Georgian Mathematical Journal Volume 10 (2003), Number 1, 17–36

INVESTIGATION OF TWO-DIMENSIONAL MODELS OF ELASTIC PRISMATIC SHELL

M. AVALISHVILI AND D. GORDEZIANI

Abstract. Statical and dynamical two-dimensional models of a prismatic elastic shell are constructed. The existence and uniqueness of solutions of the corresponding boundary and initial boundary value problems are proved, the rate of approximation of the solution of a three-dimensional problem by the vector-function restored from the solution of a two-dimensional one is estimated.

2000 Mathematics Subject Classification: 42C10, 65M60, 74K20. Key words and phrases: Boundary and initial boundary value problems for elastic prismatic shell, Fourier–Legendre series, a priori error estimation.

1. Introduction

In mathematical physics and theory of elasticity one of the urgent issues is constructing and investigating lower-dimensional models. I. Vekua proposed one of the methods of constructing two-dimensional models of prismatic shells in [1]. It must be pointed out that in [1] boundary value problems are considered in C^{k} spaces and the convergence of sequences of approximations to exact solutions of the corresponding three-dimensional problems is not investigated. In the statical case the existence and uniqueness of a solution of the reduced twodimensional problem in Sobolev spaces were investigated in [2] and the rate of approximation of an exact solution of a three-dimensional problem by the vector-function restored from the solution of the reduced problem in C^{k} spaces was estimated in [3]. Later, various types of lower-dimensional models were constructed and investigated in [4–18].

In the present paper we consider static equilibrium of a prismatic elastic shell and a dynamical problem of vibration of a shell. Due to I. Vekua’s reduction method we construct a statical two-dimensional model of the plate and investigate the obtained boundary value problem. Moreover, if the solution of the original problem satisfies additional regularity properties, we estimate the accuracy of its approximation by the vector-function restored from the solution of a two-dimensional problem. We reduce the dynamical three-dimensional problem for a prismatic shell to the two-dimensional one, prove the existence and uniqueness of the solution of the corresponding initial boundary value problem and show that the vector-function restored from the latter problem approximates the solution of the original problem. Also, under the regularity conditions on the solution of the original problem we obtain the rate of its approximation.

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Let us consider a prismatic elastic shell of variable thickness and initial configuration \overline{Ω} ⊂ ℝ^{3}; x = (x_{1}, x_{2}, x_{3}) ∈ \overline{Ω}, \overline{Ω} denotes the closure of the domain Ω ⊂ ℝ^{3} (the domain is a bounded open connected set with a Lipschitzcontinuous boundary, the set being locally on one side of the boundary [25]). Assume that the ruling of the lateral surface Γ of the plate is parallel to the Ox_{3}- axis (Ox_{1}x_{2}x_{3} is a system of Cartesian coordinates in ℝ^{3}) and the upper Γ^{+}, lower Γ^{−} surfaces of the plate are defined by the equations x_{3} = h^{+}(x_{1}, x_{2}), x_{3} = h^{−}(x_{1}, x_{2}), h^{+}(x_{1}, x_{2}) > h^{−}(x_{1}, x_{2}), x_{1}, x_{2} ∈ ω̄, h^{−}, h^{+} ∈ C^{1}(ω̄), where ω ⊂ ℝ^{2} is a domain with boundary γ. Let γ_{0} denote the Lipschitz-continuous part of γ with positive length.

In order to simplify the notation, we assume that the indices i, j, p, q take their values in the set {1, 2, 3}, while the indices α, β vary in the set {1, 2} and the summation convention with respect to repeated indices is used. Also, the

partial derivative with respect to the p-th argument \frac{∂}{∂x_{p}} is denoted by ∂_{p}. For

any domain D ⊂ ℝ^{s}, L^{2}(D) denote the space of square-integrable functions in D in the Lebesgue sense, H^{m}(D) = W^{m,2}(D) denotes the Sobolev space of order m and the space of vector-functions denote with H^{m}(D) = [H^{m}(D)]^{3}, L^{2}(D) = [L^{2}(D)]^{3}, s, m ∈ ℕ.

Let us suppose that the material of the plate is elastic, homogeneous, isotropic and λ, μ are the Lamé constants of the material. The applied body force density is denoted by f = (f_{i}) : Ω × (0, T) → ℝ^{3} and the surface force densities on the surfaces Γ^{+}, Γ^{−} by g^{+} and g^{−}, respectively, g^{±} = (g_{i}^{±}) : Γ^{±} × [0, T] → ℝ^{3}. The plate is clamped on the part Γ^{0} = {(x_{1}, x_{2}, x_{3}) ∈ ℝ^{3}; (x_{1}, x_{2}) ∈ γ_{0}, h^{−}(x_{1}, x_{2}) ≤ x_{3} ≤ h^{+}(x_{1}, x_{2})} of its lateral surface Γ, while the surface Γ^{1} = Γ\Γ^{0} is free. For the stress-strain state of the plate we have the following initial boundary value problem:

\frac{∂^{2}u_{i}}{∂t^{2}} − ∑_{j=1}^{3} \frac{∂}{∂x_{j}} \{λe_{pp}(u)δ_{ij} + 2μe_{ij}(u)\} = f_{i}(x, t), (x, t) ∈ Ω_{T} ,  (1.1)

u(x, 0) = φ(x), \frac{∂u}{∂t}(x, 0) = ψ(x), x ∈ Ω,  (1.2)
u = 0 on Γ_{T}^{0} = Γ^{0} × [0, T],

∑_{j=1}^{3} (λe_{pp}(u)δ_{ij} + 2μe_{ij}(u))n_{j} = \begin{cases}g_{i}^{+} & on Γ_{T}^{+} = Γ^{+} × [0, T], \\ g_{i}^{−} & on Γ_{T}^{−} = Γ^{−} × [0, T], \\ 0 & on Γ_{T}^{1} = Γ^{1} × [0, T], \end{cases}  (1.3)

where Ω_{T} = Ω × (0, T), u = (u_{i}) : \overline{Ω}_{T} → ℝ^{3} is the unknown displacement vector-function, φ, ψ : Ω → ℝ^{3} are initial displacement and velocity of the plate, n = (n_{j}) denotes the unit outer normal vector along the boundary ∂Ω. δ_{ij} is the Kronecker symbol and e(u) = {e_{ij}(u)} is the deformation tensor

e_{ij}(u) = \frac{1}{2}(\frac{∂u_{i}}{∂x_{j}} + \frac{∂u_{j}}{∂x_{i}} ), i, j = 1, 2, 3.

In Section 2 we study the statical case of problem (1.1)–(1.3) and construct a statical two-dimensional model of the plate, investigate the convergence of the sequence of vector-functions restored from the solutions of the corresponding boundary value problems to the solution of the original three-dimensional problem. In Section 3 we consider problem (1.1)–(1.3) in the suitable functional spaces, construct and investigate a hierarchic dynamical two-dimensional model of a prismatic shell.

2. Statical Boundary Value Problem

As we have mentioned in the introduction, in this section we study the statical case of problem (1.1)–(1.3). In this case the latter problem admits the following variational formulation: find a vector-function u ∈ V (Ω) = {v = (v_{i}) ∈ H^{1}(Ω), v = 0 on Γ^{0}}, which satisfies the equation

B^{Ω}(u, v) = L^{Ω}(v), ∀ v ∈ V (Ω),  (2.1)

where

B^{Ω}(u, v) = ∫_{Ω} (λe_{pp}(u)e_{qq}(v) + 2μe_{ij}(u)e_{ij}(v))dx,
L^{Ω}(v) = ∫_{Ω} f_{i}v_{i}dx + ∫_{Γ^{+}} g_{i}^{+}v_{i}dΓ + ∫_{Γ^{−}} g_{i}^{−}v_{i}dΓ.

The variational problem (2.1) has a unique solution if λ ≥ 0, μ > 0, f ∈ L^{2}(Ω), g^{±} ∈ L^{2}(Γ^{±}), which is a unique solution of the following minimization problem: find u ∈ V (Ω) such that

J^{Ω}(u) = inf_{v∈V (Ω} J^{Ω}(v), J^{Ω}(v) = \frac{1}{2}B^{Ω}(v, v) − L^{Ω}(v), ∀ v ∈ V (Ω).

