Lobachevskii Journal of Mathematics

http://ljm.ksu.ru

Vol. 16, 2004, 79 – 89 \begin{matrix}c \\ ◯ \end{matrix} A. Rashid

Abdur Rashid

THE PSEUDOSPECTRAL METHOD FOR THERMOTROPIC PRIMITIVE EQUATION AND ITS

ERROR ESTIMATION

(submitted by A. Lapin)

Abstract. In this paper, a pseudospectral method is proposed for solving the periodic problem of thermotropic primitive equation. The strict error estimation is proved.

1. Introduction

Thermotropic primitive equation is governed by the following differential equations^{[1]}:

\{ \begin{matrix}\frac{∂U}{∂t} + U \frac{∂U}{∂x} + V \frac{∂U}{∂y} + \frac{∂ϕ}{∂x} − νΔU − F V = 0, \\ \frac{∂V}{∂t} + U \frac{∂V}{∂x} + V \frac{∂V}{∂y} + \frac{∂ϕ}{∂y} − νΔV + F U = 0, \\ \frac{∂ϕ}{∂t} + U \frac{∂ϕ}{∂x} + V \frac{∂ϕ}{∂y} + ϕ (\frac{∂U}{∂x} + \frac{∂V}{∂x} ) = 0, \end{matrix}  (1.1)

where U,V are the components of the speed in x, y directions respectively, g is the acceleration of gravity, H is the height of the geopotential surface, ϕ = gH, F is coriolis parameter and ν is the coefficient of friction.

There has been a rapid development in the spectral methods for the last two decades. They have become important tools for numerical solutions of partial differential equations, and have been widely applied to

scheme, error estimation.

2000 Mathematical Subject Classification. 35Q35, 65M70,65N30..

Key words and phrases. Thermotropic primitive equation, pseudospectral

This work is supported by Gomal University, D.I.Khan, Pakistan..

numerical simulations in various fields [2-5]. Although the pseudospectral methods are easier to implement for nonlinear partial differential equations, they are not stable as the spectral ones due to ’aliasing’. Therefore some author proposed the filtering technique [10-11] to remedy the deficiency of instability. Some papers have also been devoted to theoretical study and numerical solutions of (1.1) [6-9].

The aim of this paper is to consider the periodic initial boundaryvalue problem for thermotropic primitive equation. A pseudospectral scheme with restraint operator in combination with first order time differencing technique is considered for thermotropic primitive equation. The stability and rate of convergence for the approximate problem are proved.

2. The Pseudospectral Scheme

Let Ω = {(x, y)| − π < x, y < π} and all functions have the period 2π for the variable x and y. The norm of the space L^{q}(Ω) is denoted by ∥·∥_{L^{q}(Ω)}. In particular, the scaler product and the norm of L^{2}(Ω) are denoted by (·, ·) and ∥·∥_{L^{2}(Ω)} respectively. Let m_{1}, m_{2} and N be integers and m = \sqrt{m_{1}^{2} + m_{2}^{2}}. Define

V_{N} = Span {e^{i(m_{1}x+m_{2}y)}| |m| ≤ N} , N > 0.

Let P_{N} be the orthogonal projection operator, i.e.

(P_{N}η, ψ) = (η, ψ) , ∀ψ ∈ V_{N}.

For the pseudospectral approximation, we put the nodes

(x_{j_{1}}, y_{j_{2}}) = (\frac{2πj_{1}}{2N + 1}, \frac{2πj_{2}}{2N + 1}) , −N ≤ j_{1}, j_{2} ≤ N,

and let P̃_{c} be the interpolation operator, i.e. for η(x, y) ∈ C(Ω)

P̃_{c}η (x_{j_{1}}, y_{j_{2}}) = η (x_{j_{1}}, y_{j_{2}}) , −N ≤ j_{1}, j_{2} ≤ N.

