Citation: C. N. Panagopoulos, E. P. Georgiou, D.A. Lagaris, V. Antonakaki. The effect of nanocrystalline Ni-W coating on the tensile properties of copper[J]. AIMS Materials Science, 2016, 3(2): 324-338. doi: 10.3934/matersci.2016.2.324
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Since Hirsch and Smale[1] proposed the necessity of structural stability, this topic has received sufficient attention from scholars. This type of research focuses on whether small disturbances in the coefficients, initial data, and geometric models in the equations will cause significant disturbances in the solutions. At the beginning, people were mainly keen on dealing with the continuous dependence and convergence of fluid in porous media defined in two-dimensional or three-dimensional bounded regions. Freitas et al. [2] studied the long-term behavior of porous-elastic systems and proved that solutions depend continuously on the initial data. Payne and Straughan[3] established a prior bounds and maximum principles for the solutions and obtained the structural stability of Darcy fluid in porous media, where they assumed that the temperature satisfies Newton's cooling conditions at the boundary. Scott[4] considered the situation where Darcy fluid undergoes exothermic reactions at the boundary and obtained the continuous dependence of the solutions on the boundary parameters. Li et al.[5] studied the interface connection between Brinkman–Forchheimer fluid and Darcy fluid in a bounded region, and obtained the continuous dependence on the heat source and Forchheimer coefficient. For more papers, on can see [6,7,8,9,10].
With the continuous development of technology and progress in the field of engineering, the necessity of studying the structural stability of fluid equations on a semi-infinite cylinder is even more urgent. The semi-infinite cylinder refers to a cylinder whose generatrix is parallel to the coordinate axis and its base is located on the coordinate plane, i.e.,
$ R = \Big\{(x_1, x_2, x_3)\Big|(x_1, x_2)\in D,\ x_3\geq0\Big\}, $ |
where $ D $ is a bounded domain on $ x_1Ox_2 $.
Li et al. have already done some work on this topic. Li and Lin[11] proved the continuous dependence on the Forchheimer coefficient of the Brinkman–Forchheimer equations in $ R $. Papers [12] and [13] obtained structural stability for Forchheimer fluid and temperature-dependent bidispersive flow in $ R $, respectively.
In this paper, we introduce a new cylinder with a disturbed base, which has been considered in [14]. Let $ D(f) $ represent the disturbed base, i.e.,
$ D(f) = \Big\{(x_1, x_2, x_3)\Big|x_3 = f(x_1, x_2)\geq0,\ (x_1, x_2)\in D\Big\}, $ |
where the given function $ f $ satisfies
$ |f(x1,x2)|<ϵ, ϵ>0. $
|
$ \epsilon $ is called the perturbation parameter. The cylinder with a disturbed base is defined as
$ R(f) = \Big\{(x_1, x_2, x_3)\Big|(x_1, x_2)\in D,\ x_3\geq f(x_1, x_2)\geq0\Big\}. $ |
Different from [14], we study the heat conduction equation applicable to the study of layered composite materials in binary mixtures[15]
$ b1ut=k1△u−γ(u−v), in R×{t>0}, $
|
(1.1) |
$ b2vt=k2△v+γ(u−v), in R×{t>0}, $
|
(1.2) |
$ u=v=0, on ∂D×{x3>0}×{t>0}, $
|
(1.3) |
$ u=v=0, in R×{t=0}, $
|
(1.4) |
where $ k_1, k_2, b_1, b_2 $ and $ \gamma $ are positive constants. $ u $ and $ v $ are the temperature fields in each constituent. Papers [16,17,18] further discussed and generalized the application of Eqs (1.1) and (1.2).
In this paper, we shall also use the notations
$ R(z) = \Big\{(x_1, x_2, x_3)\Big| (x_1, x_2)\in D, x_3\geq z\geq 0\Big\}, $ |
$ D(z) = \Big\{(x_1, x_2, x_3)\Big| (x_1, x_2)\in D, x_3 = z\geq 0\Big\}. $ |
The main work of this article investigates the continuous dependence of solutions to Eqs (1.1)–(1.4) on perturbation parameters and base data. Due to many practical constraints, it is very common for the base of the cylinder to experience minor disturbance. Therefore, studying the effects of these disturbances is essential. To this end, we assume that $ u^* $ and $ v^* $ are perturbed solutions of Eqs (1.1)–(1.4) on $ R(f) $, and then prove that the difference between the unperturbed solutions and the perturbed solutions satisfies a first-order differential inequality. By solving this inequality, we can obtain the continuous dependence of the solution.
On the finite end $ D $, we assume that the solutions to (1.1)–(1.4) satisfy
$ u(t,x)=L11(t,x1,x2), v(t,x)=L12(t,x1,x2),t>0, x3=0, (x1,x2)∈D(0), $
|
(2.1) |
$ u∗(t,x)=L21(t,x1,x2), v∗(t,x)=L22(t,x1,x2),t>0, x3=f(x1,x2), (x1,x2)∈D(0). $
|
(2.2) |
In (2.1) and (2.2), the known functions $ L_{ij} (i, j = 1, 2) $ satisfy the compatibility conditions on $ \partial D $.