In order to reduce the three-dimensional problem (2.1) to a two-dimensional one, let us consider equation (2.1) on the subspace of V (Ω), which consists of polynomials of degree N with respect to the variable x_{3}, i.e.,

v_{N} = ∑_{r=0}^{N} a(r + \frac{1}{2}) v^{r} P_{r}(ax_{3} − b), v^{r}= (v_{i}^{r}), r = \overline{0, N},

where a = \frac{2}{h^{+} − h^{−}} , b = \frac{h^{+} + h^{−}}{h^{+} − h^{−}}, and P_{r} is a Legendre polynomial of degree r. Thus we get the following problem:

B^{Ω}(w_{N}, v_{N}) = L^{Ω}(v_{N}), ∀ v_{N} ∈ V_{N}(Ω),  (2.2)

V_{N}(Ω) = \{v_{N} = ∑_{r=0}^{N} a(r + \frac{1}{2}) v^{r} P_{r}(ax_{3} − b);

v^{r}∈ H^{1}(ω), v^{r}= 0 on γ_{0}, r = \overline{0, N}\}.

In problem (2.2) the unknown function w_{N} ∈ V_{N}(Ω) is of the form

w_{N} = ∑_{r=0}^{N} a(r + \frac{1}{2}) w^{r} P_{r}(ax_{3} − b),

so we have to find the vector-function w⃗_{N} = (w^{0}, …, w^{N}),

w⃗_{N} ∈ V⃗_{N}(ω) = \{v⃗_{N} = (v^{0}, …, v^{N}); v^{r}∈ H^{1}(ω), v^{r}= 0 on γ_{0}, r = \overline{0, N}\},

such that the corresponding w_{N} is a solution of problem (2.2).

Taking into account properties of the Legendre polynomials [19] equation (2.2) can be written in the following form

∑_{r=0}^{N}(r + \frac{1}{2}) ∫_{ω} a(λ θ^{r} (w⃗_{N}) θ^{r} (v⃗_{N}) + 2μ e^{r}_{ij} (w⃗_{N}) e^{r}_{ij} (v⃗_{N}))dx_{1}dx_{2}

= ∑_{r=0}^{N} (r + \frac{1}{2}) ∫_{ω} a f^{r}_{i}v^{r}_{i} dx_{1}dx_{2} + ∑_{r=0}^{N} (r + \frac{1}{2}) ∫_{ω} ag̃_{i}^{+}k_{+} v^{r}_{i} dx_{1}dx_{2}

+ ∑_{r=0}^{N} (r + \frac{1}{2}) ∫_{ω} ag̃_{i}^{−}k_{−}(−1)^{r} v^{r}_{i} dx_{1}dx_{2}, ∀ v⃗_{N} ∈ V⃗_{N}(ω),  (2.3)

where g̃_{i}^{±}(x_{1}, x_{2}) = g_{i}^{±}(x_{1}, x_{2}, h^{±}(x_{1}, x_{2})), (x_{1}, x_{2}) ∈ ω, i = \overline{1, 3},

θ^{r} (v⃗_{N}) =e^{r}_{ii} (v⃗_{N}),

e^{r}_{ij} (v⃗_{N}) = \frac{1}{2}(\frac{∂ v^{r}_{i}}{∂x_{j}} + \frac{∂ v^{r}_{j}}{∂x_{i}} ) + \frac{1}{2} ∑_{s=r}^{N}(b^{r}_{is}v^{s}_{j} + b^{r}_{js}v^{s}_{i}),

b^{r}_{αr}= −(r + 1)\frac{∂_{α}h^{+} − ∂_{α}h^{−}}{h^{+} − h^{−}} , b^{r}_{3r}= 0,

b^{r}_{js}= \begin{cases}0, & s < r, \\ −(2s + 1)\frac{∂_{α}h^{+} − (−1)^{s+r}∂_{α}h^{−}}{h^{+} − h^{−}} , & j = α, s > r, \\ (2s + 1)\frac{1 − (−1)^{s+r}}{h^{+} − h^{−}} , & j = 3, s > r, \end{cases}

k_{±} = \sqrt{1 + (\frac{∂h^{±}}{∂x_{1}} )^{2} + (\frac{∂h^{±}}{∂x_{2}} )^{2}}, f^{r}_{i}= ∫_{h^{−}}^{h^{+}} f_{i}P_{r}(ax_{3} − b)dx_{3}, r, s = \overline{0, N}.

Thus three-dimensional problem (2.1) we have reduced to two-dimensional one. For the last problem (2.3) we obtain the existence and uniqueness of its solution. First we prove the inequalities of Korn’s type in this case.

Theorem 2.1. Assume that ω ⊂ ℝ^{2} is a bounded domain with Lipschitz boundary γ = ∂ω.

INVESTIGATION OF TWO-DIMENSIONAL MODELS 21

a) There exists a constant α > 0, which depends only on ω, such that

∑_{r=0}^{N} (r + \frac{1}{2}) ∫_{ω} a e^{r}_{ij} (v⃗_{N}) e^{r}_{ij} (v⃗_{N})dx_{1}dx_{2} + ∑_{r=0}^{N} ∫_{ω} v^{r}_{i}v^{r}_{i} dx_{1}dx_{2} ≥ α∥v⃗_{N}∥_{ℌ^{1}(ω)}^{2},

for any v⃗_{N} ∈ ℌ^{1}(ω), where ℌ^{1}(ω) = [H^{1}(ω)]^{N+1}.

b) There exists a constant α > 0, which depends only on ω and γ_{0}, such that

∑_{r=0}^{N} (r + \frac{1}{2}) ∫_{ω} a e^{r}_{ij} (v⃗_{N}) e^{r}_{ij} (v⃗_{N})dx_{1}dx_{2} ≥ α∥v⃗_{N}∥_{ℌ^{1}(ω)}^{2}, ∀ v⃗_{N} ∈ V⃗_{N}(ω).

Proof. Let us introduce the space

E(ω) = \{v⃗_{N} = (v^{0}, …, v^{N}) ∈ 𝔏^{2}(ω) = [L^{2}(ω)]^{N+1}, e^{r}_{ij} (v⃗_{N}) ∈ L^{2}(ω), r = \overline{0, N}\}.

Then, equipped with the norm ∥.∥ defined by

∥v⃗_{N}∥ = {|v⃗_{N}|_{0,ω}^{2} + |e(v⃗_{N})|_{0,ω}^{2}}^{1/2},

where

|v⃗_{N}|_{0,ω}^{2} = ∑_{r=0}^{N} ∫_{ω} v^{r}_{i}v^{r}_{i} dx_{1}dx_{2},

|e(v⃗_{N})|_{0,ω}^{2} = ∑_{r=0}^{N} (r + \frac{1}{2}) ∫_{ω} a e^{r}_{ij} (v⃗_{N}) e^{r}_{ij} (v⃗_{N})dx_{1}dx_{2},

the space E(ω) is a Hilbert space. Indeed, let us consider the Cauchy sequence {v_{N}^{(k)}}_{k=1}^{∞} in the space E(ω). By the definition of the norm ∥.∥ there exists

v^{r}_{i}∈ L^{2}(ω) and e^{r}_{ij}∈ L^{2}(ω) such that

v^{r}_{i}^{(k)}→v^{r}_{i}, e^{r}_{ij} (v_{N}^{(k)}) →e^{r}_{ij} in L^{2}(ω), as k → ∞.

Moreover, for any φ ∈ D(ω) (D(ω) is a space of infinitely differentiable functions with compact support in ω) the following equality is valid:

∫_{ω} e^{r}_{ij} (v_{N}^{(k)})φdx_{1}dx_{2} = \frac{1}{2} ∫_{ω} (− v^{r}_{i}^{(k)} ∂_{j}φ− v^{r}_{j}^{(k)} ∂_{i}φ

+ ∑_{s=r}^{N}(b^{r}_{is}v^{s}_{j}^{(k)} + b^{r}_{js}v^{s}_{i}^{(k)})φ)dx_{1}dx_{2}, ∀ k ∈ ℕ.

Hence, passing to the limit as k → ∞, we obtain e^{r}_{ij}=e^{r}_{ij} (v⃗_{N}).

Let us show that the spaces E(ω) and ℌ^{1}(ω) are isomorphic. It is clear that ℌ^{1}(ω) ⊂ E(ω). Moreover, if we take v⃗_{N} ∈ E(ω), then for any 1 ≤ i, j, p ≤ 3 and r = \overline{0, N} we get

∂_{p} v^{r}_{i}∈ H^{−1}(ω),

∂_{j}(∂_{p} v^{r}_{i}) = ∂_{j} e^{r}_{ip} (v⃗_{N}) + ∂_{p} e^{r}_{ij} (v⃗_{N}) − ∂_{i} e^{r}_{jp} (v⃗_{N})

− \frac{1}{2} ∑_{s=r}^{N} (∂_{j}(b^{r}_{is}v^{s}_{p} + b^{r}_{ps}v^{s}_{i}) + ∂_{p}(b^{r}_{is}v^{s}_{j} + b^{r}_{js}v^{s}_{i}) − ∂_{i}(b^{r}_{ps}v^{s}_{j} + b^{r}_{js}v^{s}_{p})) ∈ H^{−1}(ω)

since if y ∈ L^{2}(ω), then ∂_{p}y ∈ H^{−1}(ω). By virtue of the lemma of Lions [20–22] we have ∂_{p} v^{r}_{i}∈ L^{2}(ω), and therefore the spaces E(ω) and ℌ^{1}(ω) coincide.