Define P_{c} = P_{N} P̃_{c}. To weaken the nonlinear instability of computation, we follow the work of [11] to adopt the filtering operator R_{γ} with γ > 1, i.e. if

η(x, y) = ∑_{|m|≤N} η_{m_{1},m_{2}}e^{i(m_{1}x+m_{2}y)},

then

R_{γ}η(x, y) = ∑_{|m|≤N} (1 − (\frac{|m|}{N})^{γ}) η_{m_{1},m_{2}}e^{i(m_{1}x+m_{2}y)}.

THERMOTROPIC PRIMITIVE EQUATION 81

Let τ be the mesh spacing of the variable t and define

S_{τ} = {t = kτ|k = 0, 1, 2, ⋯} .

η_{t}(t) = \frac{η(t + τ) − η(t)}{τ}.

To approximate the nonlinear terms, we define

d_{α} (η, u, v) = αd^{(1)} (η, u, v) + (1 − α)d^{(2)} (η, u, v) , 0 ≤ α ≤ 1,

d^{(1)} (η, u, v) = P_{c} (u\frac{∂η}{∂x} + v \frac{∂η}{∂y} ) ,

d^{(2)} (η, u, v) = \frac{∂}{∂x}P_{c}(uη) + \frac{∂}{∂y} P_{c}(vη).

Let u^{N}, v^{N}, φ^{N} be the approximations to U, V and ϕ respectively, where for all (x, y) ∈ Ω and t ∈ S_{τ},

η^{N}(x, y, t) = ∑_{|m|,|n|≤N} η_{m,n}^{N}(t)e^{i(mx+ny)}, η = u, v, φ.

The pseudospectral scheme for solving (1.1) is

\{ \begin{matrix}u_{t}^{N} + R_{γ}d_{1/2} (R_{γ}(u^{N} + δτu_{t}^{N}), u^{N}, v^{N}) + \frac{∂}{∂x}φ^{N} − νΔ(u^{N} + στu_{t}^{N}) \\ −F (v^{N} + δτv_{t}^{N}) = 0, \\ v_{t}^{N} + R_{γ}d_{1/2} (R_{γ}(v^{N} + δτv_{t}^{N}), u^{N}, v^{N}) + \frac{∂}{∂y}φ^{N} − νΔ(v^{N} + στv_{t}^{N}) \\ +F (u^{N} + δτu_{t}^{N}) = 0, \\ φ_{t}^{N} + R_{γ}d_{0} (R_{γ}(φ^{N} + δτφ_{t}^{N}), u^{N}, v^{N}) \\ +A(φ^{N} + δτφ_{t}^{N}, u^{N} + δτu_{t}^{N}, v^{N} + δτv_{t}^{N}) = 0, \end{matrix}

(2.1)

where 0 ≤ δ ≤ 1, 0 ≤ σ ≤ 1 and A(η, ξ, η^{∗}) = P_{c} [η (\frac{∂ξ}{∂x} + \frac{∂η^{∗}}{∂x} )] .

3. Some Lemmas

Lemma 1. [1]. For all η(x, y, t)

2 (η(t), η_{t}(t)) = (∥η(t)∥^{2})_{t} − τ ∥η_{t}(t)∥^{2} .

Lemma 2. [5]. For all η(x, y, t) ∈ V_{N}, then

∥\frac{∂η}{∂x}∥^{2} ≤ N^{2} ∥η(t)∥^{2} , ∥\frac{∂η}{∂y} ∥^{2} ≤ N^{2} ∥η(t)∥^{2} .

Lemma 3. [10]. For all η(x, y, t) ∈ V_{N} and ξ(x, y, t) ∈ V_{N}, then ∥η(t)ξ(t)∥^{2} ≤ (2N + 1)^{2} ∥η(t)∥^{2} ∥ξ(t)∥^{2} .