We let that $ \mathcal{H}_1(t, \boldsymbol{x}) $ and $ \mathcal{H}_2(t, \boldsymbol{x}) $ are specific functions who have the same boundary conditions as $ u^* $ and $ v^* $, respectively. That is
$ H1(t,x)=L21(t,x1,x2)exp{−σ(x3−f)}, H2(t,x)=L22(t,x1,x2)exp{−σ(x3−f)}, $
|
(2.3) |
where $ \sigma > 0 $.
We now derive some lemmas.
Lemma 2.1. If $ L_{21}, L_{22}\in H^1([0, \infty)\times D(f)) $, then
$ ∫t0exp{−η1τ}[k1||∇u∗(τ)||2L2(R(f))+k2||∇v∗(τ)||2L2(R(f))]dτ≤d1(t), $
|
where
$ d1(t)=∫t0exp{−η1τ}[k1||∇H1||2L2(R(f))+k2||∇H2||2L2(R(f))]dτ+exp{−η1t}[b1||H1(t)||2L2(R(f))+b2||H2(t)||2L2(R(f))]+12∫t0exp{−η1τ}[b1η1||H1,τ(τ)||2L2(R(f))+b2η1||H2,τ(τ)||2L2(R(f))]dτ+12γ∫t0exp{−η1τ}||(H1−H2)(τ)||2L2(R(f))dτ. $
|
(2.4) |
Proof. Using (1.1)–(1.4), we begin with
$ ∫t0∫R(f)exp{−η1τ}[b1u∗τ−k1△u∗+γ(u∗−v∗)]u∗dxdτ=0,∫t0∫R(f)exp{−η1τ}[b2v∗τ−k2△v∗−γ(u∗−v∗)]v∗dxdτ=0. $
|
We compute
$ 12exp{−η1t}[b1||u∗(t)||2L2(R(f))+b2||v∗(t)||2L2(R(f))]+∫t0exp{−η1τ}[b1η1||u∗(τ)||2L2(R(f))+b2η1||v∗(τ)||2L2(R(f))]dτ+∫t0exp{−η1τ}[k1||∇u∗(τ)||2L2(R(f))+k2||∇v∗(τ)||2L2(R(f))]dτ+γ∫t0exp{−η1τ}||(u∗−v∗)(τ)||2L2(R(f))dτ=−∫t0∫D(f)exp{−η1τ}[k1∂u∗∂x3u∗+k2∂v∗∂x3v∗]dAdτ. $
|
(2.5) |
On the other hand, we use (2.3) to compute
$ −∫t0∫D(f)exp{−η1τ}[k1∂u∗∂x3u∗+k2∂v∗∂x3v∗]dAdτ=−∫t0∫D(f)exp{−η1τ}[k1∂u∗∂x3H1+k2∂v∗∂x3H2]dAdτ=∫t0∫R(f)exp{−η1τ}[k1∇⋅(∇u∗H1)+k2∇⋅(∇v∗H2)dxdτ=∫t0∫R(f)exp{−η1τ}[k1∇u∗⋅∇H1+k2∇v∗⋅∇H2]dxdτ+exp{−η1t}∫R(f)[b1u∗H1+b2v∗H2]dx+η1∫t0∫R(f)exp{−η1τ}[b1u∗H1,τ+b2v∗H2,τ]dxdτ+γ∫t0∫R(f)exp{−η1τ}(u∗−v∗)(H1−H2)dxdτ≐F1+F2+F3+F4. $
|
(2.6) |
An application of the Schwarz inequality leads to
$ F1≤12∫t0exp{−η1τ}[k1||∇u∗(τ)||2L2(R(f))+k2||∇v∗(τ)||2L2(R(f))]dτ+12∫t0exp{−η1τ}[k1||∇H1||2L2(R(f))+k2||∇H2||2L2(R(f))]dτ, $
|
(2.7) |
$ F2≤12exp{−η1t}[b1||u∗(t)||2L2(R(f))+b2||v∗(t)||2L2(R(f))]+12exp{−η1t}[b1||H1(t)||2L2(R(f))+b2||H2(t)||2L2(R(f))], $
|
(2.8) |
$ F3≤∫t0exp{−η1τ}[b1η1||u∗(τ)||2L2(R(f))+b2η1||v∗(τ)||2L2(R(f))]dτ+14∫t0exp{−η1τ}[b1η1||H1,τ(τ)||2L2(R(f))+b2η1||H2,τ(τ)||2L2(R(f))]dτ, $
|
(2.9) |
$ F4≤γ∫t0exp{−η1τ}||(u∗−v∗)(τ)||2L2(R(f))dτ+14γ∫t0exp{−η1τ}||(H1−H2)(τ)||2L2(R(f))dτ. $
|
(2.10) |
Inserting Eqs (2.7)–(2.10) into (2.6) and combining (2.5), it can be obtained
$ ∫t0exp{−η1τ}[k1||∇u∗(τ)||2L2(R(f))+k2||∇v∗(τ)||2L2(R(f))]dτ≤∫t0exp{−η1τ}[k1||∇H1||2L2(R(f))+k2||∇H2||2L2(R(f))]dτ+exp{−η1t}[b1||H1(t)||2L2(R(f))+b2||H2(t)||2L2(R(f))]+12∫t0exp{−η1τ}[b1η1||H1,τ(τ)||2L2(R(f))+b2η1||H2,τ(τ)||2L2(R(f))]dτ+12γ∫t0exp{−η1τ}||(H1−H2)(τ)||2L2(R(f))dτ. $
|
(2.11) |
From (2.11), we can conclude that Lemma 2.1 holds.