To prove the item a) of the theorem note that the identity mapping from ℌ^{1}(ω) into E(ω) is injective, continuous and, by virtue of the preceding result, is surjective. Since both spaces are complete, the closed graph theorem [23] shows that the inverse mapping is also continuous, which proves the desired inequality.

Now we will prove the item b) of the theorem. Notice that the semi-norm |.| defined by |v⃗_{N}| = |e(v⃗_{N})|_{0,ω} is the norm in the space V⃗_{N}(ω) when the measure of γ_{0} is positive. Indeed, if |e(v⃗_{N})|_{0,ω}^{2} = 0, then [4]

v^{0}_{1} (x_{1}, x_{2}) = \frac{1}{a}(b_{3}x_{2} + \frac{1}{2}(h^{+} + h^{−})b_{1} + c_{1}),

\begin{matrix}v^{0}_{2} (x_{1}, x_{2}) = \frac{1}{a}( − b_{3}x_{1} + \frac{1}{2}(h^{+} + h^{−})b_{2} + c_{2}), \\ v^{0}_{3} (x_{1}, x_{2}) = \frac{1}{a}( − b_{1}x_{1} − b_{2}x_{2} + c_{3}), \end{matrix} (x_{1}, x_{2}) ∈ ω,

v^{1}_{1} (x_{1}, x_{2}) = \frac{b_{1}}{3a^{2}} , v^{1}_{2} (x_{1}, x_{2}) = \frac{b_{2}}{3a^{2}} , v^{1}_{3}= 0, v^{r}_{i}= 0, i = \overline{1, 3}, r = \overline{2, N,}

for any real constants b_{1}, b_{2}, b_{3}, c_{1}, c_{2}, c_{3}. Since v⃗_{N} = 0 on γ_{0} and the measure of γ_{0} is positive, we get v⃗_{N} = 0 on ω.

To prove the inequality of the item b) we argue by contradiction. Then there exists a sequence {v_{N}^{k}}_{k=1}^{∞}, v_{N}^{k} ∈ V⃗_{N}(ω) such that ∥v_{N}^{k}∥_{ℌ^{1}(ω)} = 1 for all k ∈ ℕ, lim_{k→∞} |e(v_{N}^{k})|_{0,ω} = 0. Since the sequence {v_{N}^{k}}_{k=1}^{∞} is bounded in the space

ℌ^{1}(ω), a subsequence {v_{N}^{k_{l}}}_{l=1}^{∞} converges in 𝔏^{2}(ω) by the Rellich–Kondrašov theorem. Each sequence \{e^{r}_{ij} (v_{N}^{k_{l}})\}_{l=1}^{∞}, r = \overline{0, N}, also converges in L^{2}(ω). The subsequence {v_{N}^{k_{l}}}_{l=1}^{∞} is thus a Cauchy sequence with respect to the norm ∥.∥.

According to the inequality of the item a) we have that the subsequence {v_{N}^{k_{l}}}_{l=1}^{∞} is a Cauchy sequence with respect to the norm ∥.∥_{ℌ^{1}(ω)} too. The space V⃗_{N}(ω) is complete as a closed subspace of ℌ^{1}(ω) and hence there exists v⃗_{N} ∈ V⃗_{N}(ω) such that

v_{N}^{k_{l}} → v⃗_{N} in ℌ^{1}(ω), as l → ∞,

and |e(v⃗_{N})|_{0,ω} = lim_{l→∞} |(v_{N}^{k_{l}})|_{0,ω} = 0. Therefore v⃗_{N} = 0, which contradicts the

relations ∥v_{N}^{k_{l}}∥_{ℌ^{1}(ω)} = 1 for all l ≥ 1. □

On the basis of Theorem 2.1 we prove the theorem on the existence and uniqueness of a solution of problem (2.3).

Theorem 2.2. Assume that the Lamé constants λ ≥ 0, μ > 0, f ∈ L^{2}(Ω), g^{±} ∈ L^{2}(Γ^{±}), then the symmetric bilinear form B_{N}^{Ω} : V⃗_{N}(ω) × V⃗_{N}(ω) → ℝ,

B_{N}^{Ω}(u⃗_{N}, v⃗_{N}) = ∑_{r=0}^{N} (r + \frac{1}{2}) ∫_{ω} a(λ θ^{r} (u⃗_{N}) θ^{r} (v⃗_{N}) + 2μ e^{r}_{ij} (u⃗_{N}) e^{r}_{ij} (v⃗_{N}))dx_{1}dx_{2}

is continuous and coercive, the linear form L_{N}^{Ω} : V⃗_{N}(ω) → ℝ,

L_{N}^{Ω}(v⃗_{N}) = ∑_{r=0}^{N} (r + \frac{1}{2}) ∫_{ω} a f^{r}_{i}v^{r}_{i} dx_{1}dx_{2} + ∑_{r=0}^{N} (r + \frac{1}{2}) ∫_{ω} ag̃_{i}^{+}k_{+} v^{r}_{i} dx_{1}dx_{2}

+ ∑_{r=0}^{N} (r + \frac{1}{2}) ∫_{ω} ag̃_{i}^{−}k_{−}(−1)^{r} v^{r}_{i} dx_{1}dx_{2}

is continuous. The two-dimensional problem (2.3) has a unique solution w⃗_{N} ∈ V⃗_{N}(ω), which is also a unique solution to the following minimization problem:

w⃗_{N} ∈ V⃗_{N}(ω), J_{N}(w⃗_{N}) = inf_{v⃗_{N} ∈V⃗_{N} (ω)} J_{N}(v⃗_{N}),

J_{N}(v⃗_{N}) = \frac{1}{2}B_{N}^{Ω}(v⃗_{N}, v⃗_{N}) − L_{N}^{Ω}(v⃗_{N}), ∀ v⃗_{N} ∈ V⃗_{N}(ω).

Proof. By the inequality of the item b) of Theorem 2.1, the bilinear form B_{N}^{Ω} is coercive

B_{N}^{Ω}(v⃗_{N}, v⃗_{N}) ≥ 2μ ∑_{r=0}^{N} (r + \frac{1}{2}) ∫_{ω} a e^{r}_{ij} (v⃗_{N}) e^{r}_{ij} (v⃗_{N})dx_{1}dx_{2} ≥ 2μα∥v⃗_{N}∥_{ℌ^{1}(ω)}^{2}.

Therefore, applying the Lax–Milgram theorem we obtain that problem (2.3) has a unique solution w⃗_{N}, which can be equivalently characterized as a solution of the minimization problem of the energy functional J_{N}(v⃗_{N}). □

Thus we have reduced the three-dimensional problem (2.1) to the two-dimen-sional one and for the latter problem proved the existence and uniqueness of its solution. For the reduced two-dimensional problem (2.3) the following theorem is true.

Theorem 2.3. If all the conditions of Theorem 2.2 hold, then the vector-

function w_{N} = ∑_{r=0}^{N} a(r + \frac{1}{2}) w^{r} P_{r}(ax_{3} − b) corresponding to the solution w⃗_{N} =

(w^{0}, …, w^{N}) of the reduced problem (2.3) tends to the solution u of the threedimensional problem (2.1) w_{N} → u in the space H^{1}(Ω) as N → ∞. Moreover, if u ∈ H^{s}(Ω), s ≥ 2, then the following estimate is valid:

\begin{matrix}∥u − w_{N}∥_{H^{1}(Ω)}^{2} ≤ \frac{1}{N^{2s−3}} q_{1}(h^{+}, h^{−}, N), \\ q_{1}(h^{+}, h^{−}, N) → 0 as N → ∞. \end{matrix}  (2.4)

If, additionally, ∥u∥_{H^{s}(Ω)} ≤c, where c is independent of h = max_{(x_{1},x_{2})∈ω̄}(h^{+}(x_{1}, x_{2})−

h^{−}(x_{1}, x_{2})), then the following estimate holds:

∥u − w_{N}∥_{E(Ω)}^{2} ≤ \frac{h^{2(s−1)}}{N^{2s−3}} q_{2}(N), q_{2}(N) → 0 as N → ∞,

where ∥v∥_{E(Ω)} = \sqrt{B^{Ω}(v, v)}, ∀ v ∈ V (Ω).

Proof. By virtue of Theorem 2.2 w⃗_{N} is a solution of the minimization problem of the energy functional J_{N}, i.e.