Lemma 4. [15]. For all η(x, y, t) ∈ H^{β}(Ω) and ξ(x, y, t) ∈ V_{N}, then

∥P_{N}η(t) − η(t)∥_{H^{S}(Ω)} ≤ C_{1}N^{S−β} ∥η(t)∥_{H^{β}(Ω)} , 0 ≤ s ≤ β,
∥P_{C}η(t) − η(t)∥_{H^{S}(Ω)} ≤ C_{2}N^{S−β} ∥η(t)∥_{H^{β}(Ω)} , 0 ≤ s ≤ β, β > 1,
∥R_{γ}ξ(t) − ξ(t)∥_{H^{S}(Ω)} ≤ C_{3}N^{S−β} ∥ξ(t)∥_{H^{β}(Ω)} , 0 ≤ s ≤ β, γ > β − s,
∥R_{γ}P_{N}η(t) − η(t)∥_{H^{S}(Ω)} ≤ C_{4}N^{S−β} ∥η(t)∥_{H^{β}(Ω)} , 0 ≤ s ≤ β, γ > β − s,

where C_{1} − C_{4} are positive constants.

Lemma 5. [9]. Assume that the following conditions are fulfilled:

(i) ξ(t) and η(t) are non-negative functions defined on S_{τ}; (ii) ρ, a, M_{1}, M_{2}, and M_{3} are nonnegative constants; (iii) A(x) is a function such that, if x ≤ M_{3}, then A(x) ≤ 0; (iv) ξ(t) ≤ ρ + τ ∑_{t′=0}^{t−τ} [M_{1}ξ(t′) + M_{2}N^{a}ξ^{2}(t′) + A(ξ(t′))η(t′)] ;

(v) ρe^{(M_{1}+M_{2})T} ≤ min (M_{3}, \frac{1}{N^{a}} ), ξ(0) ≤ ρ, t ≤ T. Then

ξ(t) ≤ ρe^{(M_{1}+M_{2})t}.

In particular, if M_{2} = 0 and A(ξ(t′)) = 0, then for all ρ and n.

ξ(t) ≤ ρe^{M_{1}t}.

4. Error Estimation

For simplicity, we take δ = 0, let U^{N} = P_{N}U, V^{N} = P_{N}V, and ϕ^{N} = P_{N}ϕ, then (1.1) leads to

\{ \begin{matrix}U_{t}^{N} + R_{γ}d_{1/2} (R_{γ}U^{N}, U^{N}, V^{N}) + \frac{∂}{∂x}ϕ^{N} − νΔ(U^{N} + στU_{t}^{N}) − F V^{N} = 0, \\ V_{t}^{N} + R_{γ}d_{1/2} (R_{γ}V^{N}, U^{N}, V^{N}) + \frac{∂}{∂y}ϕ^{N} − νΔ(V^{N} + στV_{t}^{N}) + F U^{N} = 0, \\ ϕ_{t}^{N} + R_{γ}d_{0} (R_{γ}ϕ^{N}, U^{N}, V^{N}) + A(ϕ^{N}, U^{N}, V^{N}) = 0, \end{matrix}

(4.1)

THERMOTROPIC PRIMITIVE EQUATION 83

where

G_{1}^{N} = U_{t}^{N} − \frac{∂U^{N}}{∂t} ,
G_{2}^{N} = R_{γ}d_{1/2} (R_{γ}U^{N}, U^{N}, V^{N}) − P_{N} [U \frac{∂U}{∂x} + V \frac{∂U}{∂y} ] ,
G_{3}^{N} = −νστΔU_{t}^{N},
G_{4}^{N} = V_{t}^{N} − \frac{∂V^{N}}{∂t} ,
G_{5}^{N} = R_{γ}d_{1/2} (R_{γ}V^{N}, U^{N}, V^{N}) − P_{N} [U \frac{∂V}{∂x} + V \frac{∂V}{∂y} ] ,
G_{6}^{N} = −νστΔV_{t}^{N},
G_{7}^{N} = ϕ_{t}^{N} − \frac{∂ϕ^{N}}{∂t} ,
G_{8}^{N} = R_{γ}d_{0} (R_{γ}ϕ^{N}, U^{N}, V^{N}) − P_{N} [U \frac{∂ϕ}{∂x} + V \frac{∂ϕ}{∂y}] ,
G_{9}^{N} = A (ϕ^{N}, U^{N}, V^{N}) − P_{N} [ϕ\frac{∂U}{∂x} + ϕ\frac{∂V}{∂x} ] .