We not only need a prior bounds for $ v $ and $ v^* $, but also for $ u $ and $ u^* $. Since $ u $ and $ u^* $ are undisturbed solutions of Eqs (1.1)–(1.4), in Lemma 2.1 we only need to set $ f = 0 $ and replace $ L_{21} $ and $ L_{22} $ with $ L_{11} $ and $ L_{12} $, respectively, and then we can obtain the a prior bounds for $ u $ and $ u^* $.
Lemma 2.2. If $ L_{11}, L_{12}\in H^1([0, \infty)\times D) $, then
$ ∫t0exp{−η1τ}[k1||∇u(τ)||L2(R)+k2||∇v(τ)||L2(R)]dτ≤d2(t), $
|
where
$ d2(t)=∫t0exp{−η1τ}[k1||∇H3||2L2(R)+k2||∇H4||2L2(R)]dτ+exp{−η1t}[b1||H3(t)||2L2(R)+b2||H4(t)||2L2(R)]+12∫t0exp{−η1τ}[b1η1||H3,τ(τ)||2L2(R)+b2η1||H4,τ(τ)||2L2(R)]dτ+12γ∫t0exp{−η1τ}||(H3−H4)(τ)||2L2(R)dτ $
|
and
$ \mathcal{H}_3(t, {\boldsymbol{x}}) = L_{11}(t, x_1, x_2)\exp\{-\sigma x_3\},\ \mathcal{H}_4(t, {\boldsymbol{x}}) = L_{12}(t, x_1, x_2)\exp\{-\sigma x_3\}. $ |
Remark 2.1. Lemmas 2.1 and 2.2 will provide a priori estimates for the proof of the lemmas in the next section.
Let $ w $ and $ s $ represent the difference between the perturbed solutions and the unperturbed solutions, i.e.,
$ w=u−u∗, s=v−v∗, $
|
(3.1) |
then $ w $ and $ s $ satisfy
$ b1wt=k1△w−γ(w−s), in R(ϵ)×{t>0}, $
|
(3.2) |
$ b2st=k2△s+γ(w−s), in R(ϵ)×{t>0}, $
|
(3.3) |
$ w=s=0, on ∂D×{x3>ϵ}×{t>0}, $
|
(3.4) |
$ w=s=0, in R(ϵ)×{t=0}. $
|
(3.5) |
To obtain the continuous dependence of the solution on the perturbation parameter, we establish a new energy function
$ V(t,x3)=∫t0[||w(τ)||2L2(R(x3))+||s(τ)||2L2(R(x3))]dτ, x3≥ϵ. $
|
(3.6) |
Noting the definition of $ R(x_3) $, we can obtain the derivative of $ V(t, x_3) $ as follows:
$ −∂∂x3V(t,x3)=∫t0[||w(τ)||2L2(D(x3))+||s(τ)||2L2(D(x3))]dτ. $
|
We introduce two auxiliary functions $ \varphi $ and $ \psi $ such that
$ b1φτ+k1△φ=−w, b2ψτ+k2△ψ=−s, in R(x3),0<τ<t, $
|
(3.7) |
$ φ(τ,x1,x2,x3)=ψ(τ,x1,x2,x3)=0, on ∂D×{x3},0<τ<t, $
|
(3.8) |
$ φ(τ,x1,x2,x3)=ψ(τ,x1,x2,x3)=0, (x1,x2)∈D,0<τ<t, $
|
(3.9) |
$ φ(t,x)=ψ(t,x)=0, in R(x3), $
|
(3.10) |
$ φ,∇φ,ψ,∇ψ→0(uniformly in x1,x2,τ) as x3→∞, $
|
(3.11) |
where $ x_3 > \epsilon $.
Next, we will derive some necessary properties of the auxiliary functions, which will play a crucial role in proving the continuous dependence of the solutions.
Lemma 3.1. If $ \varphi, \psi\in H^1([0, t]\times R(x_3)) $, then
$ ∫t0[b1||φτ(τ)||2L2(R(x3))+b2||ψτ(τ)||2L2(R(x3))]dτ≤a1V(t,x3), x3≥ϵ, $
|
where $ a_1 = \max\{b_1^{-1}, b_2^{-1}\} $.