J_{N}(w⃗_{N}) = \frac{1}{2}B_{N}^{Ω}(w⃗_{N}, w⃗_{N}) − L_{N}^{Ω}(w⃗_{N})
≤ J_{N}(v⃗_{N}) = \frac{1}{2}B_{N}^{Ω}(v⃗_{N}, v⃗_{N}) − L_{N}^{Ω}(v⃗_{N}), ∀ v⃗_{N} ∈ V⃗_{N}(ω).  (2.5)

Taking into account that

B_{N}^{Ω}(v⃗_{N}, v⃗_{N}) = B^{Ω}(v_{N}, v_{N}), L_{N}^{Ω}(v⃗_{N}) = L^{Ω}(v_{N}), ∀ v⃗_{N} ∈ V⃗_{N}(ω),

where v_{N} = ∑_{r=0}^{N} a(r + \frac{1}{2}) v^{r} P_{r}(ax_{3} − b), and applying (2.5), we obtain

B^{Ω}(u − w_{N}, u − w_{N}) ≤ B^{Ω}(u, u) − 2L^{Ω}(v_{N}) + B^{Ω}(v_{N}, v_{N}).

From the latter inequality we have

B^{Ω}(u − w_{N}, u − w_{N}) ≤ B^{Ω}(u − v_{N}, u − v_{N}), ∀ v_{N} ∈ V_{N}(Ω).  (2.6)

Since γ_{0} is Lipschitz-continuous, by the trace theorems for Sobolev spaces [24], for any v ∈ H^{1}(Ω), v = 0 on Γ^{0}, there exists a continuation ṽ ∈ H_{0}^{1}(Ω_{1}) of the function v, where Ω_{1} ⊃ Ω, ∂Ω_{1} ⊃ Γ^{0}. From the density of D(Ω_{1}) in H_{0}^{1}(Ω_{1}) we obtain that the space of infinitely differentiable functions in Ω, which are equal to zero on Γ^{0}, is dense in V (Ω), and since the set of polynomials is dense in L^{2}(−1, 1), we conclude that ⋃_{N≥1} V_{N}(Ω) is dense in V (Ω), and therefore

w_{N} → u in the space H^{1}(Ω) as N → ∞.

Let us prove estimate (2.4). Suppose that u ∈ H^{s}(Ω), s ≥ 2, then

ε_{N} = u − u_{N} = u − ∑_{r=0}^{N} a(r + \frac{1}{2}) u^{r} P_{r}(ax_{3} − b), u^{r}= ∫_{h^{−}}^{h^{+}} uP_{r}(ax_{3} − b)dx_{3}.

Applying the properties of Legendre polynomials [19], we obtain

∥ε_{N}∥_{L^{2}(Ω)}^{2} = ∑_{r=N+1}^{∞} ∫_{ω} a(r + \frac{1}{2})(u^{r}_{i})^{2}dx_{1}dx_{2},

∥\frac{∂ε_{N}}{∂x_{3}} ∥_{L^{2}(Ω)}^{2} = ∑_{r=N}^{∞} ∫_{ω} a(r + \frac{1}{2})(∂_{3}u^{r}_{i})^{2}dx_{1}dx_{2} + ∫_{ω} a\frac{N(N − 1)}{4}(∂_{3}u^{N}_{i})^{2}dx_{1}dx_{2}

+ ∫_{ω} a\frac{N(N + 1)}{4}(∂_{3}u^{N+1}_{i})^{2}dx_{1}dx_{2},

∥\frac{∂ε_{N}}{∂x_{α}} ∥_{L^{2}(Ω}^{2} ≤ 2( ∑_{r=N+1}^{∞} ∫_{ω} a(r + \frac{1}{2})(∂_{α}u^{r}_{i})^{2}dx_{1}dx_{2}

+ ∫_{ω} a\frac{N + 1}{4}(N(\frac{∂h̃}{∂x_{α}} )^{2} + (N + 2)(\frac{∂h̄}{∂x_{α}} )^{2})(∂_{3}u^{N}_{i})^{2}dx_{1}dx_{2}

+ ∫_{ω} a\frac{N + 1}{4}((N + 2)(\frac{∂h̃}{∂x_{α}} )^{2} + N(\frac{∂h̄}{∂x_{α}} )^{2})(∂_{3}u^{N+1}_{i})^{2}dx_{1}dx_{2}),

where α = 1, 2, h̄ = \frac{1}{2}(h^{+} + h^{−}), h̃ = \frac{1}{2}(h^{+} − h^{−}).

It should be mentioned that

u^{r}= ∫_{h^{−}}^{h^{+}} uP_{r}(ax_{3} − b)dx_{3} = \frac{1}{a(2r + 1)}( ∂_{3}u^{r−1} − ∂_{3}u^{r+1} ), r = 1, 2, …, N,

and thus we have

∥ u^{r} ∥_{L^{2}(ω)}^{2} ≤ \frac{c}{r^{2s}} ∑_{k=r−s}^{r+s} ∥\frac{1}{a^{s}} \underbrace{∂_{3} … ∂_{3}^{k}}_{s} u∥_{L^{2}(ω)}^{2},  (2.7)

where c = const is independent of r, h^{+}, h^{−}. Therefore from (2.7) we get

\begin{aligned}∥ε_{N}∥_{L^{2}(Ω)}^{2} & ≤ \frac{1}{N^{2s}} q(h^{+}, h^{−}, N), \\ ∥\frac{∂ε_{N}}{∂x_{i}} ∥_{L^{2}(Ω)}^{2} & ≤ \frac{1}{N^{2s−3}} q(h^{+}, h^{−}, N), \end{aligned} q(h^{+}, h^{−}, N) → 0 as N → ∞,

where i = 1, 2, 3. Taking into account (2.6) and the coerciveness of B^{Ω}, we obtain

∥u − w_{N}∥_{H^{1}(Ω)}^{2} ≤ \frac{1}{N^{2s−3}} q_{1}(h^{+}, h^{−}, N), q_{1}(h^{+}, h^{−}, N) → 0 as N → ∞. (2.8)

From (2.7) we also have

\begin{aligned}∥ε_{N}∥_{L^{2}(Ω)}^{2} & ≤ \frac{h^{2s}}{N^{2s}} q̄(N), \\ ∥\frac{∂ε_{N}}{∂x_{i}} ∥_{L^{2}(Ω)}^{2} & ≤ \frac{h^{2(s−1)}}{N^{2s−3}} q̄(N), \end{aligned} q̄(N) → 0 as N → ∞, i = 1, 2, 3, (2.9)

where h = max_{(x_{1},x_{2})∈ω̄} 2h̃(x_{1}, x_{2}).

From inequalities (2.9) we obtain the second inequality of the theorem

∥u − w_{N}∥_{E(Ω)}^{2} ≤ \frac{h^{2(s−1)}}{N^{2s−3}} q_{2}(N), q_{2}(N) → 0 as N → ∞.

3. Dynamical Initial Boundary Value Problem

Now we proceed to studying the dynamical problem (1.1)–(1.3), constructing and investigating a dynamical two-dimensional model of a prismatic shell. This problem admits the following variational formulation: find u ∈ C^{0}([0, T]; V (Ω)), u′ ∈ C^{0}([0, T]; L^{2}(Ω)), which satisfies the equation

\frac{d}{dt}(u′, v)_{L^{2}(Ω)} + B^{Ω}(u, v) = L^{Ω}(v), ∀ v ∈ V (Ω)  (3.1)

in the sense of distributions in (0, T) and the initial conditions

u(0) = φ, u′(0) = ψ,  (3.2)

where φ ∈ V (Ω), ψ ∈ L^{2}(Ω).

Note that the formulated three-dimensional dynamical problem (3.1), (3.2)

has a unique solution u for λ ≥ 0, μ > 0, f ∈ L^{2}(Ω × (0, T)), g^{±}, \frac{∂g^{±}}{∂t} ∈

L^{2}(Γ^{±} × (0, T)), which satisfies the following energetical identity: ∀ t ∈ [0, T],

(u′(t), u′(t))_{L^{2}(Ω)} + B^{Ω}(u(t), u(t)) = (ψ, ψ)_{L^{2}(Ω)} + B^{Ω}(φ, φ) + 2L̃^{Ω}(u)(t),

where

L̃^{Ω}(u)(t) = ∫_{0}^{t} (f(τ), u′(τ))_{L^{2}(Ω)}dτ + (g^{+}(t), u(t))_{L^{2}(Γ^{+})}

+ (g^{−}(t), u(t))_{L^{2}(Γ^{−})} − (g^{+}(0), u(0))_{L^{2}(Γ^{+})} − (g^{−}(0), u(0))_{L^{2}(Γ^{−})}

− ∫_{0}^{t} (\frac{∂g^{+}}{∂t} (τ), u(τ))_{L^{2}(Γ^{+})}dτ −∫_{0}^{t} (\frac{∂g^{−}}{∂t} (τ), u(τ))_{L^{2}(Γ^{−})}dτ, ∀ t ∈ [0, T].

