The well-posedness and regularity of a rotating blades equation

1.   Introduction

Rotating blades (thin-walled beam) are important structures widely used in mechanical and aerospace engineering as aviation engine blades, various cooling fans, windmill blades, helicopter rotor blades, airplane propellers etc. The study of the dynamics of rotating blades is important to design purposes, optimization, and control.

If the shear effect is not considered, the Euler-Bernoulli beam equation is used to model vibration of thin-walled beam. Chen et al.[6] studied the boundary feedback stabilization of a linear Euler-Bernoulli beam equation, they proved that the total energy of the equation decays uniformly and exponentially with $t$, when the beam is clamped at the left end and subjected to a feedback boundary conditions at the right end. Then Guo et al.[12] proved the well-posedness and stability of the system proposed by Chen et al.[14] considered a nonlinear Euler-Bernoulli beam equation with a feedback force applied at the free end, the existence of the weak and classical solutions were proved. Other relevant studies are referred to Refs. [4,18,19,17,13]. Note that the above literatures studied the longitudinal vibration in one direction.

A very extensive work devoted to the longitudinal vibration in two directions were done by Librescu and Song[21,32,33] and their co-workers[25,28]. Under assumption of the cross-section to be rigid in its own plane, they modelled the rotating blades by 1-D linear governing equations. The influence of many factors on rotating blades, such as the anisotropy and heterogeneity of constituent materials, functionally graded materials (FGM), temperature, shear effects, primary and secondary warping phenomena (Vlasov effect), centrifugal and Coriolis forces etc have been taken into account in the 1-D linear governing equations.

Following Librescu's approach, various blades models were derived. Georgiades et al.[11] modelled a rotating blades by means of linear strain-displacement relationships, considering arbitrary pitch (presetting) angle and non-constant rotating speed. Choi et al.[8] studied bending vibration control of the pre-twisted rotating composite blades, who emphasized the important of piezoelectric effect in single cell composite blades. Fazelzadeh et al.[9] considered a thin-walled blades made of FGM which is used in turbomachinery under aerothermodynamics loading. In the paper, quasi-steady aerodynamic pressure loadings was determined by the first-order piston theory, and steady beam surface temperature was obtained from gas dynamics theory. Fazelzadeh et al.[10] studied the governing equations which included the effects of the presetting angle and the rotary inertia. The effects of steady wall temperature and quasi-steady aerodynamic pressure loadings due to flow motion were also taken into account.

The models in Refs.[21,32,33,25,28,11,8,9,10] are linear. When the engine blades rotate at a low speed, the linear approximation can completely meet the needs of practical application. However, when the blade rotate at a high speed, the simple linear approximation can not accurately describe the dynamic behavior of the system. So the non-linear analysis of rotating blades has attracted considerable attention.

The nonlinear governing equations of a rotating blades at constant angular velocity was presented by Anderson[1], and the author linearized the equation under the assumption that a small perturbed motion occurred at an initially stressed equilibrium configuration. Chen et al.[7] considered the effects of geometric non-linearity, shear deformation and rotary inertia. Arvin et al.[2] builded a nonlinear governing equation for rotating blades considering centrifugal forces by means of von-Karmans strain-displacement relationships under assumption of the constant rotation speed and zero pitch (presetting) angle. Yao et al.[39] employed the Hamilton's principle to derive the nonlinear governing equations with periodic rotating speed, arbitrary pitch (presetting) angle and linear pre-twist angle. Under the assumption that the location of shear centre is different from the centre of gravity, Avramov et al.[3] obtained results of the investigations on flexural-flexural-torsional nonlinear vibrations of twisted rotating blades described by the model of three nonlinear integro-differential equations. Other nonlinear models can be found in Refs.[34,29,27,37,15,31,36].

To the best of the author's knowledge, all the above literatures about the longitudinal vibration in two directions skipped the existence proof of solutions to the governing equations, and directly used the finite element method to study the influence of various parameters on blades vibration. To address this situation, we first try to model a governing equations of the blades with arbitrary rotating speed, arbitrary pre-setting angle and pre-twist angle. In the process of building the model, we take into account the free vibration at the right end of the blade and the nonlinear relationship between stress and strain. In the paper we aim to investigate the well-posedness and regularity of the governing equations. The well-posedness of other nonlinear blade vibration models can be found in Refs. [5,38,26,35,16].

