Evaluating the nonlinear relationship between nonfinancial corporate sector leverage and financial stability in the post crisis era

1.   Introduction

There have been many literatures on continuous dependence and structural stability for the past few years, including those of Aulisa et al. ^[1], Celebi et al. ^[2,3], Liu et al. ^[4,5,6], Chen et al. ^[7,8], Ames and Payne ^[9,10], Ames and Straughan ^[11], Ciarletta and Straughan ^[12], Franchi and Straughan ^{[13,14,15,16]}, Lin and Payne ^[17,18], Li et al. ^[19,20,21], Straughan et al. ^[22,23] and Zhou et al. ^[24,25]. Particularly, most researches focus on the continuous dependence on the boundary data, domain geometry, initial time geometry, and the model itself. Hirsch and Smale ^[26] pointed out the necessity of studying the continuous dependence of solutions. They emphasized the physical significance of this type of research. This means that changes in the coefficients of partial differential equations may be physically reflected through changes in constitutive parameters. We trust that mathematical analysis of these equations will help to disclose their applicability in physics. Since inevitable errors occur in both numerical calculations and physical measurements of data, continuous correlation results are very important. It is relevant to understand the extent to which such errors affect the solution.

Harfash ^[27] researched a system of equations to describe the double-diffusion convection in Darcy flow with magnetic field effect. The author assumed the magnetic fields with only the vertical component which was a specific magnetic field. By establishing a priori results, the author illustrates that the solution of the equations depends continuously on changes in the magnetic force and gravity vector coefficients. Some authors have paid attentions to similar problems. By employing Payne's ^[28] highly innovative procedure for obtaining a priori estimates, Ames and Payne ^[9] have established a similar result for the Navier-Stokes equations. But it is necessary to restrict the size of the interval or the size of the initial data in their result. A similar result for a Brinkman porous material and for the Darcy equations of flow in porous media has been derived by Franchi and Straughan ^[29] and Payne and Straughan ^[30], respectively.

In this paper, we assume that the Darcy flow with magnetic field effect occupies a bounded region $\Omega$ in $R^3$ and that the boundary of the region is denoted by $\partial\Omega$ which is sufficient smooth to use the divergence theorem. The variables $v_i$, $T$, $C$ and $p$ are the fluid velocity vector, the temperature, the salt concentration and the pressure, respectively. The governing equations for Darcy flow with magnetic field effect may be written as

where $g_i$ and $h_i$ are gravity vector terms arising in the density equation of state, $\Delta$ is Laplacian operator, $\gamma$ is the Soret coefficient, $\sigma$ is magnetic coefficient, and ${\textbf{B}}_0 = (0, 0, B_0)$ is a magnetic field with only the vertical component and ${\textbf{v}} = (v_1, v_2, v_3)$. In (1.1), we take a particular magnetic field, as in ^[27,31].

On the boundary, we impose

where $F$ and $G$ are positive functions, $n_i$ is the unit outward normal to $\partial\Omega$ and $k$ and $\tau$ are positive constants. Equation $(1.5)$ may be thought of as expressing Newton's law of cooling with inhomogeneous outside temperature or inhomogeneous outside salt concentration, i.e.

where $T_a$ and $C_a$ are the ambient outside temperature and the ambient outside salt concentration, respectively. The initial conditions are written as

for prescribed functions $T_0$ and $C_0$.

In our work, we still consider the same particular equations as in ^[27]. But our boundary conditions is Newton's law of cooling type with inhomogeneous outside temperature. Thus, the Sobolev inequalities which are used in ^[27] are not available in our paper. Compared with ^[9], we no longer need to impose special restrictions on the region $\Omega$. So their method fails to handle the system in this paper. In this paper, we derive the upper bounds of $\int_\Omega T^4dx$ and $\int_\Omega C^4dx$ which are difficulty to obtain. By using the these priori results, we derive the continuous dependence on the magnetic coefficient and the boundary parameter. Throughout this paper, the usual summation convention is employed with repeated Latin subscripts summed from 1 to 3. The comma is used to indicate partial differentiation, i.e. $u_{i, j} = \frac{\partial u_i}{\partial x_j}$, $u_{i, j}u_{i, j} = \Sigma_{i, j = 1}^3\frac{\partial u_i}{\partial x_j}$.