As in the statical case, to reduce the three-dimensional problem (3.1), (3.2) to a two-dimensional one, let us consider equation (3.1) on the subspace V_{N}(Ω) (V_{N}(Ω) is defined in Section 2), and choose φ, ψ as elements of V_{N}(Ω) and H_{N}(Ω), respectively, where

H_{N}(Ω) = \{v_{N} = ∑_{k=0}^{N} a(k + \frac{1}{2}) v^{k} P_{k}(ax_{3} − b); v^{k}∈ L^{2}(ω), k = 0, …, N\}.

Consequently, we consider the following variational problem: find w_{N} ∈ C^{0}([0, T]; V_{N}(Ω)), w′_{N} ∈ C^{0}([0, T]; H_{N}(Ω)), which satisfies the equation

\frac{d}{dt}(w′_{N}, v_{N})_{L^{2}(Ω)} + B^{Ω}(w_{N}, v_{N}) = L^{Ω}(v_{N}), ∀ v_{N} ∈ V_{N}(Ω),  (3.3)

in the sense of distributions in (0, T) and the initial conditions

w_{N}(0) = φ_{N}, w′_{N}(0) = ψ_{N},  (3.4)

where φ_{N} ∈ V_{N}(Ω), ψ_{N} ∈ H_{N}(Ω).

It must be pointed out that in problem (3.3), (3.4) the unknown is the vector-

function w_{N}(t) = ∑_{k=0}^{N} a(k + \frac{1}{2}) w^{k} (t)P_{k}(ax_{3} − b) and so this problem is equivalent to the following one: find w⃗_{N} = (w^{0}, …, w^{N}) ∈ C^{0}([0, T]; V⃗_{N}(ω)), w′_{N} ∈ C^{0}([0, T]; [L^{2}(ω)]^{N+1}), which satisfies the equation

\frac{d}{dt}(P w′_{N}, v⃗_{N})_{[L^{2}(ω)]^{N+1}} + B_{N}^{Ω}(w⃗_{N}, v⃗_{N}) = L_{N}^{Ω}(v⃗_{N}), ∀ v⃗_{N} ∈ V⃗_{N}(ω),  (3.5)

in the sense of distributions in (0, T) and the initial condition

w⃗_{N}(0) = φ⃗_{N}, w′_{N}(0) = ψ⃗_{N},  (3.6)

where φ⃗_{N} = (φ^{0}, …, φ^{N}) ∈ V⃗_{N}(ω), ψ⃗_{N} = (ψ^{0}, …, ψ^{N}) ∈ [L^{2}(ω)]^{N+1},

φ_{N} = ∑_{k=0}^{N} a(k + \frac{1}{2}) φ^{k} P_{k}(ax_{3} − b), ψ_{N} = ∑_{k=0}^{N} a(k + \frac{1}{2}) ψ^{k} P_{k}(ax_{3} − b),

P = (P_{rl}), P_{rl} = a(r + \frac{1}{2})δ_{rl}, r, l = 0, …, N,

and B_{N}^{Ω}, L_{N}^{Ω} are defined in Section 2.

Thus we get a two-dimensional model of the prismatic shell. To investigate problem (3.5), (3.6) let us consider a more general variational problem and formulate the theorem on the existence and uniqueness of its solution, from which we obtain the corresponding result for (3.5), (3.6).

Let us suppose that V and H are Hilbert spaces, V is dense in H and continuously imbedded into it. The dual space of V is denoted by V′ and H is identified with its dual with respect to the scalar product, then

V ↪ H ↪ V′

with continuous and dense imbeddings. The scalar product and the norm in the space V is denoted by ((., .)), ∥.∥, while in the space H by (., .) and |.|. Denote the norm in the space V′ by ∥.∥_{∗}, and the dual relation between the spaces V′ and V by ⟨., .⟩.

Assume that A, B, L are linear continuous operators such that

B = B_{1} + B_{2}, B_{1} ∈ L(V, V′), B_{2} ∈ L(V, H) ∩ L(H, V′), A, L ∈ L(H, H),

B_{1} is self-adjoint and B_{1}+λI is coercive for some real number λ, A is self-adjoint and coercive, i.e.,

\begin{matrix}\begin{aligned}b_{1}(u, v) & = b_{1}(v, u), |b_{1}(u, v)| ≤ c_{b_{1}}∥u∥∥v∥, \\ b_{1}(u, u) & ≥ β∥u∥^{2} − λ|u|^{2}, β > 0, \end{aligned} ∀ u, v ∈ V, \\ |b_{2}(ũ, ṽ)| ≤ \begin{cases}c_{b_{2}}∥ũ∥|ṽ|, & ∀ ũ ∈ V, ṽ ∈ H, \\ c_{b_{2}}|ũ|∥ṽ∥, & ∀ ũ ∈ H, ṽ ∈ V, \end{cases} \\ \begin{aligned}a(u_{1}, v_{1}) & = a(v_{1}, u_{1}), a(u_{1}, u_{1}) ≥ α|u_{1}|^{2}, α > 0, \\ |a(u_{1}, v_{1})| & ≤ c_{a}|u_{1}||v_{1}|, |l(u_{1}, v_{1})| ≤ c_{l}|u_{1}||v_{1}|, \end{aligned} ∀ u_{1}, v_{1} ∈ H, \end{matrix}  (3.7)

where b_{1}(u, v) = ⟨B_{1}u, v⟩, b_{2}(u, v) = ⟨B_{2}u, v⟩, l(u_{1}, v_{1}) = (Lu_{1}, v_{1}), a(u_{1}, v_{1}) = (Au_{1}, v_{1}), b(u, v) = b_{1}(u, v) + b_{2}(u, v), ∀ u, v ∈ V, u_{1}, v_{1} ∈ H.

Let us consider the following variational problem: find z ∈ C^{0}([0, T]; V), z′ ∈ C^{0}([0, T]; H), which satisfies the equation

\frac{d}{dt}a(z′, v) + b(z, v) + l(z′, v) = (F, v) + ⟨G, v⟩, ∀ v ∈ V,  (3.8)

in the sense of distributions in (0, T) and the initial conditions

z(0) = z_{0}, z′(0) = z_{1},  (3.9)

where z_{0} ∈ V, z_{1} ∈ H, F ∈ L^{2}(0, T; H), G, G′ ∈ L^{2}(0, T; V′).

For the latter problem the following theorem is true.

Theorem 3.1. If all the conditions (3.7) hold, then problem (3.8), (3.9) has a unique solution which satisfies the equality

a(z′(t), z′(t)) + b_{1}(z(t), z(t)) + 2 ∫_{0}^{t} b_{2}(z(τ), z′(τ))dτ + 2 ∫_{0}^{t} l(z′(τ), z′(τ))dτ

= a(z_{1}, z_{1}) + b_{1}(z_{0}, z_{0}) + 2 ∫_{0}^{t}(F (τ), z′(τ))dτ + 2⟨G(t), z(t)⟩ − 2⟨G(0), z_{0}⟩

−2 ∫_{0}^{t}⟨G′(τ), z(τ)⟩dτ, ∀ t ∈ [0, T].

In Theorem 3.1 the existence and uniqueness of the solution can be proved in a standard way applying the Faedo–Galerkin method [25], while the energetical equality is obtained by regularization and passing to the limit.

For the reduced problem (3.5), (3.6), from Theorem 3.1 we obtain

Theorem 3.2. Assume that Lamé constants λ ≥ 0, μ > 0, f ∈ L^{2}(Ω ×

(0, T)), g^{±}, \frac{∂g^{±}}{∂t} ∈ L^{2}(Γ^{±} × (0, T)), φ⃗_{N} ∈ V⃗_{N}(ω), ψ⃗_{N} ∈ [L^{2}(ω)]^{N+1}, then

problem (3.5), (3.6) has a unique solution w⃗_{N}(t) and the following energetical identity is valid:

(w′_{N}(t), w′_{N}(t))_{L^{2}(Ω)} + B^{Ω}(w_{N}(t), w_{N}(t)) = (ψ_{N}, ψ_{N})_{L^{2}(Ω)}
+B^{Ω}(φ_{N}, φ_{N}) + 2L̃^{Ω}(w_{N})(t), ∀ t ∈ [0, T].  (3.10)

Proof. This theorem is just a consequence of Theorem 3.1. Indeed, it is sufficient to take V = V⃗_{N}(ω), H = [L^{2}(ω)]^{N+1}, z(t) = w⃗_{N}(t), v = v⃗_{N},

a(w⃗_{N}, v⃗_{N})=(P w⃗_{N}, v⃗_{N})_{[L^{2}(ω)]^{N+1}}, b_{1}(w⃗_{N}, v⃗_{N})=B_{N}^{Ω}(w⃗_{N}, v⃗_{N}),

b_{2} ≡ 0, l ≡ 0, z_{0} =φ⃗_{N}, z_{1} =ψ⃗_{N},

⟨G, v⃗_{N}⟩ = (Q⃗^{+}, v⃗_{N})_{[L^{2}(ω)]^{N+1}} + (Q⃗^{−}, v⃗_{N})_{[L^{2}(ω)]^{N+1}}, ∀ v⃗_{N} ∈ V⃗_{N}(ω),

Q⃗^{±} = (Q^{±}^{r})_{r=0}^{N}, Q^{+}^{r}= a(r + \frac{1}{2})g̃^{+}k_{+}, Q^{−}^{r}= a(−1)^{r}(r + \frac{1}{2})g̃^{−}k_{−},

g̃^{±}(x_{1}, x_{2}, t) = g^{±}(x_{1}, x_{2}, h^{±}(x_{1}, x_{2}), t) , (x_{1}, x_{2}, t) ∈ ω × (0, T),

F = P f⃗_{N}, f⃗_{N} = (f^{r})_{r=0}^{N}, f^{r}= ∫_{h^{−}}^{h^{+}} fP_{r}(ax_{3} − b)dx_{3}, r = \overline{0, N}.