Put

ũ = u − U^{N}, ṽ = v − V^{N}, φ̃ = φ − ϕ^{N}.

Then from (1.1) and (2.1), we obtain

\{ \begin{matrix}ũ_{t}^{N} + ξ_{1}^{N} + ξ_{2}^{N} + \frac{∂}{∂x}φ^{N} − νΔ (ũ^{N} + στũ_{t}^{N}) − F ṽ^{N} = ∑_{l=1}^{3} G_{l}^{N}, \\ ṽ_{t}^{N} + ξ_{3}^{N} + ξ_{4}^{N} + \frac{∂}{∂y}φ^{N} − νΔ (ṽ^{N} + στṽ_{t}^{N}) + F ũ^{N} = ∑_{l=4}^{6} G_{l}^{N}, \\ φ̃_{t}^{N} + ∑_{k=5}^{9} ξ_{k}^{N} = ∑_{l=7}^{9} G_{l}^{N}, \end{matrix}

(4.2)

where

ξ_{1}^{N} = R_{γ}d_{1/2} (R_{γ}ũ^{N}, U^{N}, v^{N}) + R_{γ}d_{1/2} (R_{γ}U^{N}, ũ^{N}, ṽ^{N}) ,
ξ_{2}^{N} = R_{γ}d_{1/2} (R_{γ}ũ^{N}, ũ^{N}, ṽ^{N}) ,
ξ_{3}^{N} = R_{γ}d_{1/2} (R_{γ}ṽ^{N}, U^{N}, V^{N}) + R_{γ}d_{1/2} (R_{γ}V^{N}, ũ^{N}, ṽ^{N}) ,
ξ_{4}^{N} = R_{γ}d_{1/2} (R_{γ}ṽ^{N}, ũ^{N}, ṽ^{N}) ,
ξ_{5}^{N} = R_{γ}d_{0} (R_{γ}φ̃^{N}, U^{N}, V^{N}) + R_{γ}d_{0} (R_{γ}ϕ^{N}, ũ^{N}, ṽ^{N}) ,
ξ_{6}^{N} = R_{γ}d_{0} (R_{γ}φ̃^{N}, ũ^{N}, ṽ^{N}) ,
ξ_{7}^{N} = −A (φ̃^{N}, ũ^{N}, ṽ^{N}) ,
ξ_{8}^{N} = −A (φ̃^{N}, U^{N}, V^{N}) ,
ξ_{9}^{N} = −A (ϕ^{N}, ũ^{N}, ṽ^{N}) .

We shall use the following notations

E = (U, V, ϕ) , E^{N} = (U^{N}, V^{N}, ϕ^{N}) , Ẽ^{N} = (ũ^{N}, ṽ^{N}, φ̃^{N}) ,
∥Ẽ^{N}(t)∥^{2} = ∥ũ^{N}(t)∥^{2} + ∥ṽ^{N}(t)∥^{2} + ∥φ̃^{N}(t)∥^{2} ,
∥Ẽ_{t}^{N}(t)∥^{2} = ∥ũ_{t}^{N}(t)∥^{2} + ∥ṽ_{t}^{N}(t)∥^{2} + ∥φ̃_{t}^{N}(t)∥^{2} ,
|Ẽ^{N}(t)|_{1}^{2} = |ũ^{N}(t)|_{1}^{2} + |ṽ^{N}(t)|_{1}^{2} + |φ̃^{N}(t)|_{1}^{2} .

Let H^{β}(Ω) be the Sobolev space equipped with the norm ∥·∥_{H^{β}(Ω)}. In particular L^{2}(Ω) = H^{0}(Ω), we define E_{1} = (U, V, ϕ)

∥E_{1}∥_{H^{β}(Ω)}^{2} = ∥U(t)∥_{H^{β}(Ω)}^{2} + ∥V (t)∥_{H^{β}(Ω)}^{2} + ∥ϕ(t)∥_{H^{β}(Ω)}^{2} ,

∥|E_{1}∥ |_{β} = max_{0≤t≤τ} ∥E_{1}(t)∥_{H^{β}(Ω)} .