Proof. We begin with
$ ∫t0∫R(x3)φτ[b1φτ+k1△φ+w]dxdτ=0,∫t0∫R(x3)ψτ[b2ψτ+k2△ψ+s]dxdτ=0. $
|
Using the divergence theorem $ \oint_{\partial R(x_3)}Fds = \int_{R(x_3)}div F dx $ and (3.8)–(3.11), we have
$ b1∫t0||φτ(τ)||2L2(R(x3))dτ=−12k1||∇φ(0)||2L2(R(x3))+∫t0∫R(x3)wφτdxdτ≤[∫t0||φτ(τ)||2L2(R(x3))dτ∫t0||w(τ)||2L2(R(x3))dτ]12, $
|
(3.12) |
and
$ b2∫t0||ψτ(τ)||2L2(R(x3))dτ≤[∫t0||ψτ(τ)||2L2(R(x3))dτ∫t0||s(τ)||2L2(R(x3))dτ]12. $
|
(3.13) |
Using the Schwarz inequality, (3.12) and (3.13), Lemma 3.1 can be obtained.
Lemma 3.2. If $ \varphi, \psi\in H^1(R(x_3)) $, then
$ ∫t0[k1||∇φ(τ)||2L2(R(x3))+k2||∇ψ(τ)||2L2(R(x3))]dτ≤a2V(t,x3), $
|
where $ a_2 = \frac{1}{\lambda}\max\{k_1^{-1}, k_2^{-1}\} $.
Proof. We begin with
$ ∫t0∫R(x3)φ[b1φτ+k1△φ+w]dxdτ=0,∫t0∫R(x3)φ[b2ψτ+k2△ψ+s]dxdτ=0. $
|
Using the divergence theorem and Lemma 2.2, we have
$ k1∫t0||∇φ(τ)||2L2(R(x3))dτ=−12b1||φ(0)||2L2(R(x3))+∫t0∫R(x3)wφdxdτ≤[∫t0||φ(τ)||2L2(R(x3))dτ∫t0||w(τ)||2L2(R(x3))dτ]12≤1√λ[∫t0||∇2φ(τ)||2L2(R(x3))dτ∫t0||w(τ)||2L2(R(x3))dτ]12 $
|
(3.14) |
and
$ k2∫t0||∇ψ(τ)||2L2(R(x3))dτ≤1√λ[∫t0||∇2ψ(τ)||2L2(R(x3))dτ∫t0||s(τ)||2L2(R(x3))dτ]12. $
|
(3.15) |
Using the following inequality
$ √ab+√cd≤√(a+c)(b+d), for a,b,c,d>0, $
|
(3.16) |
the Young inequality and Lemma 3.1, we can have from (3.14) and (3.15)
$ ∫t0[k1||∇φ(τ)||2L2(R(x3))+k2||∇ψ(τ)||2L2(R(x3))]dτ≤1√λ{∫t0[k1||∇2φ(τ)||2L2(R(x3))+k2||∇2ψ(τ)||2L2(R(x3))]dτ⋅∫t0[k−11||w(τ)||2L2(R(x3))+k−12||s(τ)||2L2(R(x3))]dτ}12. $
|
(3.17) |
From (3.17) we can obtain Lemma 3.2.
Lemma 3.3. If $ \varphi, \psi\in H^1(R(x_3)) $, then
$ k1∫t0||∂φ∂x3(τ)||2L2(D(x3))dτ+k2∫t0||∂ψ∂x3(τ)||2L2(D(x3))dτ≤a3V(t,x3), $
|
where $ a_3 $ is a positive constant.
Proof. Letting $ \delta $ be a positive constant. We compute
$ ∫t0∫R(x3)[∂φ∂x3−δφτ][b1φτ+k1△φ+w]dxdτ=0, $
|
(3.18) |
$ ∫t0∫R(x3)[∂ψ∂x3−δψτ][b2ψτ+k2△ψ+s]dxdτ=0. $
|
(3.19) |
Using the divergence theorem and (3.8)–(3.10) in (3.18) and (3.19), we obtain
$ 12k1δ||∇φ(0)||2L2(R(x3))dτ+b1δ∫t0||φτ(τ)||2L2(R(x3))dτ+12k1∫t0||∂φ∂x3(τ)||2L2(D(x3))dτ=∫t0∫R(x3)∂φ∂x3φτdxdτ+∫t0∫R(x3)[∂φ∂x3−δφτ]wdxdτ. $
|
(3.20) |
Using the Schwarz inequality, we obtain
$ ∫t0∫R(x3)∂φ∂x3φτdxdτ≤[∫t0||∂φ∂x3(τ)||2L2(R(x3))dτ∫t0||φτ(τ)||2L2(R(x3))dτ]12, $
|
(3.21) |
$ ∫t0∫R(x3)∂φ∂x3wdxdτ≤[∫t0||∂φ∂x3(τ)||2L2(R(x3))dτ∫t0||w(τ)||2L2(R(x3))dτ]12, $
|
(3.22) |
$ −δ∫t0∫R(x3)φτwdxdτ≤δ[∫t0||φτ(τ)||2L2(R(x3))dτ∫t0||w(τ)||2L2(R(x3))dτ]12. $
|
(3.23) |