It should be mentioned that all the conditions of Theorem 3.1 are valid and therefore problem (3.5), (3.6) has a unique solution which satisfies the energetical identity

(P w′_{N}(t), w′_{N}(t))_{[L^{2}(ω)]^{N+1}} + B_{N}^{Ω}(w⃗_{N}(t), w⃗_{N}(t)) = (P ψ⃗_{N}, ψ⃗_{N})_{[L^{2}(ω)]^{N+1}}

+ 2 ∫_{0}^{t} (P f⃗_{N}(τ), w′_{N}(τ))_{[L^{2}(ω)]^{N+1}}dτ + 2(Q⃗^{+}(t) + Q⃗^{−}(t), w⃗_{N}(t))_{[L^{2}(ω)]^{N+1}}

+ B_{N}^{Ω}(φ⃗_{N}, φ⃗_{N}) − 2(Q⃗^{+}(0) + Q⃗^{−}(0), φ⃗_{N})_{[L^{2}(ω)]^{N+1}}

− 2 ∫_{0}^{t} (\frac{∂Q⃗^{+}}{∂t} (τ) + \frac{∂Q⃗^{−}}{∂t} (τ), w⃗_{N}(τ))_{[L^{2}(ω)]^{N+1}}dτ, ∀ t ∈ [0, T],

which is equivalent to identity (3.10). □

Thus we have reduced the three-dimensional problem (3.1), (3.2) to the twodimensional one (3.5), (3.6) and proved that it has a unique solution. Now, let us estimate the rate of approximation of the exact solution u of the threedimensional problem by the vector-function w_{N}(t) restored from the solution w⃗_{N}(t) of the reduced problem. In order to simplify the notation, the norms in spaces V (Ω) and L^{2}(Ω) are denoted by ∥.∥ and |.|, respectively, and the scalar product in L^{2}(Ω) is denoted by (., .). The following theorem is true.

Theorem 3.3. If all the conditions of Theorem 3.2 are valid and φ_{N}, ψ_{N} corresponding to φ⃗_{N}, ψ⃗_{N} tend to φ, ψ in the spaces V (Ω) and L^{2}(Ω), respectively, then the vector-function w_{N}(t) corresponding to the solution w⃗_{N}(t) = (w^{0} (t), …, w^{N} (t)) of the two-dimensional problem (3.5), (3.6) tends to the solution u(t) of the three-dimensional problem (3.1), (3.2) in the space V (Ω)

w_{N}(t) → u(t) strongly in V (Ω), w′_{N}(t) → u′(t) strongly in L^{2}(Ω) as N → ∞, ∀ t ∈ [0, T].

Moreover, if components of φ⃗_{N}, ψ⃗_{N} are moments of φ, ψ with respect to Le-

gendre polynomials, i.e. φ⃗_{N} = (φ^{0}, …, φ^{N}), ψ⃗_{N} = (ψ^{0}, …, ψ^{N}),

φ^{k}= ∫_{h^{−}}^{h^{+}} φP_{k}(ax_{3} − b)dx_{3}, ψ^{k}= ∫_{h^{−}}^{h^{+}} ψP_{k}(ax_{3} − b)dx_{3}, k = \overline{0, N},

and u satisfies additional regularity properties with respect to the spatial variables u ∈ L^{2}(0, T; H^{s_{0}}(Ω)), u′ ∈ L^{2}(0, T; H^{s_{1}}(Ω)), u′′ ∈ L^{2}(0, T; H^{s_{2}}(Ω)), s_{0} ≥ s_{1} ≥ s_{2} ≥ 1, s_{1} ≥ 2, then the following estimate is valid: s = min{s_{2}, s_{1} − 3/2},

|u′ − w′_{N}|^{2} + ∥u − w_{N}∥^{2} ≤ \frac{1}{N^{2s}} q(Ω, Γ^{0}, N), q(Ω, Γ^{0}, N) → 0 as N → ∞.

If additionally ∥u∥_{L^{2}(0,T;H^{s_{0}}(Ω))} ≤ c, ∥u′∥_{L^{2}(0,T;H^{s_{1}}(Ω))} ≤ c, ∥u′′∥_{L^{2}(0,T;H^{s_{2}}(Ω))} ≤ c, where c is independent of h = max_{(x_{1},x_{2})∈ω̄}(h^{+}(x_{1}, x_{2}) − h^{−}(x_{1}, x_{2})), then

|u′ − w′_{N}|^{2} + ∥u − w_{N}∥^{2} ≤ q_{1}(Ω, Γ^{0})\frac{h^{2s̄}}{N^{2s}} q_{2}(N), q_{2}(N) → 0 as N → ∞,

where s̄ = min{s_{2}, s_{1} − 1}.

Proof. By virtue of Theorem 3.2 the vector-function w_{N}(t) corresponding to the solution w⃗_{N}(t) of the reduced problem (3.5), (3.6) satisfies the energetical equality (3.10) and since φ_{N} → φ in V (Ω) and ψ_{N} → ψ in L^{2}(Ω), we get: ∀ t ∈ [0, T],

|w′_{N}(t)|^{2} + ∥w_{N}(t)∥^{2} ≤ c(|ψ|^{2} + ∥φ∥^{2} + ∫_{0}^{t} |f(τ)|^{2}dτ + ∥g^{+}(t)∥_{L^{2}(Γ^{+})}^{2}

+∥g^{−}(t)∥_{L^{2}(Γ^{−})}^{2} + ∥g^{+}(0)∥_{L^{2}(Γ^{+})}^{2} + ∥g^{−}(0)∥_{L^{2}(Γ^{−})}^{2} + ∫_{0}^{t} ∥\frac{∂g^{+}}{∂t} (τ)∥_{L^{2}(Γ^{+})}^{2}dτ

+ ∫_{0}^{t} ∥\frac{∂g^{−}}{∂t} (τ)∥_{L^{2}(Γ^{−})}^{2}dτ) + c ∫_{0}^{t} (|w′_{N}(τ)|^{2} + ∥w_{N}(τ)∥^{2})dτ.

Applying Gronwall’s lemma [26], from the latter inequality we have

|w′_{N}(t)|^{2} + ∥w_{N}(t)∥^{2} < c_{1}, ∀ N ∈ ℕ, t ∈ [0, T].  (3.11)

Hence {w_{N}(t)} belongs to the bounded set of L^{∞}(0, T; V (Ω)) ∩ L^{2}(0, T; V (Ω)), while {w′_{N}(t)} belongs to the bounded set of L^{∞}(0, T; L^{2}(Ω)) ∩ L^{2}(0, T; L^{2}(Ω)). Consequently, there exists a subsequence {w_{ν}} of {w_{N}} such that when ν → ∞,

w_{ν} → ũ weakly in L^{2}(0, T; V (Ω)), weakly- ∗ in L^{∞}(0, T; V (Ω)), w′_{ν} → ũ′ weakly in L^{2}(0, T; L^{2}(Ω)), weakly- ∗ in L^{∞}(0, T; L^{2}(Ω)).