Now we suppose

ρ(t) = ∥Ẽ^{N}(0)∥^{2}) + ντ(σ + \frac{q}{2})|Ẽ^{N}(0)|_{1}^{2} + τ ∑_{t′=0}^{t−τ} ∥G_{l}^{N}(t′)∥^{2}, l = 1, ..., 9

\begin{aligned}Ẽ(t) & = ∥Ẽ^{N}(t)∥^{2} + ντ(σ + \frac{q}{2})|Ẽ^{N}(0)|_{1}^{2} \\ & + τ ∑_{t′=0}^{t−τ} r_{1}τ [∥Ẽ_{t}^{N}(t′)∥^{2} + ν(2 − 7ε)|Ẽ^{N}(t′)|_{1}^{2}] . \end{aligned}

Theorem 1. Suppose the following conditions are fulfilled

(i) δ = 0, τN^{2} < ∞, (ii) σ > 1/2 or τN^{2} < \frac{2}{v(1−2σ)}, (iii) for suitably small positive constant M_{1} and all t ≤ T , such that ρ(T) ≤ \frac{M_{1}}{N^{2}} . Then there exists a positive constant M_{2} such that for all t ∈ S_{τ}, t ≤ T, we have

Ẽ(t) ≤ ρ(t)e^{M_{2}t}.

Theorem 2. Assume that the conditions (i), (ii) of Theorem 1 are satisfied. In addition E ∈ C^{2}(0, T; m_{0}^{0}(Ω)), E_{1} ∈ C(0, T; H_{\frac{5}{2} +r}(Ω) ⋂ m_{0}^{B+1}(Ω)), r > 0, β ≥ 1, then

Ẽ(t) ≤ M_{3}(τ^{2} + N^{−2β})e^{M_{4}t}.

M_{ℓ} being positive constants depending only on ∥|E_{1}∥|_{\frac{5}{2} +r} and ν.

Now we define

∥η(t)∥_{m_{r}^{β}} = max_{0≤s≤q} (\frac{1}{4π^{2}} ∑_{a+b=0}^{β} ∫∫_{Ω} (\frac{∂^{a+b}η(x, y, t)}{∂x^{a}∂y^{b}})^{2} dxdy)^{1/2} ,

∥η∥_{C^{q}(0,T;m_{r}^{β}(Ω))} = max_{0≤s≤q} max_{0≤t≤T} ∥\frac{∂^{s}η(t)}{∂t^{s}} ∥_{m_{r}^{β}(Ω)} ,

C^{q} (0, T; m_{r}^{β}(Ω)) = (η(x, y, t)| ∥η∥_{C^{q}(0,T;m_{r}^{β}(Ω))} < ∞) .

5. The Proof of Theorem 1

Let c be a positive constant which may be different in different cases, q denote an undetermined positive constant and ε > 0. Taking the scaler product (4.2) with 2ũ^{N} + qτũ_{t}^{N}, we have

\begin{aligned}(∥ũ^{N}(t)∥^{2})_{t} & + τ(q − 1 − ε)∥ũ_{t}^{N}(t)∥^{2} \\ & + (2ũ^{N}(t) + qτũ_{t}^{N}(t), ξ̃_{1}^{N}(t) + ξ̃_{2}^{N}(t) \\ & + \frac{∂ϕ̃^{N}}{∂x} (t) + F ṽ^{N}(t)) + 2ν|ũ^{N}(t)|_{1}^{2} \\ & + ντ(σ + \frac{q}{2})(|ũ(t)|_{1}^{2})_{t} + ντ^{2}(σq − σ − \frac{q}{2})|ũ_{t}^{N}(t)|_{1}^{2} \\ & ≤ c∥ũ^{N}(t)∥^{2} + c(1 + \frac{q^{2}τ}{4ε}) ∑_{l=1}^{3} ∥G_{l}^{N}(t)∥^{2}. \end{aligned} (5.1)