Inserting (3.21)–(3.23) into (3.20) and dropping the first two terms in the left of (3.20), we have
$ 12k1∫t0||∂φ∂x3(τ)||2L2(D(x3))dτ≤[∫t0||∂φ∂x3(τ)||2L2(R(x3))dτ∫t0||φτ(τ)||2L2(R(x3))dτ]12+[∫t0||∂φ∂x3(τ)||2L2(R(x3))dτ∫t0||w(τ)||2L2(R(x3))dτ]12+δ[∫t0||φτ(τ)||2L2(R(x3))dτ∫t0||w(τ)||2L2(R(x3))dτ]12. $
|
(3.24) |
Similar, we can also have from (3.19)
$ 12k2∫t0||∂ψ∂x3(τ)||2L2(D(x3))dτ≤[∫t0||∂ψ∂x3(τ)||2L2(R(x3))dτ∫t0||ψτ(τ)||2L2(R(x3))dτ]12+[∫t0||∂ψ∂x3(τ)||2L2(R(x3))dτ∫t0||s(τ)||2L2(R(x3))dτ]12+δ[∫t0||ψτ(τ)||2L2(R(x3))dτ∫t0||s(τ)||2L2(R(x3))dτ]12. $
|
(3.25) |
Using (3.16) and Lemmas 3.1 and 3.2, we obtain
$ k1∫t0||∂φ∂x3(τ)||2L2(D(x3))dτ+k2∫t0||∂ψ∂x3(τ)||2L2(D(x3))dτ≤2a1a2{∫t0[b1||∂φ∂x3(τ)||2L2(R(x3))+b2||∂ψ∂x3(τ)||2L2(R(x3))]dτ⋅∫t0[k1||φτ(τ)||2L2(R(x3))+k2||ψτ(τ)||2L2(R(x3))]dτ}12+2a2{∫t0[b1||∂φ∂x3(τ)||2L2(R(x3))+b2||∂ψ∂x3(τ)||2L2(R(x3))]dτ⋅∫t0[||w(τ)||2L2(R(x3))+||s(τ)||2L2(R(x3))]dτ}12+2a1δ{∫t0[k1||φτ(τ)||2L2(R(x3))+k2||ψτ(τ)||2L2(R(x3))]dτ⋅∫t0[||w(τ)||2L2(R(x3))+||s(τ)||2L2(R(x3))]dτ}12≤a3V(t,x3), $
|
(3.26) |
where $ a_3 = 2a_1^2a_2^2+2a_2^2+2a_1^2 $.
In the next section, we will use Lemma 3.3 to derive the continuous dependence of the solutions.
In this section, we first derive a bound for $ V(t, \epsilon) $. To do this, we define
$ u(t,x)=L11(t,x1,x2), v(t,x)=L12(t,x1,x2), −ϵ≤x3≤0,(x1,x2)∈D,t∈[0,+∞), $
|
(4.1) |
$ u∗(t,x)=L21(t,x1,x2), v∗(t,x)=L22(t,x1,x2), −ϵ≤x3≤f(x1,x2),(x1,x2)∈D,t∈[0,+∞). $
|
(4.2) |
When $ -\epsilon\leq x_3\leq\epsilon $, we let
$ w(t,x)=u(t,x)−u∗(t,x), s(t,x)=v(t,x)−v∗(t,x),(x1,x2)∈D,t∈[0,+∞). $
|
(4.3) |
In view of (3.1) and (4.3), using the triangle inequality, it can be obtained that
$ k1∫t0∫R(−ϵ)(∂w∂x3)2dxdτ+k2∫t0∫R(−ϵ)(∂s∂x3)2dxdτ≤∫t0∫R(−ϵ)[k1(∂u∂x3)2+k2(∂v∂x3)2]dxdτ+∫t0∫R(−ϵ)[k1(∂u∗∂x3)2+k2(∂v∗∂x3)2]dxdτ. $
|
(4.4) |
Using Lemmas 2.1 and 2.2, (4.1) and (4.2), from (4.4), we obtain
$ k1∫t0∫R(−ϵ)(∂w∂x3)2dxdτ+k2∫t0∫R(−ϵ)(∂s∂x3)2dxdτ≤∫t0∫R[k1(∂u∂x3)2+k2(∂v∂x3)2]dxdτ+∫t0∫R(f)[k1(∂u∗∂x3)2+k2(∂v∗∂x3)2]dxdτ.≤eη1t[d1(t)+d2(t)]≐d3(t). $
|
(4.5) |
Now, we write the main theorem as:
Theorem 4.1. If $ L_{11}, L_{12}\in H^1([0, \infty)\times R), L_{21}, L_{22}\in H^1([0, \infty)\times R(f)) $ and $ t < \frac{\pi}{4a_1\gamma} $, then
$ V(t,x3)≤exp{−d4(x3−ϵ)}{32d4πmax{1k1,1k2}d3(t)ϵ+d5∫t0[||(L11−L21)(τ)||2L2(D)+||(L12−L22)(τ)||2L2(D)]dτ},x3≥ϵ $
|
holds, where $ d_4 = a_3^{-1}\max\{k_1, k_2\}^{-1} $ and $ d_5 = \frac{d_4\pi}{2}+\frac{2}{d_4} $.
