h→0lim 1 hg − h =∂tg.

Moreover if ∂tg is weakly continuous with the values in X , then for every t ∈(0, τ), ∂hg(t) → ∂tg(t), weakly in

X , and 1 hg − h(τ) → ∂tg(τ); 1 hg + h(0) → ∂tg(0); weakly in X.

(iii) In addition to previous assumptions, let V ⊂ X ⊂ V0, ∂t tg ∈ L2([0, τ[, V0); g ∈ L2([0, τ[, V ), then lim h→0 Z τ 0 (∂t tg(t), ∂hg(t))Xdt = 1 2[k∂tg(τ)k 2 X− k∂tg(0)k2X].

Proof. The proof ofProposition 6is elementary and the main steps are detailed in [18]. _ We return to the proof ofLemma 11.

Proof of Lemma 11. Eq.(47)gives EF_{(t) ≡ E(t) −} l

2 Z

Γ(C BM

e, BMe). (124)

Recall that the definition of EF includes the kinetic energy of the shell as well as the flexural terms from the potential energy Ep(see(46)and(35)for a definition of these quantities).

EF_{(t) =} ρl 2 Z Γ |∂_te_Γ|2+ |∂_tw|2+ρlγ 2 Z Γ |D2b∂teΓ|2+ |∇Γ∂tw|2+ |D2b∂teΓ − ∇Γ∂tw|2 +λγ l 2 h1Γw, 1Γwiˆ Γ +µγ l Z Γ tr((S_Γw + G_Γw)(S_Γw + Gˆ _Γw)).ˆ (125) We will use the variational form(16)with the test functions∂he =∂heˆΓ+∂hw∇b ∈ Vˆ 1(Γ ). For reasons of readability

we omit the details of the calculation, but this choice of test functions, dividing by 2, and the first three formulas of

Proposition 6easily give the following identity: 1 2E F_{(τ) + lim} h→0 Z τ 0 Xhdt = 1 2E F_{(0) (126)}

where, in terms of the tensor BM, we have

Xh≡ Z Γ C BM(e) · · (ε(∂heΓ) + VΓ(∂heΓ) + D2b∂hw + ∇Γw ⊗ ∇Γ∂hw). (127) We note that Z Γ C BMe · ·∂hBMe = Z Γ C BM(e) · · (ε(∂heΓ) + VΓ(∂heΓ) + D2b∂hw + ∇Γw ⊗ ∇Γ∂hw) (128)

so that we can rewrite Xhas

Xh= Z Γ C BMe · ·∂hBMe − Z Γ C BMe · ·  1 2∂h(∇Γw ⊗ ∇Γw) − ∇Γw ⊗ ∇Γ∂hw  . (129)

Since C(BMe) is a symmetric tensor we have C BM(e) · ·  1 2∂h(∇Γw(t) ⊗ ∇Γw(t)) − ∇Γw ⊗ ∇Γ∂hw(t)  =1 2C BMe · ·(∂h∇Γw(t) ⊗ ∇Γw(t + h) − ∇Γw ⊗ ∇Γ∂hw(t)) +1 2C BMe · ·(∇Γw(t − h) ⊗ ∂h∇Γw(t) − ∇Γw ⊗ ∇Γ∂hw(t)) =1 2C BMe · ·(∂h∇Γw(t) ⊗ ∇Γ(w(t + h) − ∇Γw(t))) +1 2C BMe · ·(∂h∇Γw(t) ⊗ ∇Γ(w(t − h) − ∇Γw(t))) =1 2C BMe · ·(∂h∇Γw(t) ⊗ (∇Γw + h(t) − ∇Γw − h(t))).

From these equations we see that

Xh= Z Γ C BMe · ·∂hBMe − 1 2 Z Γ C BMe · ·(∂h∇Γw(t) ⊗ (∇Γw+h(t) − ∇Γw−h(t))) (130)

and applying the first identity ofProposition 6together with weak continuity of BMe we have

lim h→0 Z τ 0 Xhdt = 1 2kC 1/2_B Me(τ)k2L2(Γ )− 1 2kC 1/2_B Me(0)k2L2(Γ )− 1 2h→0lim Z τ 0 Yhdt (131) with Yh≡ Z Γ C BMe · ·(∂h∇Γw(t) ⊗ (∇Γwh+(t) − ∇Γw − h(t))). (132)