2.   Model building

Let us consider a slender, straight blades mounted on a rigid hub of radius $R_0$ rotating with the angular velocity $\omega(t)$. The length of the blades is denoted by $l$, its wall thickness by $h$, the length and the width of the cross section of the blades are $a$ and $b$, respectively. The sum of pre-twist angle $\alpha(x)$ and pre-setting angle $\beta$ is defined by $\theta(x)$, as shown in Figure 1 and 2.

2.1.   Assumptions

To derive the model of the rotating blades, the following kinematic and static assumptions are postulated:

(ⅰ) The blades is perfectly elastic bodies, the blades material is isotropic and is not affected by temperature,

(ⅱ) The cross section of the blades is rectangular and all its geometrical dimensions remain invariant in its plane,

(ⅲ) The ratio of wall thickness $h$ to the radius of curvature $r$ at any point of the blade wall is negligibly small while compared to unity,

(ⅳ) The transverse shear effect of the cross section is neglected,

(ⅴ) The axial displacement $w$ is much smaller than $u$ or $v$ and the derivatives of $w$ can be neglected in the strain-displacement relations. where $u, v, w$ represent the displacement of the middle line of cross-sections along the $y, z, x$ axis, respectively.

2.2.   Four coordinate systems are defined to describe the motion of the blades

Inertial Cartesian coordinate systems $(X, Y, Z)$ attached to the center of the hub $O$, the unit vectors of the inertial coordinate systems $(X, Y, Z)$ is defined as $(I, J, K)$.

Rotating coordinate systems $(x, y, z)$ located at the blade root, the origin $o$ of the rotating coordinate systems is set at the center of the beam cross section, the unit vectors of the inertial coordinate systems $(x, y, z)$ is defined as $(i, j, k)$.

Transformation between $XYZ$ and $xyz$ frames can be written in the form

Local coordinates systems $(x^p, y^p, z^p)$ also located at the blade root and oriented with respect to plane of rotation $(y, z)$ at angle $\omega(t)$. So $(x, y, z)' = \mathrm B (x^p, y^p, z^p)'$, where the rotation matrix $\mathrm B$ is

Local, curvilinear coordinate systems $(x, n, s)$ related to blade cross section (Figure 2). Its origin is set conveniently at the point on a mid-line contour. $s$ and $n$ are the circumferential and thickness coordinate variables, the unit vectors of axis $n$ and $s$ are defined as $\mathbf e_n, \mathbf e_t$, respectively.

In order to determine the relationship between the two coordinate systems $(x, y, z)$ and $(x, n, s)$, one defines the position vector $\mathbf r(\equiv \mathbf r(s, x))$ from the reference $x$-axis of the blades to an arbitrary point $A$ located on the middle surface as

The position vector $\mathbf{r^*}$ of an arbitrary point $A^*$ off the mid-surface of the blades can be expressed as

As a result

Moreover, in order to avoid confusion, the notation $(x, y, z)$ represents the points associated with the middle surface, the notation $A^* = (x^*, y^*, z^*)$ represents the points off the middle surface, the two notations $(x, y, z)$ and $(x^*, y^*, z^*)$ are presented as follows:

2.3.   Displacement field

Based on the assumptions (ⅳ) and (ⅴ), the axial displacement $D_x, D_y, D_z$ (see Figure 3) are expressed as (see Ref. [21])

where $\phi_y, \phi_z$ denote rotation of the cross-section about $y$ and $z$ axis. For non-shearable blades, The expression of $\phi_y$ and $\phi_z$ can be written in the form

where $u_x, v_x$ denote derivatives with respect to $x$.

2.4.   Strains and stress

Based on the assumptions (ⅴ), the displacement-strain relationships is expressed as follows[20]:

Thanks to (8), we can get

where

The shear strain $\varepsilon_{nx}$ can be expressed as

where $U_n$ represents the components of $\big(D_x, D_y, D_z\big)$ along the $n$ axis,

Substituting (7) and (12) into (11), we deduce $\varepsilon_{nx} = 0.$ Using the coordinate transformation of strain component, taking into account assumptions (iv), the shear strain $\varepsilon_{sx}$ can be expressed as

where $\gamma_{xy}, \gamma_{xz}$ are the transverse shear effect of the cross section.

Since these materials are isotropic, the corresponding thermoelastic constitutive law adapted to the case of structures is expressed as

Herein, the reduced thermoelastic coefficients are defined as:

where $E$ is Young's modulus, $\nu$ is Poisson's ratio, $k^2$ is the transverse shear correction factor.