Proof. Let $ x_3\geq\epsilon $ be a fixed point on the coordinate axis $ x_ 3 $. Using (3.7)–(3.11) and the divergence theorem, we can have
$ V(x3,t)=−∫t0∫R(x3)w[b1φτ+k1△φ]dxdτ−∫t0∫R(x3)s[b2ψτ+k2△ψ]dxdτ=−∫t0∫R(x3)[b1φτw+b2ψτs]dxdτ+∫t0∫R(x3)[k1∇w⋅∇φ+k2∇s⋅∇ψ]dxdτ+∫t0∫D(x3)[k1w∂φ∂x3+k2s∂ψ∂x3]dAdτ=−∫t0∫R(x3)[b1φτw+b2ψτs]dxdτ−∫t0∫R(x3)[k1△wφ+k2△sψ]dxdτ+∫t0∫D(x3)[k1w∂φ∂x3+k2s∂ψ∂x3]dAdτ=−∫t0∫R(x3)[b1φτw+b2ψτs]dxdτ−∫t0∫R(x3)[b1φwτ+b2ψsτ]dxdτ+∫t0∫D(x3)[k1w∂φ∂x3+k2s∂ψ∂x3]dAdτ−γ∫t0∫R(x3)(φ−ψ)(w−s)dxdτ. $
|
(4.6) |
In light of (1.4) and (3.10), it is clear that
$ ∫t0∫R(x3)[b1φτw+b1φwτ]dxdτ=0, ∫t0∫R(x3)[b2ψτs+b2ψsτ]dxdτ=0. $
|
(4.7) |
A combination of the Hölder inequality, (3.16) and Lemma 3.3 leads to
$ ∫t0∫D(x3)[k1w∂φ∂x3+k2s∂ψ∂x3]dAdτ≤k1[∫t0||∂φ∂x3(τ)||2L2(D(x3))dτ∫t0||w(τ)||2L2(D(x3))dτ]12+k2[∫t0||∂ψ∂x3(τ)||2L2(D(x3))dτ∫t0||s(τ)||2L2(D(x3))dτ]12≤max{√k1,√k2}[∫t0(k1||∂φ∂x3(τ)||2L2(D(x3))+k2||∂ψ∂x3(τ)||2L2(D(x3)))dτ]12⋅[∫t0(||w(τ)||2L2(D(x3))+||s(τ)||2L2(D(x3)))dτ]12≤√a3max{√k1,√k2}√V(t,x3)[−∂∂x3V(t,x3)]12. $
|
(4.8) |
For the fourth term in the right of (4.6), we compute
$ −γ∫t0∫R(x3)(φ−ψ)(w−s)dxdτ≤γ[∫t0(||φ(τ)||2L2(R(x3))+||ψ(τ)||2L2(R(x3)))dτ⋅∫t0(||w(τ)||2L2(R(x3))+||s(τ)||2L2(R(x3)))dτ]12. $
|
(4.9) |
Using the inequality (see p182 in [19])
$ ∫10ϕ2dx≤4π2∫10(ϕ′)2dx, for ϕ(0)=0, $
|
(4.10) |
we have from (4.9)
$ −γ∫t0∫R(x3)(φ−ψ)(w−s)dxdτ≤γ2tπ[∫t0(||φτ(τ)||2L2(R(x3))+||ψτ(τ)||2L2(R(x3)))dτV(t,x3)]12≤γ2tπa1V(t,x3), $
|
(4.11) |
where we have also used Lemma 3.1. Combining (4.6), (4.7), (4.8) and (4.11) and choosing $ t < \frac{\pi}{4a_1\gamma} $, we can have
$ V(t,x3)≤−1d4∂∂x3V(t,x3),x3>ϵ. $
|
(4.12) |
Integrating (4.12) from $ \epsilon $ to $ x_3 $, we have
$ V(t,x3)≤V(t,ϵ)exp{−d4(x3−ϵ)},x3≥ϵ. $
|
(4.13) |
Equation (4.13) only indicates that the solutions to (1.1)–(1.4) decay exponentially as $ x_3\rightarrow \infty $. This decay result is not rigorous because we do not yet know whether $ V(t, \epsilon) $ depends on the perturbation parameter $ \epsilon $. Therefore, we derive the explicit bound of $ V(t, \epsilon) $ in terms of $ \epsilon $ and $ L_{ij}(ij = 1, 2) $.