2.   A priori bounds

In this section, we want to derive bounds for various norms of $v_i$, $T$ and $C$ in term of known data which will be used in the next sections. Before we derive these bounds, we prove some lemmas firstly.

Lemma 2.1. Let functions $f_i, (i = 1, 2, 3)$, defined on $\partial\Omega$, be some functions such that

and

where $f_0 > 0$ is a constant and $m_1$, $m_2$ are both positiveconstants. Then,

for a function $\varphi$ which is defined on the closure of thedomain $\Omega$. In (2.3), $\alpha > 0$ is an arbitrary constant which may be very small and $m_3 = (m_1+\frac{m_2^2}{\alpha})$.

Proof. We began with the identity

Integrating (2.4) over $\Omega$, using (2.1) and the divergence theorem, we have

The Hölder inequality and (2.2) allow us to obtain

from which it follows that

Lemma 2.2. Let $T, {\boldsymbol{v}}\in H^1(\Omega)$, $T_0\in L^{2P}(\Omega)$ and $F\in L^{2P}(\partial\Omega)$. Then, the solution for $(1.2)$ satisfies

where $T_m = \max\{|T_0|, |F|\}.$

Proof. We began with

Using $(1.2)$, the divergence theorem and the Young inequality, we are leaded to

An integration of this inequality allows that

Allowing $p\rightarrow \infty$, we obtain

where $T_m$ depends on the initial-boundary conditions of $T$.

Lemma 2.3. Let $T, {{\boldsymbol{v}}}\in H^1(\Omega)$ and $C$ be thesolutions for (1.2) and (1.3) and $T_0, C_0\in C^2(\Omega)$, $F, G\in C^2({\partial\Omega\times\{t > 0\}})$. Then,

where $A_1(t)$ and $A_2(t)$ are positive functions which will be given later.

Proof. Using $(1.2)$ and the divergence theorem, we compute

By the Hölder inequality and the Young inequality, from (2.9) we have

Integrating (2.10) from 0 to $t$, we have

From the identity

we get

Upon employing the Cauchy-Schwarz inequality and the arithmetic-geometric mean inequality, we can get

We use these inequalities together with (2.12) to arrive at

Letting $\varphi = T$ in Lemma 2.1 and using (2.11), we have

Thus, (2.13) can be rewritten as

with $\alpha = \frac{f_0\tau}{k^2}$. An integration of (2.15) leads to

In light of (2.11), we have

Lemma 2.4. Let $T$ and $C$ be the solutions for(1.2) and (1.3), and $T, {\boldsymbol{v}}\in H^1(\Omega)$, $T_0, C_0\in C^4(\Omega)$, $F, G\in C^4({\partial\Omega\times\{t > 0\})}$. Then,

where $A_3(t)$ and $A_4(t)$ will be given later.

Proof. We first let $H$ be a solution of the problem

Using $(2.19)$ and the divergence theorem, we find

By the Hölder inequality, we have

From (2.21), it is clear that $\int_\Omega H^4dx$ can be bounded by known data. Now, we set

Then, $\psi$ satisfies the initial-boundary condition problem

Next, we also define a new function

where $\delta_1$ and $\delta_2$ are positive constants to be determined later. Now, it is easy to see that

from which we may get that

Now using the arithmetic-geometric mean and the Schwarz inequalities, we find that

and

where $T_m$ is defined in Lemma 2.2. Furthermore,

Inserting (2.26)–(2.28) into (2.25), and using the Hölder and the Young inequalities to the integrals on the boundary, we have

where $\varepsilon_i\ (i = 1, 2, \cdots, 13)$ are positive constants to be determined. To ensure that the coefficients of the first three terms and the sixth and seventh terms to be non-positive, we choose that

We drop the non-positive terms in (2.29) to have

Using the arithmetic-geometric mean inequality and integrating the above formula from 0 to $t$, we obtain

where $\widetilde{m}_1 = \delta_2T_m^2\gamma+6\gamma^2\varepsilon_4T_m^2$, $\widetilde{m}_2 = \delta_2T_m^2\gamma$ and $\widetilde{m}_3 = (\delta_1\varepsilon_5^{-3}+\frac{\delta_2}{2\varepsilon_6\varepsilon_7}+\frac{\delta_2}{2\varepsilon_8\varepsilon_9} +\gamma\varepsilon_{12}^{-3})$.