According to assumption (ⅱ), the cross section is rigid, then we can derive $\sigma_{ns} = 0$. Considering the assumption of the hoop stress $\sigma_{ss}$ to be negligible, we can get

2.5.   The centrifugal force and the perturbed gas pressure

The centrifugal force can be represented as

where $\rho$ is the density of the blades, $\bar A$ is the cross section area of the blades, $R(x) = R_0(l-x)+\frac{1}{2}(l^2-x^2)$.

In the paper, we use the first-order piston theory (see Ref.[24]) to evaluate the perturbed gas pressure. The pressure on the principal plane of the blade can be obtained as

where

$C_{\infty}$ represents the speed of sound, $\rho_{\infty}$ and $U_{\infty}$, respectively, denote the density and the velocity of the free stream air, $U_{y^p}^t$ and $U_{z^p}^t$ are, respectively, the tangential components of the fluid velocity on the positive $y^p$ and $z^p$ plane, and $u^p$ and $v^p$ denote the displacement components along the principal axes $y^p$ and $z^p$, respectively.

The transformation relationship between $(u^p, v^p)$ and $(u, v)$ is given by

Therefore, the external forces per unit axial length in the $y$ direction and the $z$ direction can be obtained as

Combining (14), (15), (16), $p_y, p_z$ can be expressed as a linear function of $u, v, u_t, v_t, u_x, v_x$, respectively.

where

2.6.   The velocity vector and the acceleration

In order to calculate the kinetic energy, the velocity vector and the acceleration should be given first. Based on the assumption (ⅴ), the position vector $\mathbf R$ of a point $A^{**}$ of the deformed blades is expressed in the form

Keep in mind that the rotation takes place solely in the $XY$ plan, it results

Then the velocity vector and the acceleration of an arbitrary point $A^{**}$ are obtained:

where subscript $t$ denotes derivatives with respect to the time.

2.7.   Rotating blades model

In order to derive the blades model and the associated boundary conditions, the extended Hamilton's principle is used. This can be formulated as

where $K$ and $U$, respectively, denote the kinetic energy and the strain energy, $W$ is the virtual work of external forces, $\delta$ is the variation operator.

Thanks to the cross section of the blades is rectangular, we get

Utilizing (23), the kinetic energy is obtained

where $\mathrm d \tau$ denotes the differential volume element. Then

Due to the rotating motion of the blades, the total strain energy consists of two parts. The strain energy caused by the centrifugal force can be obtained as

where $\rho$ is the bulk density of the blades, $\int_{x}^l \rho\omega^2(R_0+\varsigma) \mathrm d\varsigma$ is the centrifugal force per unit cross section. Then,

where $\bar A$ is the area of the cross section of the blades, $R(x) = R_0(l-x)+\frac{1}{2}(l^2-x^2).$

Thanks to (23), the variation of the strain energy caused by the centrifugal force can be rewritten as

The strain energy induced by the deformation of the rotating blades can be expressed as

Substituting the expressions of the stress and strain resultants into (28) yields

where

The work of the non-conservative external forces can be obtained as

Then

where

Inserting variation of potential energy (27) and (30), variation of kinetic energy (24) and variation of external work equation (32) into the extended Hamilton's principle (22), collecting the terms associated with the same variations, invoking the stationarity of the functional within the time interval $[t_0, t_1]$, and the fact that the variations $(\delta u, \delta v)$ are independent and arbitrary, their coefficients in the two integrands must vanish independently [see [21]]. The partial differential equations with respect to variation of problem's independent variables and the associated boundary conditions are obtained as

where

The following two types of boundary conditions are generated due to the different design of engine blades.

and

The conditions (38)-(39) represent C-C boundary condition. The conditions (40)-(44) represent C-F boundary condition. Now we study the well-posedness and regularity of the solution for C-C and C-F blades.

3.   Preliminary

We write $Q = \Omega \times (0, T),$ where $\Omega = (0, l)$ and $T>0,$

We list Gagliardo-Nirenberg inequality for bounded domains (see [23]) to be used in the subsequent sections

Lemma 3.1. Let $\Omega \subset R^n$ be a bounded domain with smooth boundary. Let $1\leq p, q, r \leq \infty$ be real numbers and $j\leq m$ be non-negative integers. If a real number $\alpha$ satisfies

Then

where $s >0$, and the constants $C_1$ and $C_2$ depend upon $\Omega$ and the indices $p, q, r, m, j, s$ only.

Now, we give the Aubin-Lions Lemma (see [22]).