(3.12)

Let us prove that ũ is a solution of problem (3.1), (3.2). Since ⋃_{N≥0} V_{N}(Ω) is

dense in V (Ω), for any v ∈ V (Ω) there exists v_{N} ∈ V_{N}(Ω) such that v_{N} → v in V (Ω) as N → ∞. For any ζ ∈ D((0, T)), θ = ζv, θ_{N} = ζv_{N}, we have

θ_{N} → θ strongly in L^{2}(0, T; V (Ω)), θ′_{N} → θ′ strongly in L^{2}(0, T; V (Ω)) as N → ∞.  (3.13)

Taking into account (3.13), from (3.3) we obtain

∫_{0}^{T} B^{Ω}(w_{ν}(t), θ_{ν}(t))dt − ∫_{0}^{T} (w′_{ν}(t), θ′_{ν}(t))dt = ∫_{0}^{T} (f(t), θ_{ν}(t))dt
+ ∫_{0}^{T} (g^{+}(t), θ_{ν}(t))_{L^{2}(Γ^{+})}dt + ∫_{0}^{T} (g^{−}(t), θ_{ν}(t))_{L^{2}(Γ^{−})}dt.

Passing to the limit as ν → ∞ and applying (3.12), (3.13) from this equality we obtain

∫_{0}^{T} B^{Ω}(ũ(t), θ(t))dt − ∫_{0}^{T} (ũ′(t), θ′(t))dt = ∫_{0}^{T} (f(t), θ(t))dt
+ ∫_{0}^{T} (g^{+}(t), θ(t))_{L^{2}(Γ^{+})}dt + ∫_{0}^{T} (g^{−}(t), θ(t))_{L^{2}(Γ^{−})}dt.

Hence, ũ ∈ L^{∞}(0, T; V (Ω)), ũ′ ∈ L^{∞}(0, T; L^{2}(Ω)) and satisfies equation (3.1).

Therefore ũ ∈ C^{0}([0, T]; L^{2}(Ω)) ∩ L^{∞}(0, T; V (Ω)), ũ′ ∈ C^{0}([0, T]; V′(Ω)) ∩ L^{∞}(0, T; L^{2}(Ω)) (V′(Ω) denotes the dual space of V (Ω)) and consequently ũ, ũ′ are scalarly continuous functions from [0, T] to spaces V (Ω) and L^{2}(Ω), respectively [25]. Note that ũ satisfies the energetical identity, from which we obtain the continuity of ũ and ũ′, ũ ∈ C^{0}([0, T]; V (Ω)), ũ′ ∈ C^{0}([0, T]; L^{2}(Ω)), and as φ_{N} → φ in V, ψ_{N} → ψ in L^{2}(Ω), ũ satisfies the initial conditions (3.2).

Since problem (3.1), (3.2) has a unique solution u, we get that ũ = u and the whole sequence w_{N} has property (3.12), i.e. when N → ∞,

w_{N} → u weakly in L^{2}(0, T; V (Ω)), weakly- ∗ in L^{∞}(0, T; V (Ω)), w′_{N} → u′ weakly in L^{2}(0, T; L^{2}(Ω)), weakly- ∗ in L^{∞}(0, T; L^{2}(Ω)).

(3.14)

Let us prove that {w_{N}} satisfies the convergence properties formulated in the theorem. Applying the energetical equalities for u(t) and w_{N}(t), we obtain the following equality for the difference δ_{N}(t) = u(t) − w_{N}(t):

(δ′_{N}(t), δ′_{N}(t)) + B^{Ω}(δ_{N}(t), δ_{N}(t)) + 2B^{Ω}(u(t), w_{N}(t))

+ 2(u′(t), w′_{N}(t)) = (δ′_{N}(0), δ′_{N}(0)) + B^{Ω}(δ_{N}(0), δ_{N}(0))

+ 2(u′(0), w′_{N}(0)) + 2B^{Ω}(u(0), w_{N}(0)) + 2L̃^{Ω}(δ_{N})(t) + 4L̃^{Ω}(w_{N})(t). (3.15)

Denote

J_{N}(t) = (u′(0), w′_{N}(0)) + B^{Ω}(u(0), w_{N}(0)) − B^{Ω}(u(t), w_{N}(t))
− (u′(t), w′_{N}(t)) + 2L̃^{Ω}(w_{N})(t).

Then from (3.15) we obtain

(δ′_{N}(t), δ′_{N}(t)) + B^{Ω}(δ_{N}(t), δ_{N}(t)) = (δ′_{N}(0), δ′_{N}(0))
+B^{Ω}(δ_{N}(0), δ_{N}(0)) + 2L̃^{Ω}(δ_{N})(t) + 2J_{N}(t).  (3.16)

From (3.11) it follows that for each t ∈ (0, T] there exists a sequence {w_{ν_{1}}(t)} such that

w_{ν_{1}}(t) → χ_{1} weakly in V (Ω), w′_{ν_{1}}(t) → χ_{2} weakly in L^{2}(Ω) as ν_{1} → ∞.  (3.17)

Let us take ζ ∈ C^{1}([0, T]), ζ(0) = 0, ζ(t) ≠ 0 and consider the above-mentioned vector-functions θ = ζv, θ_{N} = ζv_{N}, which also have property (3.13). By the integration by parts we have

∫_{0}^{t}(w′_{ν_{1}}(τ), θ_{ν_{1}}(τ))dτ = (w_{ν_{1}}(t), θ_{ν_{1}}(t)) − ∫_{0}^{t}(w_{ν_{1}}(τ), θ′_{ν_{1}}(τ))dτ.

Applying (3.14), (3.17) and passing to the limit as ν_{1} → ∞, from the latter equality we obtain

∫_{0}^{t}(u′(τ), θ(τ))dτ = (χ_{1}, θ(t)) − ∫_{0}^{t}(u(τ), θ′(τ))dτ.

On the other hand,

∫_{0}^{t}(u′(τ), θ(τ))dτ = (u(t), θ(t)) − ∫_{0}^{t}(u(τ), θ′(τ))dτ.

From the latter equalities we have (u(t), v) = (χ_{1}, v), ∀ v ∈ V (Ω), and therefore χ_{1} = u(t).

Hence since χ_{1} is unique and is equal to u(t), then for any fixed t the whole sequence {w_{N}(t)} converges to u(t) weakly in V (Ω).

Also, since w_{N} satisfies equation (3.3), we have

− ∫_{0}^{t}(w′_{ν_{1}}(τ), θ′_{ν_{1}}(τ))dτ + ∫_{0}^{t} B^{Ω}(w_{ν_{1}}(τ), θ_{ν_{1}}(τ))dτ
= −(w′_{ν_{1}}(t), θ_{ν_{1}}(t)) + ∫_{0}^{t} L^{Ω}(θ_{ν_{1}}(τ))dτ,

from which, passing to the limit as ν_{1} → ∞ and applying (3.14), (3.17), we obtain

− ∫_{0}^{t}(u′(τ), θ′(τ))dτ + ∫_{0}^{t} B^{Ω}(u(τ), θ(τ))dτ = −(χ_{2}, θ(t)) + ∫_{0}^{t} L^{Ω}(θ(τ))dτ.

Since u is a solution of problem (3.1), (3.2), we get

− ∫_{0}^{t}(u′(τ), θ′(τ))dτ + ∫_{0}^{t} B^{Ω}(u(τ), θ(τ))dτ = −(u′(t), θ(t)) + ∫_{0}^{t} L^{Ω}(θ(τ))dτ,

and therefore

(u′(t), v) = (χ_{2}, v), ∀ v ∈ V (Ω).

From the density of V (Ω) in L^{2}(Ω), it follows that χ_{2} = u′(t). As in the case of w_{ν_{1}}(t), we obtain that the whole sequence w′_{N}(t) converges to u′(t) weakly in L^{2}(Ω). Thus, for any t ∈ [0, T],

w_{N}(t) → u(t) weakly in V (Ω), w′_{N}(t) → u′(t) weakly in L^{2}(Ω) as N → ∞.

Therefore passing to the limit in J_{N}(t) as N → ∞, by the energetical identity we obtain

J_{N}(t) → (u′(0), u′(0)) + B^{Ω}(u(0), u(0)) + 2L̃^{Ω}(u)(t)
− (u′(t), u′(t)) − B^{Ω}(u(t), u(t)) = 0.  (3.18)

So, from (3.16) we have

|δ′_{N}(t)|^{2} + ∥δ_{N}(t)∥^{2} ≤ c(2J_{N}(t) + (δ′_{N}(0), δ′_{N}(0))
+ B^{Ω}(δ_{N}(0), δ_{N}(0)) + 2L̃^{Ω}(δ_{N})(t)).  (3.19)

From the conditions of the theorem we have that δ_{N}(0) → 0 strongly in V (Ω) and δ′_{N}(0) → 0 strongly in L^{2}(Ω). Thus due to (3.14), (3.18) we obtain

(δ′_{N}(0), δ′_{N}(0)) + B^{Ω}(δ_{N}(0), δ_{N}(0)) + 2J_{N}(t) + 2L̃^{Ω}(δ_{N})(t) → 0 as N → ∞.

Therefore from (3.19) we have

|δ′_{N}(t)|^{2} + ∥δ_{N}(t)∥^{2} → 0 as N → ∞.