Next, we multiply $(2.22)_1$ with $\psi$, integrate in $\Omega$ and use Cauchy-Schwarz's inequality to obtain

In light of (2.14), (2.31) yields that

Integrating (2.32) from 0 to $t$, we have

With the aid of (2.11), inequality (2.33) can be rewritten as

Inserting (2.34) into (2.30) and using (2.11) again, we have

where

Recalling the definition of $\Phi(t)$ in (2.23), we may get

By the triangle inequality, we have

Combining (2.21) and (2.36), we have

where

Next, we pay our attention to seek the bound for $L_2$ norm of $v_i$ as well as $\nabla v$. We obtain the following lemma which will be used in the continuous dependence proof.

Lemma 2.5. Let $v_i$, $T$ and $C$ are the solutions of(1.1)–(1.3) with the initial-boundary conditions (1.5) and(1.6), and $T_0, C_0\in C^4(\Omega)$, $F, G\in C^4({\partial\Omega\times\{t > 0\})}$. Then,

where $A_5(t)$ and $A_6(t)$ are positive functions which will bederived later.

Proof. We start with the identity

Since ${\textbf{B}}_0 = (0, 0, B_0)$, it is clear that $[({\textbf{v}}\times{\textbf{B}}_0)\times{\textbf{B}}_0]_i = B_0^2(\overline{k}_iv_3-v_i)$, where $\overline{{\textbf{k}}} = (\overline{k}_1, \overline{k}_2, \overline{k}_3) = (0, 0, 1)$. Obviously,

so by the Hölder inequality and the arithmetic-geometric mean inequality, we have

Combining (2.8) and Lemma 2.3, we obtain

We commence bounding the $L_2$ norm for the velocity gradient. To do this, we split the velocity into symmetric and skew parts. We write

To bound the first term of (2.41), we use the Eq $(1.1)$ to have

Using Hölder inequality and arithmetic-geometric inequality again in (2.42), we arrive at

Similarly, we also have

In view of $\overline{{\textbf{k}}} = (0, 0, 1)$, the third term of (2.42) yields

Inserting (2.43)–(2.45) into (2.42), we have

To handle the second term of (2.41), we use the divergence theorem and integrate by parts to obtain

The first term of (2.47) is zero, since $v_{i}n_i = 0$ on $\partial\Omega$. If the region $\Omega$ is convex, Lin and Payne ^[18] state $\int_{\partial\Omega}v_iv_{j}n_{i, j}dA\geq0$ which leads to

For non-convex $\Omega$,

Using Lemma 2.1 with $\varphi = v_i$, we conclude that

Choosing $\alpha = \frac{f_0}{4k_0}$ and then inserting (2.46) and (2.48) into (2.41), we have

from which it follows that

By (2.11), (2.19) and (2.48), we have

where we have used (2.11), (2.17) and (2.40).

3.   Continuous dependence on the coefficient $\sigma$

Let $(v_i, p, T, C)$ and $(v^*_i, p^*, T^*, C^*)$ be the solutions to the problem (1.1)–(1.6) for the same initial-boundary data, but for different magnetic coefficients $\sigma_1$ and $\sigma_2$, respectively. Differential variables $w_i$, $\pi$, $\theta$, $\Sigma$ and $\sigma$ are defined by

Then,

with the initial-boundary conditions

We have the following theorem.

Theorem 3.1. If $T_0, C_0\in L^\infty(\Omega)$, $F, G\in C^4({\partial\Omega\times\{t > 0\}})$, then the solutions of (1.1)–(1.6)depend continuously on the magnetic coefficient $\sigma$, asshown explicit in inequalities (3.26) and (3.27) whichderives a relation of the form

and

where $L_1$ and $L_2$ are priori constants and $\beta > 0$ is acomputable constant.

Proof. Multiplying $(3.16)$ with $w_i$ and integrating over $\Omega$, then using Cauchy-Schwarz's inequality and the arithmetic-geometric mean inequality, we obtain

where $g = \max\{\sqrt{g_ig_i}\}, \ h = \max\{\sqrt{h_ih_i}\}.$ Since $\overline{{\textbf{k}}} = (0, 0, 1)$, it is easy to find

as in (2.39). By the Cauchy-Schwarz inequality, we have

Inserting (3.8) and (3.9) into (3.7) and applying the arithmetic-geometric mean inequality, we have

In view of (2.38) in Lemma 2.5, from (3.10) we have

Next, we compute

Using Cauchy-Schwarz inequality and the arithmetic-geometric mean inequality and Lemma 2.4, we have

and

Inserting these two inequalities into (3.12) and using the Cauchy-Schwarz inequality in the last two terms on the right of (3.12), we have

for some arbitrary positive constants $\beta_1$ and $\beta_2$.