Similarly from the second and third formulas of (4.2), we have

\begin{aligned}(∥ṽ^{N}(t)∥^{2})_{t} & + τ(q − 1 − ε)∥ṽ_{t}^{N}(t)∥^{2} \\ & + (2ṽ^{N}(t) + qτṽ_{t}^{N}(t), ξ̃_{3}^{N}(t) + ξ̃_{4}^{N}(t) \\ & + \frac{∂ϕ̃^{N}}{∂y} (t) + F ũ^{N}(t)) + 2ν|ṽ^{N}(t)|_{1}^{2} \\ & + ντ(σ + \frac{q}{2})(|ṽ(t)|_{1}^{2})_{t} + ντ^{2}(σq − σ − \frac{q}{2})|ṽ_{t}^{N}(t)|_{1}^{2} \\ & ≤ c∥ṽ^{N}(t)∥^{2} + c(1 + \frac{q^{2}τ}{4ε}) ∑_{l=4}^{6} ∥G_{l}^{N}(t)∥^{2} \end{aligned} (5.2)

\begin{aligned}(∥φ̃^{N}(t)∥^{2})_{t} & + τ(q − 1 − ε)∥φ̃_{t}^{N}(t)∥^{2} \\ & + (2φ̃^{N}(t) + qτφ̃_{t}^{N}(t), ∑_{l=5}^{9} ξ̃_{l}^{N}(t)) \\ & ≤ c∥φ̃^{N}(t)∥^{2} + c(1 + \frac{q^{2}τ}{4ε}) ∑_{l=7}^{9} ∥G_{l}^{N}(t)∥^{2}. \end{aligned}  (5.3)

Putting (5.1)-(5.3) together, we get

\begin{aligned}(∥Ẽ^{N}(t)∥^{2})_{t} & + τ(q − 1 − ε)∥Ẽ_{t}^{N}(t)∥^{2} \\ & + 2ν|Ẽ^{N}(t)|_{1}^{2} + ντ(σ + \frac{q}{2})(|Ẽ(t)|_{1}^{2})_{t} \\ & + ντ^{2}(σq − σ − \frac{q}{2})|Ẽ_{t}^{N}(t)|_{1}^{2} + ∑_{l=1}^{6} M_{l}^{N}(t) \\ & ≤ c∥Ẽ^{N}(t)∥^{2} + c(1 + \frac{q^{2}τ}{4ε}) ∑_{l=1}^{9} ∥G_{l}^{N}(t)∥^{2}, \end{aligned}  (5.4)

where

M_{1}^{N}(t) = (2ũ^{N}(t) + qτũ_{t}^{N}(t), ξ_{1}^{N}(t)) + (2ṽ^{N}(t) + qτṽ_{t}^{N}(t), ξ_{3}^{N}(t))
+ (2φ̃^{N}(t) + qτφ̃_{t}^{N}(t), ξ_{5}^{N}(t)) − qτ(ũ_{t}^{N}(t), F ṽ^{N}(t))
+ qτ(ṽ_{t}^{N}(t), F ũ^{N}(t)),
M_{2}^{N}(t) = (2ũ^{N}(t) + qτũ_{t}^{N}(t), ξ_{2}^{N}(t)) + (2ṽ^{N}(t) + qτṽ_{t}^{N}(t), ξ_{4}^{N}(t))
+ (2φ̃^{N}(t) + qτφ̃_{t}^{N}(t), ξ_{6}^{N}(t)),
M_{3}^{N}(t) = (2ũ^{N}(t) + qτũ_{t}^{N}(t), \frac{∂}{∂x}φ^{N}(t)) + (2ṽ^{N}(t) + qτṽ_{t}^{N}(t), \frac{∂}{∂y} φ̃^{N}(t)),
M_{4}^{N}(t) = (2φ̃^{N}(t) + qτφ̃_{t}^{N}(t), ξ_{7}^{N}(t)),
M_{5}^{N}(t) = (2φ̃^{N}(t) + qτφ̃_{t}^{N}(t), ξ_{8}^{N}(t)),
M_{4}^{N}(t) = (2φ̃^{N}(t) + qτφ̃_{t}^{N}(t), ξ_{9}^{N}(t)).