Our goal is to show that

lim

h→0

Z τ

0

Yh=0. (133)

Once this is shown, the energy identity will follow from combining(126)and(131). Eqs.(131)and(133)give Z τ 0 Xhdt → 1 2kC 1/2_B Me(τ)k2_{L2(Γ )}− 1 2kC 1/2_B Me(0)k2_{L2(Γ )} (134)

and thus from(126)we have EF_{(τ) + kC}1/2_B

Me(τ)k2_{L2(Γ )}=EF(0) + kC1/2BMe(0)k2_{L2(Γ )} (135)

which gives the result ofLemma 11for points s = 0, t =τ. Other points in the interval are treated the same way, due to the fact that the argument is local.

It remains only to show that the Eq.(133) holds. The a priori regularity of weak solutions gives that BMe ∈

B([0, τ[; L2(Γ )) so that the assertion follows if

Z τ 0 k∂_h∇_Γw ⊗ [∇_Γw+ h − ∇Γw − h]kL2(Γ )→0 as h → 0. (136)

Applying Cauchy–Schwarz lets us define

Zh= Z τ 0 hk∂h∇Γwk2L4(Γ )+h −1_(k∇ Γwh+k 2 L4(Γ )+ k∇Γw − hk 2 L4(Γ )) dt. (137)

The fact that Zh→0 as h → 0 follows by a density argument after we show that

We apply Sobolev’s inequality kgkL4(Γ ) ≤C kgkW1,1_{(Γ )}which is known to be true for the tangential derivatives on the shell mid-surface [11]. Letting g = k∂h∇Γwk2gives

hk∂h∇Γwk2L4 ≤C hk∂h∇Γwk 2 W1,1 ≤C hk∇Γ|∂h∇Γw|2kL1+C hk∂h∇Γwk 2 L1 ≤C h Z Γ |∇_Γh∂_h∇_Γw, ∂_h∇_Γwi| + Ch Z Γ k∂_h∇_Γwk2 ≤C hk∂h∇ΓwkL2(Γ )k∂h∇ΓwkH1_{(Γ )}+C h Z Γ k∂_h∇_Γwk2. Using the fact that h∂his bounded then gives that

hk∂h∇Γwk2L4 ≤2Ck∇ΓwkH1(Γ )k∂h∇ΓwkL2(Γ ). (139)

The argument is similar for the other terms. _ 8.2. Completion of the proof

Let us define

x(t) = (e, ∂te) = (eΓ(t), w(t), ∂teΓ(t), ∂tw(t))

where e = e_Γ(t) + w(t)∇b is a weak solution to the small finite deflections shell problem at the time t due to some initial data e(0).

We start with initial data x(0) ∈ H such that xn(0) → x(0) in H as n → ∞. Our aim is to prove that

xn(0) → x(0) in C([0, τ[, H). (140)

We will follow the road map presented in [18].

The potential energy defined by(35)only differs from the potential energy used in [5] by lower order terms. As a consequence, it is topologically equivalent to V1(Γ ). In particular, using Sobolev’s embedding and Korn’s inequality (both of which are valid for the tangential calculus [11]) we have the inequalities

keΓkH1_{(Γ )} ≤C [kBMekL2(Γ )+ k∇Γwk 2 L4(Γ )+ keΓkL2(Γ )] ≤C [kBMekL2(Γ )+ kwk 2 H2_{(Γ )}+ keΓkL2(Γ )]. (141)

ByLemma 11and the inequality(141), we have

kxkH≤CE(x(t)) (142)

so that

ken(t)kH≤C(ken(0)kH) ≤ C(ke(0)kH). (143)

Hence, on a subsequence denoted by the same index we have xn(t) → x∗(t) weakly∗in L∞(0, τ; H).

By using the variational equality together with weak continuity of nonlinear terms, we can show that x∗(t) coincides with a weak solution to(16)due to the initial data x(0). By uniqueness of the weak solution x∗(t) = x(t). Hence we have

xn(t) → x(t) weakly∗in L∞(0, τ; H). (144) In view of this, to prove the theorem it is enough to show the norm convergence of

kxn(t)kH→ kx(t)kH in C(0, τ).

We will use the equality in the energy relation from Lemma 11to show this. We denote by E(x(t)) the energy corresponding to the solution x(t). FromLemma 11we have