Lemma 3.2. Suppose $B_0, B, B_1$ are Banach Space, if

(i) $\{u_i\}_{i = 1}^{\infty}$ is bounded in $L^{p_0}(0, T; B_0)$;

(ii) $\{u_{i, t}\}_{i = 1}^{\infty}$ is bounded in $L^{p_1}(0, T; B_1)$;

(iii) $B_0\hookrightarrow \hookrightarrow B \hookrightarrow B_1$,

then $\{u_i\}_{i = 1}^{\infty}$ admits a strongly converging subsequence in $L^{p_0}(0, T; B)$, provided $p_0< \infty, p_1 > 1.$

4.   Mathematical results of C-C blades

Without loss of generality, we assume $a_5 = 1.$ For the sake of brevity, the initial boundary-value problems of C-C blades are rewritten as:

where

Definition 4.1. We say function $(u, v)$, $u, v\in L^{\infty}(0, T; H_0^2(\Omega))$, with

is a weak solution of the initial boundary value problem (45) provided

(ⅰ)

for each $\varphi \in H_0^2(\Omega)$, and a.e. time $0\leq t\leq T$, and

(ⅱ)

Remark 1. From the definition, we know $u, v\in C ( [0, T]; L^2(\Omega) )$ and $u_t, v_t \in C ( [0, T]; H^{-2}(\Omega) )$.

Remark 2. By the product Minkowshi inequality, we can obtain:

Theorem 4.2. (Existence for weak solution of (45)) Assume

there exists a weak solution of (45).

We now briefly outline the proof of Theorem 4.2 in the following:

Step 1. employing Galerkin's method to construct solutions of certain finite-dimensional approximations to (45) (correspond to Lemma 6.1 in chapter 6);

Step 2. using the energy method to find the uniform estimates of the finite-dimensional approximations solutions (correspond to Lemma 6.2 in chapter 6);

Step 3. using compactness method to obtain the weak solutions of (45).

Now we give the smoothness of weak solutions of (45).

Theorem 4.3. (Improved regularity) Assume

the weak solution of (45) satisfies

Theorem 4.4. (Interior regularity) Under the condition (52), for any $\varphi(x)\in C_0^{\infty}(\Omega), \varphi(x)>0, x \in \Omega,$ then the weak solution of (45) satisfies

Remark 3. Multiplying the first and second equation of (45) by $a_4(x)u_{tt}, a_3(x)v_{tt}$ respectively, integrating with over $\Omega'$, and summing the two equations, we can obtain that the weak solution of (45) satisfies

where $\Omega'$ is a bounded open interval and $\overline{\Omega}'\subset \Omega.$

If the pre-twist angle $\alpha(x)$ is neglected, we can get $\theta(x), a_3(x), a_4(x), a_6(x)$ are all constant. Then we give the following theorem.

Theorem 4.5. (Improved regularity when $\alpha(x) = 0$) Under the condition (52), assume $\alpha(x) = 0$, then the weak solution of (45) satisfies

Now, we study the uniqueness and stability of (45), denote operator $\pi$ satisfying

and

with the norm

Then $W(Q)$ is Banach space.

Theorem 4.6. Undering the condition (52),

is continuous.

5.   Mathematical results of C-F blades

The initial boundary-value problems of C-F blades are rewritten as :

Definition 5.1. We say function $(u, v)$, $u, v\in L^{\infty}(0, T; H_f^2(\Omega))$, with

is a weak solution of the initial boundary value problem (54) provided

(ⅰ)

for each $\varphi \in H_f^2(\Omega)$, and a.e. time $0\leq t\leq T$, and

(ⅱ)

Similar as above, we derive the following conclusion of the initial boundary value problem (54).

Theorem 5.2. (Existence for the weak solutions of (54)) Assume

there exists a weak solution of (54).

Theorem 5.3. (Regularity weak solution of (54)) Assume

Then there exists $T^*< T$ such that, the weak solution of (54) satisfies

Remark 4. If pre-twist angle $\alpha(x)$ is neglected, according to the first and second equations of (54), we can obtain that the weak solution of (54) satisfies

where $u_{tt}, v_{tt} \in L^{\infty}(0, T^*; L^2(\Omega))$ and Gagliardo-Nirenberg inequality are used.

6.   Proof of mathematical results

We construct weak solution of the initial boundary-value problem (45) by first solving a finite dimensional approximation. We thus employ Galerkin's method by selecting smooth functions $\varphi_k = \varphi_k(x)(k = 1, \cdots)$ such that

and

Fix a positive integer $m$, and write

where we intend to select the coefficients $d_{0m}^k(t), d_{1m}^k(t)(0\leq t\leq T, k = 1, \cdots, m)$ to satisfy

and

where

In order to proof Theorem 4.2, we need the following Lemma.