We now estimate |M_{l}^{N}(t)|. Because of the Schwarz inequality and embedding theorem, we have

|M_{1}^{N}(t)| ≤ εν|Ẽ^{N}(t)|_{1}^{2} + ετ∥Ẽ_{t}^{N}(t)∥^{2} + \frac{c}{εν} (1 + τq^{2}N^{2})|∥E_{1}^{N}∥|_{\frac{5}{2} +r}∥Ẽ^{N}(t)∥^{2},
|M_{2}^{N}(t)| ≤ εν|Ẽ^{N}(t)|_{1}^{2} + ετ∥Ẽ_{t}^{N}(t)∥^{2} + \frac{cN^{2}}{εν} (∥Ẽ^{N}(t)∥^{2}

+ |Ẽ^{N}(t)|_{1}^{2} + |∥E_{1}^{N}∥|_{\frac{5}{2} +r})∥Ẽ^{N}(t)∥^{2},

|M_{3}^{N}(t)| ≤ ετ|Ẽ_{t}^{N}(t)|_{1}^{2} + εν|Ẽ^{N}(t)|_{1}^{2} + \frac{c}{εν} (1 + τq^{2}N^{2})∥Ẽ^{N}(t)∥^{2},
|M_{4}^{N}(t)| ≤ ετ∥Ẽ_{t}^{N}(t)∥_{1}^{2} + εν|Ẽ^{N}(t)|_{1}^{2} + \frac{c}{εν} (1 + τq^{2}N^{2})∥Ẽ^{N}(t)∥^{2},
|M_{5}^{N}(t)| ≤ ετ∥Ẽ_{t}^{N}(t)∥_{1}^{2} + εν|Ẽ^{N}(t)|_{1}^{2} + \frac{cN^{2}}{εν} (1 + τq^{2}N^{2})∥Ẽ^{N}(t)∥^{4},
|M_{6}^{N}(t)| ≤ ετ∥Ẽ_{t}^{N}(t)∥_{1}^{2} + εν|Ẽ^{N}(t)|_{1}^{2} + \frac{c}{εν} |∥E_{1}∥|_{\frac{5}{2} +r}∥Ẽ^{N}(t)∥^{2}.

By substituting the above estimates into (5.4), we get

\begin{aligned}(∥Ẽ^{N}(t)∥^{2})_{t} & + τ(q − 1 − 7ε)∥Ẽ^{N}(t)∥^{2} + ν(2 − 6ε)|Ẽ^{N}(t)|_{1}^{2} \\ & + ντ(σ + \frac{q}{2})(|Ẽ^{N}(t)|_{1}^{2})_{t} + ντ^{2}(σq − σ − \frac{q}{2})|Ẽ^{N}(t)|_{1}^{2} \\ & ≤ H^{N}(t) + A_{1}∥Ẽ^{N}(t)∥^{2} + B_{1}∥Ẽ^{N}(t)∥^{4} + B_{2}|Ẽ^{N}(t)|_{1}^{2}, \end{aligned} (5.5)

where

A_{1} = c (1 + \frac{c}{εν} + \frac{c}{εν} (1 + τq^{2}N^{2})) |∥E_{1}∥|_{\frac{5}{2} +r},

B_{1} = \frac{cN^{2}}{εν} (1 + τq^{2}N^{2}) + \frac{cN^{2}}{εν} ,

B_{2} = \frac{cN^{2}}{εν} ∥Ẽ^{N}(t)∥^{2},

H^{N}(t) = c (1 + \frac{q^{2}τ}{4ε} ) ∑_{l=1}^{9} ∥G_{l}^{N}∥^{2}.

Now let ε be suitably small, r_{1} > 0, and

q_{1} = max [1 + r_{1} + 7ε, \frac{2σ}{2σ − 1}] ,
q_{2} = r_{1} + 1 + 7ε + \frac{ντN^{2}}{2} ,
q_{3} = (2r_{1} + 2 + 14ε + 2σντN^{2}) [2 − ντN^{2}(1 − 2σ)]^{−1} .