After letting $ x_3 = \epsilon $ in (4.12), we have
$ V(ϵ,t)≤1d4∫t0[||w(τ)||2L2(D(ϵ))+||s(τ)||2L2(D(ϵ))]dAdτ=2d4∫t0∫ϵ−ϵ∫D(x3)[w∂w∂x3+s∂s∂x3]dxdτ+2d4∫t0[||(L11−L21)(τ)||2L2(D)+||(L12−L22)(τ)||2L2(D)]dτ≤2d4[∫t0||w(τ)||2L2(D(x3)×[−ϵ,ϵ])dτ∫t0||∂w∂x3(τ)||2L2(D(x3)×[−ϵ,ϵ])dτ]12+2d4[∫t0||s(τ)||2L2(D(x3)×[−ϵ,ϵ])dτ∫t0||∂s∂x3(τ)||2L2(D(x3)×[−ϵ,ϵ])dτ]12+2d4∫t0[||(L11−L21)(τ)||2L2(D)+||(L12−L22)(τ)||2L2(D)]dτ. $
|
(4.14) |
Using (4.10) again, we have
$ ∫t0||w(τ)||2L2(D(x3)×[−ϵ,ϵ])dτ≤16ϵ2π2∫t0||∂w∂x3(τ)||2L2(D(x3)×[−ϵ,ϵ])dτ+2ϵ∫t0||(L11−L21)(τ)||2L2(D)dτ, $
|
(4.15) |
$ ∫t0||s(τ)||2L2(D(x3)×[−ϵ,ϵ])dτ≤16ϵ2π2∫t0||∂s∂x3(τ)||2L2(D(x3)×[−ϵ,ϵ])dτ+2ϵ∫t0||(L12−L22)(τ)||2L2(D)dτ. $
|
(4.16) |
Inserting (4.15) into (4.16) and combining the Schwarz inequality, we obtain
$ V(ϵ,t)≤32d4πϵ∫t0[||∂w∂x3(τ)||2L2(D(x3)×[−ϵ,ϵ])+||∂s∂x3(τ)||2L2(D(x3)×[−ϵ,ϵ])]dτ+[d4π2+2d4]∫t0[||(L11−L21)(τ)||2L2(D)+||(L12−L22)(τ)||2L2(D)]dτ≤32d4πmax{1k1,1k2}ϵ∫t0[k1||∂w∂x3(τ)||2L2(R(−ϵ))+k2||∂s∂x3(τ)||2L2(R(−ϵ))]dτ+[d4π2+2d4]∫t0[||(L11−L21)(τ)||2L2(D)+||(L12−L22)(τ)||2L2(D)]dτ. $
|
(4.17) |
In view of (4.5) and (4.13), from (4.17) we have Theorem 4.1.
Remark 4.1. Theorem 4.1 indicates that $ V(t, x_3) $ continuously depends on $ \epsilon $ and the base data. That is, when $ \epsilon $ approaches 0, then $ u(t, x_3) $ and $ v(t, x_3) $ approach 0. If $ \epsilon = 0 $, Theorem 4.1 is the Saint-Venant's principle type decay result.
Remark 4.2. In any cross-section of $ R $, the continuous dependence result can still be obtained. We compute
$ ∫t0[||w(τ)||2L2(D(x3))+||s(τ)||2L2(D(x3))]dτ=−2∫t0∫R(x3)[w∂w∂x3+s∂s∂x3]dxdτ≤2√V(x3)[∫t0[k1||∂w∂x3(τ)||2L2(R(−ϵ))+k2||∂s∂x3(τ)||2L2(R(−ϵ))]dτ]12. $
|
(4.18) |
Using (4.18) and Theorem 4.1, we can obtain the continuous dependence result.
This article adopts the methods of the a prior estimates and energy estimate to obtain the continuous dependence of the solution on the base. This method can be further extended to other linear partial differential equation systems, such as pseudo-parabolic equation
$ u_t = \Delta u+\delta \Delta u_t, $ |
where $ \delta $ is a positive constant. However, for nonlinear equations (e.g., the Darcy equations), due to the inability to control nonlinear terms and derive a prior bounds for nonlinear terms, Lemma 3.3 will be difficult to obtain. This is a difficult problem we need to solve next.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This work is supported by the Research Team Project of Guangzhou Huashang College (2021HSKT01).
The author declares there is no conflict of interest.
[1] | Edelstein AS, Cammaratra RC (1996) Nanomaterials: synthesis, properties and applications, Bristol: Institute of physics. |
[2] |
Metikoš-Huković M, Grubač Z, Radić N, et al. (2006) Sputter deposited nanocrystalline Ni and Ni-W films as catalysts for hydrogen evolution. J Mol Catal A Chem 249: 172–180. doi: 10.1016/j.molcata.2006.01.020
![]() |
[3] | Brenner A (1963) Electrodeposition of Alloys: Principles and Practice, Vol. 2, Publ. Academic Press Inc., London. |
[4] | Panagopoulos CN, Plainakis GD, Tsoutsouva MG (2015) Corrosion of nanocrystalline Ni-W coated copper. J Surf Eng Mater Adv Technol 5: 65–72. |
[5] |
Sriraman KR, Raman SGS, Seshadri SK (2006) Synthesis and evaluation of hardness and sliding wear resistance of electrodeposited nanocrystalline Ni–W alloys. Mater Sci Eng A 418: 303–311. doi: 10.1016/j.msea.2005.11.046
![]() |
[6] |
Quiroga Arganaraz MP, Ribotta SB, Folquer ME, et al. (2011) Ni–W coatings electrodeposited on carbon steel: Chemical composition, mechanical properties and corrosion resistance. Electrochim Acta 56: 5898–5903. doi: 10.1016/j.electacta.2011.04.119
![]() |
[7] | Iwasaki H, Higashi K, Nieh TG (2004) Tensile deformation and microstructure of a nanocrystalline Ni–W alloy produced by electrodeposition. Scripta Mater 50: 395–399. |
[8] |
Matsui I, Takigawa Y, Uesugi T, et al. (2013) Effect of orientation on tensile ductility of electrodeposited bulk nanocrystalline Ni–W alloys. Mater Sci Eng A 578: 318–322. doi: 10.1016/j.msea.2013.04.114