Lemma 6.1. Under the condition (51), for each integer $m = 1, 2, \cdots,$ there exists a unique function $u_m, v_m$ of the form (63) satisfying (64)-(67) for $0\leq t \leq t_m$.

Proof. Assuming $u_m, v_m$ to be given by (63), by using (62), equation (66) and (67) become the nonlinear system of ODE

subject to the initial conditions (64), (65). According to standard theory for ODE, there exists unique $C^2$ funcion $\mathbf d_{0m}(t) = (d_{0m}^1, d_{0m}^2, \cdots, d_{0m}^m), \mathbf d_{1m}(t) = (d_{1m}^1, d_{1m}^2, \cdots, d_{1m}^m)$, satisfying (64), (65), and solving (68), (69) for $0\leq t \leq t_m,$ where $t_m: = t(m)$ is a function of $m$.

We propose now to send $m$ to infinity and to show a subsequence of the solutions $u_m, v_m$ of the approximate problem (64)-(67) converges to a weak solution of (45). For this we will need the following uniform estimates.

Lemma 6.2. Undering the condition (51), there exists positive constant $C(\Omega, T)$, such that

for $m = 1, 2, \cdots.$

Proof. Multiplying equality (66) by $d_{0m, t}^k$, summing $k = 1, 2, \cdots$, we deduce

for a.e. $0\leq t\leq t_m.$

Multiplying equality (67) by $d_{1m, t}^k$, summing $k = 1, 2, \cdots,$ we deduce

for a.e. $0\leq t\leq t_m.$

To simplify the equation (71) and (72), we can get

Summing the equations (73) and (74), we discover

Since $\omega \in C^{1}(0, T)$, the equality (75) implies

where we used Young inequality.

Integrating (76) with respect to $t$, we discover

Thanks to (50), there exists a constant $C$, such that

Substituting (78) into the inequality (77), By using Poincaré inequality, we find

Then, by using Gronwall inequality, we obtain

for $m = 1, 2, \cdots.$

Remark 5. From the Energy estimates, we can obtain: $t_m \rightarrow T,$ as $m\rightarrow \infty.$

Thanks to Lemma 6.1 and Lemma 6.2, we can obtain the existence for the weak solutions of the initial boundary value problem (45).

6.1.   Proof of Theorem 4.2

Proof. (ⅰ) According to the energy estimates (70), we see that

As a consequence there exists subsequence $\{u_{\mu}, v_{\mu}\}_{m = 1}^{\infty}$ and $u, v \in L^{\infty}(0, T; H_0^2(\Omega))$, with $u_t, v_t \in L^{\infty}(0, T;L^2(\Omega))$, $u_{tt}, v_{tt} \in L^{\infty}(0, T;H^{-2}(\Omega))$, such that

(ⅱ) By Gagliardo-Nirenberg inequality, we can see

Otherwise,

Combining (84) and (85), we discover

Moreover, there exists a function $\chi \in L^{\infty}(0, T; L^{2}(\Omega))$ satisfy

By Lemma 3.2, we can find

And so

Thus

Combining (87) and (90), we can obtain

where we used the Lemma 1.3 of Chapter 1 in [22]. Furthermore, we have $\chi = (u_x^2+v_x^2)u_x.$

Meanwhile,

In the same way,

Next fix an integer $k$, we select $\mu \geq k$, from (66) we can get

Thanks to (83), we can get

From (92) and (95), we can discover

for all fixed $k$.

Note

then,

for arbitrary $\varphi \in H_0^2(\Omega)$.

In the same way,

for arbitrary $\varphi \in H_0^2(\Omega)$.

Synthesizes the above analysis, there exists $u, v$ satisfying (47), (48), and

(ⅲ) Now let's prove the initial conditions.

Since, $u_{\mu}, v_{\mu}, u_{\mu, t}, v_{\mu, t}$ are bounded in $L^{2}(Q)$, by Lemma 1.2 of Chapter 1 in [22], we can discover

Otherwise,

Combining identities (99) and (100), we can get

Next, according to (83), we can obtain

Then, we can discover

Otherwise,

Comparing identities (101) and (102), we can get

So

In the same way, we can obtain

6.2.   Proof of Theorem 4.3

Proof. Differentiating the first equation of (45) with respect to $t$, multiplying by $u_{tt}$ and integrating with respect to $x$, we discover

Differentiating the second equation of (45) with respect to $t$, multiplying by $v_{tt}$ and integrating with respect to $x$, we discover

Summing the equation (103) and (104), we discover after integrating by parts:

Obviously, by Young inequality and Hölder inequality, we have

Next integrate (106) with respect to $t$,

Since $u_1, v_1\in H_0^2(\Omega),$ we have

are bounded. On the other hand, multiplying the first equation and second equation of (45) by $u_{tt}, v_{tt}$, respectively, and integrating with respect to $x$, we discover

where the condition (52), Young inequality and Poincaré inequality are used.