If σ > 1/2, we put q = q_{1}, and it follows from (5.5) that

\begin{aligned}(∥Ẽ^{N}(t)∥^{2})_{t} & + r_{1}τ∥Ẽ^{N}(t)∥^{2} + ν(2 − 6ε)|Ẽ^{N}(t)|_{1}^{2} \\ & + ντ(σ + \frac{q}{2})(|Ẽ^{N}(t)|_{1}^{2})_{t} \\ & ≤ H^{N}(t) + A_{1}∥Ẽ^{N}(t)∥^{2} + B_{1}∥Ẽ^{N}(t)∥^{4} + B_{2}|Ẽ^{N}(t)|_{1}^{2}. \end{aligned} (5.6)

If σ = 1/2, we put q = q_{2}, and so

τ(q − 1 − 7ε)∥Ẽ_{t}^{N}∥^{2} + ντ^{2}(σq − σ − \frac{q}{2})|Ẽ_{t}^{N}(t)|_{1}^{2} ≥ r_{1}τ|Ẽ_{t}^{N}(t)|^{2}.

Therefor (5.6) is still holds. If σ < 1/2, τN^{2} < \frac{2}{ν(1−2σ)}, we put q = q_{3}, and thus (5.12) holds. By summing up (5.6) for t ∈ S_{τ} , we get

\begin{aligned}(∥Ẽ^{N}(t)∥^{2})_{t} & + ντ(σ + \frac{q}{2})(|Ẽ^{N}(t)|_{1}^{2})_{t} + τ ∑_{t′=0}^{t−τ} r_{1}τ∥Ẽ_{t}^{N}(t′)∥^{2} \\ & + ν(2 − 7ε)|Ẽ^{N}(t′)|_{1}^{2} ≤ ρ(t) \\ & + τ ∑_{t′=0}^{t−τ} A_{1}Ẽ_{1}^{N}(t′) + B_{1}Ẽ_{1}^{2}(t′) \\ & + B_{2}|Ẽ_{2}^{N}(t′)|_{1}^{2}, \end{aligned}  (5.7)

where

ρ(t) = ∥Ẽ^{N}(0)∥^{2}) + ντ(σ + \frac{q}{2})|Ẽ^{N}(0)|_{1}^{2} + τ ∑_{t′=0}^{t−τ} H^{N}(t′)

from which and Lemma 5, the proof is completed.

6. The Proof of Theorem 2

We first have

∥G_{l}^{N}(t) ≤ cτ∥E∥_{C^{2}(0,T;m_{0}^{0}(Ω))}, l = 1, 4, 7.

From Lemma 4 and the embedding theorem, we get

∥G_{l}^{N}(t)∥ ≤ c∥|E_{1}∥|_{\frac{5}{2} +r}N^{−β}∥E_{1}∥_{m_{0}^{β+1}(Ω)}, l = 2, 5, 8. It is easy to show that

∥G_{l}^{N}(t)∥ ≤ cτ (∥\frac{∂E}{∂t} (t)∥_{m_{2}^{0}(Ω)} + ∥\frac{∂E}{∂t} (t)∥_{m_{2}^{0}(Ω)}) , l = 3, 6.

We have also ∥G_{9}^{N}(t)∥ ≤ cN^{−β}∥|E_{1}∥|_{\frac{5}{2} +r}∥E(t)∥_{m_{0}^{β+1}(Ω)}, ∥Ẽ^{N}(0)∥ ≤ cN^{−β}∥E(0)∥_{m_{0}^{β}(Ω)}.

THERMOTROPIC PRIMITIVE EQUATION 89

Therefore if the conditions of Theorem 1 are fulfilled, then

ρ(t) ≤ c (τ^{2} + N^{−2β}) .

By combining the above estimations with Theorem 1, we complete the proof of Theorem 2.

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Department of Mathematics, Gomal University, D.I.Khan, Pakistan. E-mail address: rashid himat@yahoo.com

Received February 8, 2004; revised version May 27, 2004