![]() |
[9] |
Panagopoulos CN, Plainakis GD, Lagaris DA (2011) Nanocrystalline Ni–W coatings on copper. Mater Sci Eng B 176: 477–479. doi: 10.1016/j.mseb.2010.03.058
![]() |
[10] | Wu Y, Chang D, Kim D, et al. (2003) Influence of boric acid on the electrodepositing process and structures of Ni–W alloy coating. Surf Coat Technol 173: 259–264. |
[11] | Hou KH, Chang Y-F, Chang S-M, et al. (2010) The heat treatment effect on the structure and mechanical properties of electrodeposited nano grain size Ni–W alloy coatings. Thin Solid Films 518: 7535–7540. |
[12] |
Mizushima I, Tang PT, Hansen HN, et al. (2006) Residual stress in Ni–W electrodeposits. Electrochim Acta 51: 6128–6134. doi: 10.1016/j.electacta.2005.11.053
![]() |
[13] |
Panagopoulos CN, Georgiou EP (2007) The effect of hydrogen charging on the mechanical behaviour of 5083 wrought aluminum alloy. Corros Sci 49: 4443–4451. doi: 10.1016/j.corsci.2007.03.047
![]() |
[14] | Metals Handbook. 9th Ed., Alloy Phase Diagrams, ASM, USA, 1992. |
[15] | Cullity BD (1959) Elements of X-ray Diffraction, 1st Edition, Addison-Wesley Publishing Company, Massachusetts. |
[16] | Auerswald J, Fecht H-J (2010) Nanocrystalline Ni-W for wear-resistant coatings and electroforming. J Electrochem Soc 157: 199–205. |
[17] |
Mizushima I, Tang PT, Hansen HN, et al. (2006) Residual stress in Ni–W electrodeposits. Electrochim Acta 51: 6128–6134. doi: 10.1016/j.electacta.2005.11.053
![]() |
[18] | Dieter GE (1988) Mechanical Metallurgy, SI Metric Edition, Publ. McGraw-Hill, London. |
[19] |
Kumar KS, Van Swygenhoven H, Suresh S (2003) Mechanical behavior of nanocrystalline metals and alloys. Acta Mater 51: 5743–5774. doi: 10.1016/j.actamat.2003.08.032
![]() |
[20] |
Hahn H, Padmanabhan KA (1995) Mechanical response of nanostructured materials. Nanostruct Mater 6: 191–200. doi: 10.1016/0965-9773(95)00042-9
![]() |
[21] | Wang N, Wang Z, Aust KT, et al. (1995) Effect of grain size on mechanical properties of nanocrystalline materials. Acta Metall Mater 43: 519–528. |
[22] | Flinn RA, Trojan PK (1994) Engineering materials and their applications, 4th ed., Wiley-VCH, Boston. |
[23] |
Meyers MA, Mishra A, Benson DJ (2006) Mechanical properties on nanocrystalline materials. Progr Mater Sci 51: 427–556. doi: 10.1016/j.pmatsci.2005.08.003
![]() |
[24] |
Panagopoulos CN, Pelegri AA (1993) Tensile properties of zinc coated aluminium. Surf Coat Technol 57: 203–206. doi: 10.1016/0257-8972(93)90041-L
![]() |
[25] | Armstrong DEJ, Haseeb ASMA, Roberts SG, et al. (2012) Nanoindentation and micro-mechanical fracture toughness of electrodeposited nanocrystalline Ni–W alloy films. Thin Solid Films 520: 4369–4372. |
[26] | Panagopoulos CN, Papachristos VD, Sigalas G (1999) Tensile behaviour of as deposited and heat-treated electroless Ni–P deposits. J Mater Sci 34: 2587–2600. |
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36. | Tarishi Baranwal, A. Akilbasha, Economical heuristics for fully interval integer multi-objective fuzzy and non-fuzzy transportation problems, 2024, 34, 0354-0243, 743, 10.2298/YJOR240115035B | |
37. | Monika Bisht, Ali Ebrahimnejad, Four-dimensional green transportation problem considering multiple objectives and product blending in Fermatean fuzzy environment, 2025, 11, 2199-4536, 10.1007/s40747-025-01829-5 | |
38. | P. Anukokila, R. Nisanthini, B. Radhakrishnan, An application of multi-objective transportation problem in type-2 Fermatean fuzzy number incorporating the RS-MABAC technique, 2025, 27731863, 100264, 10.1016/j.fraope.2025.100264 | |
39. | Ziyan Xiang, Xiuzhen Zhang, An integrated decision support system for supplier selection and performance evaluation in global supply chains, 2025, 15684946, 113325, 10.1016/j.asoc.2025.113325 | |
40. | Asghar Khan, Saeed Islam, Muhammad Ismail, Abdulaziz Alotaibi, Development of a triangular Fermatean fuzzy EDAS model for remote patient monitoring applications, 2025, 15, 2045-2322, 10.1038/s41598-025-00914-6 |