As before, we have

Applying the Gronwall inequality to (109) gives

Moreover, the weak solutions of (45) satisfies

6.3.   Proof of Theorem 4.4

Proof. Multiplying the first equation and the second equation of (45) by $-\varphi^2u_{txx},$ $-\varphi^2v_{txx}$, respectively, summing the two equations, and integrating with respect to $x$, we discover

Thanks to integration by parts and the properties of $\varphi$, we obtain the following equality

where $I_i, i = 1, \cdots, 5,$ are given as follows respectively

To conclude, we need to estimate each of $I_i, i = 1, \cdots, 5.$ By using integration by parts and conclusion (53), we find

For $I_2$, we get

where Lemma 3.1 and (52) are used. Similarly from (114), we deduce

By using Lemma 3.1 and (52), we find

Putting (113)-(117) into (112), this yields

Integrating with respect to $t \in (0, t)$, taking into account the condition (50), we discover

Thanks to the Gronwall inequality, we deduce

6.4.   Proof of Theorem 4.5

Proof. Utilizing to the first and second equations of (45), we can deduce

Thanks to (50) and Hölder inequality, we see that

Furthermore,

where Gagliardo-Nirenberg inequality for bounded domains is used. Similarly, we deduce

Combining with the conclusions in Theorem (4.2) and Theorem (4.3), we can find that

6.5.   Proof of Theorem 4.6

Proof. Denote

Then $\eta, \zeta$ satisfy

In equation (125), the nonlinear term satisfies

By Hölder inequality and Sobolev inequality, we can obtain

In the same way, we have

On the other hand, we easily have

Substituting (127)-(129) into (125), we deduce

where we used the inequalities

which are deduced from Theorem 4.2 and Theorem 4.3.

6.6.   Proof of Theorem 5.2

Proof. Assume

and

Fix a positive integer $m$, and write

where we intend to select the coefficients $\tilde d_{0m}^k(t), \tilde d_{1m}^k(t)(0\leq t\leq T, k = 1, \cdots, m)$ to satisfy

and

As in earlier treatments of C-C boundary condition, we can conclude the following two conclude without difficulty.

(ⅰ) For each integer $m = 1, 2, \cdots,$ there exists a unique Galerkin approximations function $u_m, v_m$ of the from (133) satisfying (134)-(137) for $0\leq t \leq T$.

(ⅱ) $u(x, 0) = u_0(x), v(x, 0) = v_0(x), u_t(x, 0) = u_1(x), v_t(x, 0) = v_1(x).$

Then, we proof the following estimate

Similarly from (75), we can deduce

where $b_2 = e_1$ is used.

Then we integrate (139) with respect to $t$, to discover

By simple calculation, we deduce

where $b_1e_2\geq e_1^2$ is used.

On the other hand, according to $u_{0m}(x), v_{0m}(x) \in H_f^2(\Omega)$, by Sobolev inequality, we obtain

Thus,

Substituting (141) and (142) into (140), applying Gronwall inequality, we can deduce (138).

Now we pass to limits in our Galerkin approximations, applying estimate (138), we can discover (92), (93), (95). In order to complete the proof of the theorem, we just have to proof

where $u_{\mu}, v_{\mu}$ are the convergent subsequence of $u_m, v_m$, respectively.

To verify this, recalling (138), we observe that

where the Corollary 4 of Chapter 8 in [30] is used.

Furthermore, thanks to the conditions $u_m(0) = u(0) = 0$, we obtain

Thanks to (144), we can deduce

Similarly, we have

6.7.   Proof of Theorem 5.3

Proof. Similarly as (105), we can get

where

By calculation, we can deduce

Substituting (147), (148) and (149) into (146), we get

Then integrate (150) with respect to $t$, to discover

where

and

are used.

Similarly as (108), we discover

According to (151), we get

By Hölder inequality and Sobolev inequality, we have

Then, we obtain Theorem 5.3 by Gronwall inequality.