Consequently,

w_{N}(t) → u(t) strongly in V (Ω), w′_{N}(t) → u′(t) strongly in L^{2}(Ω) as N → ∞, ∀ t ∈ [0, T].

Now we will prove the validity of the estimates of the theorem. The solution u of the three-dimensional problem satisfies equation (3.1) for any v ∈ V (Ω) and hence satisfies it for any v_{N} ∈ V_{N}(Ω) ⊂ V (Ω), i.e.

\frac{d}{dt}(u′, v_{N}) + B^{Ω}(u, v_{N}) = L^{Ω}(v_{N}), ∀ v_{N} ∈ V_{N}(Ω).

Thus, if we take into account that w_{N} corresponds to the solution w⃗_{N} of the reduced two-dimensional problem (3.5), (3.6), then from the latter equation we have

\frac{d}{dt}((u − w_{N})′, v_{N}) + B^{Ω}(u − w_{N}, v_{N}) = 0, ∀ v_{N} ∈ V_{N}(Ω).

Let u ∈ L^{2}(0, T; H^{s_{0}}(Ω)), u′ ∈ L^{2}(0, T; H^{s_{1}}(Ω)), u′′ ∈ L^{2}(0, T; H^{s_{2}}(Ω)), s_{0} ≥ s_{1} ≥ s_{2} ≥ 1, s_{1} ≥ 2. Let us consider the series expansion of the vectorfunction u by Legendre polynomials with respect to x_{3}. Denote by u_{N} the piece of series consisting of first N + 1 terms, and by α_{N}, the remainder term, i.e.

u = u_{N} + α_{N} = ∑_{k=0}^{N} a(k + \frac{1}{2}) u^{k} P_{k}(ax_{3} − b) + α_{N}, u^{k}= ∫_{h^{−}}^{h^{+}} uP_{k}(ax_{3} − b)dx_{3},

k = \overline{0, N}. Let us take initial conditions φ⃗_{N}, ψ⃗_{N} of the reduced problem (3.5),

(3.6) such that φ⃗_{N} = (φ^{0}, …, φ^{N}), ψ⃗_{N} = (ψ^{0}, …, ψ^{N}),

φ^{k}= ∫_{h^{−}}^{h^{+}} φP_{k}(ax_{3} − b)dx_{3}, ψ^{k}= ∫_{h^{−}}^{h^{+}} ψP_{k}(ax_{3} − b)dx_{3}, k = \overline{0, N}.

Thus the vector-function Δ_{N} = u_{N} − w_{N} is a solution of the problem

\begin{matrix}\frac{d}{dt}(Δ′_{N}, v_{N})+B^{Ω}(Δ_{N}, v_{N})=((α′′_{N}, v_{N})+B^{Ω}(α_{N}, v_{N})) ∀v_{N} ∈V_{N}(Ω), \\ Δ_{N}(0) = u_{N}(0) − φ_{N} = 0, Δ′_{N}(0) = u′_{N}(0) − ψ_{N} = 0. \end{matrix} (3.20)

Applying Theorem 3.1 to problem (3.20), we obtain

(Δ′_{N}(t), Δ′_{N}(t)) + B^{Ω}(Δ_{N}(t), Δ_{N}(t)) = −2 ∫_{0}^{t}(α′′_{N}(τ), Δ′_{N}(τ))dτ
−2B^{Ω}(α_{N}(t), Δ_{N}(t)) + 2 ∫_{0}^{t} B^{Ω}(α′_{N}(τ), Δ_{N}(τ))dτ, ∀ t ∈ [0, T].

From the latter equality we get

|Δ′_{N}(t)|^{2} + ∥Δ_{N}(t)∥^{2} ≤ c ∫_{0}^{t} (|Δ′_{N}(τ)|^{2} + ∥Δ_{N}(τ)∥^{2})dτ + c( ∫_{0}^{t} |α′′_{N}(τ)|^{2}dτ

+ \frac{1}{2ε}∥α_{N}(t)∥^{2} + \frac{ε}{2}∥Δ_{N}(t)∥^{2} + ∫_{0}^{t} ∥α′_{N}(τ)∥^{2}dτ),

where c depends only on λ, μ, Ω and Γ^{0}.

So by taking ε properly we have

|Δ′_{N}(t)|^{2} + ∥Δ_{N}(t)∥^{2} ≤ c_{1}( ∫_{0}^{t} (|Δ′_{N}(τ)|^{2} + ∥Δ_{N}(τ)∥^{2})dτ
+ ∫_{0}^{t} |α′′_{N}(τ)|^{2}dτ + ∥α_{N}(t)∥^{2} + ∫_{0}^{t} ∥α′_{N}(τ)∥^{2}dτ).  (3.21)

By Gronwall’s lemma, from (3.21) we obtain that for any t ∈ [0, T],

|Δ′_{N}(t)|^{2} + ∥Δ_{N}(t)∥^{2} ≤ c_{2}( ∫_{0}^{t} |α′′_{N}(τ)|^{2}dτ + ∥α_{N}(t)∥^{2} + ∫_{0}^{t} ∥α′_{N}(τ)∥^{2}dτ).

As in the proof of Theorem 2.2 we can show that

\begin{matrix}∫_{0}^{t} |α′′_{N}(τ)|^{2}dτ ≤ \frac{1}{N^{2s_{2}}} q̄(h^{+}, h^{−}, N), ∥α_{N}(t)∥^{2} ≤ \frac{1}{N^{2s_{1}−3}} q̄(h^{+}, h^{−}, N), \\ ∫_{0}^{t} ∥α′_{N}(τ)∥^{2}dτ ≤ \frac{1}{N^{2s_{1}−3}} q̄(h^{+}, h^{−}, N), q̄(h^{+}, h^{−}, N)→0 as N →∞. \end{matrix}  (3.22)

Therefore

|Δ′_{N}(t)|^{2} + ∥Δ_{N}(t)∥^{2} ≤ \frac{1}{N^{2s}} q̂(Ω, Γ^{0}, N), q̂(Ω, Γ^{0}, N) → 0 as N → ∞,

where s = min{s_{2}, s_{1} − 3/2}.

Since

|α′_{N}(t)|^{2} ≤ \frac{1}{N^{2s_{2}}} q̃(h^{+}, h^{−}, N), q̃(h^{+}, h^{−}, N) → 0 as N → ∞,

we obtain

|u′(t) − w′_{N}(t)|^{2} + ∥u(t) − w_{N}(t)∥^{2} ≤ \frac{1}{N^{2s}} q(Ω, Γ^{0}, N),

where q(Ω, Γ^{0}, N) → 0 as N → ∞.

If ∥u∥_{L^{2}(0,T;H^{s_{0}}(Ω))} ≤ c, ∥u′∥_{L^{2}(0,T;H^{s_{1}}(Ω))} ≤ c, ∥u′′∥_{L^{2}(0,T;H^{s_{2}}(Ω))} ≤ c, where c is independent of h = max_{(x_{1},x_{2})∈ω̄}(h^{+}(x_{1}, x_{2}) − h^{−}(x_{1}, x_{2})), then instead of (3.22)

we have

∫_{0}^{t} |α′′_{N}(τ)|^{2}dτ ≤ \frac{h^{2s_{2}}}{N^{2s_{2}}} q̄_{1}(N), ∥α_{N}(t)∥^{2} ≤ \frac{h^{2(s_{1}−1)}}{N^{2s_{1}−3}} q̄_{1}(N),

∫_{0}^{t} ∥α′_{N}(τ)∥^{2}dτ ≤ \frac{h^{2(s_{1}−1)}}{N^{2s_{1}−3}} q̄_{1}(N), q̄_{1}(N) → 0 as N → ∞,

and hence

\begin{matrix}|Δ′_{N}(t)|^{2} + ∥Δ_{N}(t)∥^{2} ≤ q̂_{1}(Ω, Γ^{0})\frac{h^{2s̄}}{N^{2s}} q̂_{2}(N), \\ q̂_{2}(N) → 0 as N → ∞, \end{matrix}  (3.23)

where s̄ = min{s_{2}, s_{1} − 1}.

Taking into account that

|α′_{N}(t)|^{2} ≤ \frac{h^{2s_{2}}}{N^{2s_{2}}} q̃_{1}(N), q̃_{1}(N) → 0 as N → ∞,

from (3.23) we obtain the second estimate of the theorem. □

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(Received 22.04.2002)

Authors’ addresses:

M. Avalishvili

Faculty of Mechanics and Mathematics I. Javakhishvili Tbilisi State University 2, University St., Tbilisi 380043 Georgia E-mail: mavalish.math@tsu.edu.ge

D. Gordeziani

I. Vekua Institute of Applied Mathematics I. Javakhishvili Tbilisi State University 2, University St., Tbilisi 380043 Georgia E-mail: gord@viam.hepi.edu.ge