Now, we use the bound for $L_4$ norm of $w_i$ which has been derived in ^[18] (see (B.17)). We write here as the form

where $M$ is a positive computable constant and $\delta > 0$ is an arbitrary constant. To get the bound for $\int_\Omega w_{i, j}w_{i, j}dx$, we use a similar methods which were used in (2.41) and (2.48) with $\alpha = \frac{f_0}{2k_0}$ to have

To handle the first term of (3.17), we compute

Using the Cauchy-Schwarz inequality and the arithmetic-geometric mean inequality, we have

and

Using the Cauchy-Schwarz inequality and the arithmetic-geometric mean inequality, we have

Since $\overline {\underline {\textbf{k}} } = (0, 0, 1)$, we have

Inserting (3.19)–(3.21) and (3.22) into (3.18), we obtain

It follows from (3.17) that

Combining (3.15), (3.16) and (3.23), we conclude

where

Choosing $\beta_1 = \frac{1}{2\gamma}$, $\beta_2 = \frac{2\tau}{k\gamma}$ and $\beta = \max\{\frac{k\gamma^2}{4\tau}, 2\gamma^2 \}$, we note that $M_3 > 0$, $M_4 = 0$, $\beta-\frac{\gamma}{\beta_1} > 0$ and $1-\gamma\beta_1 > 0$. Since the constant $\delta$ is at our disposal then provided $A_3(t)$ and $A_4(t)$ are bounded, we may choose $\delta$ so large that $M_1\geq0$ and $M_2\geq0$. Dropping the non-positive terms in (3.24) and using Lemma 2.5 and (3.11), we have

where

From (3.25), we have

which follows that

This is the continuous dependence result we want to prove. By (3.11), we may obtain the continuous dependence for ${\textbf{v}}$,

4.   Continuous dependence on the cooling coefficients

In this section, we derive the continuous dependence on the cooling coefficients and we let $(u_i, p, T, C)$ and $(u^*_i, p^*, T^*, C^*)$ be the solutions to the problem (1.1)–(1.3) for the same initial-boundary data and the same $F$ and $G$, but for different the cooling coefficients $k_1$, $k_2$, $\tau_1$ and $\tau_2$, respectively. As in Section 3, we still set

Then $(w_i, \theta, \Sigma, \pi)$ satisfy

with the initial-boundary conditions

We now prove the following theorem.

Theorem 4.1. If $T_0, C_0\in L^\infty(\Omega)$, $F, G\in C^4({\partial\Omega\times\{t > 0\}})$, then the solution of Eq (1.1)–(1.3) withinitial-boundary conditions (1.5) and (1.6) dependscontinuously on the boundary parameters $k$ and $\tau$ in thesense that

Further, ${\boldsymbol{v}}$ depends continuously on $k$ and $\tau$ inthe manner

where $L_3$–$L_6$ are a priori constants.

Proof. Employing a similar methods of the last section, we have

and

By using $(4.2)$, (4.3) and the divergence theorem, as the calculation in (3.12), we get

We note that (3.13) and (3.14) are still valid in this section. We inserting them into (4.9) and use Cauchy-Schwarz inequality in the other terms on the right of (4.9) to have

We use the inequality (3.16) again and use (4.8) to have

where $M$ is a positive computable constant. Inserting (4.11) into (4.10) and letting

and then choosing $\beta$ and $\delta$ large enough such that the coefficients of the first four terms of (4.10) are non-positive, we have

where we have dropped the non-positive terms. Now, we derive bounds for the integrals on $\partial\Omega$. Using Lemma 2.1, we find

and

where we have chosen $\alpha = 1$. Inserting (4.13) and (4.14) into (4.12) and recalling (2.8) and (4.7), we have

where

It is obvious that (4.15) yields that

where we have used the fact $\exp\Big(-\int_0^t\widetilde{\mathcal{F}}_1(\eta)d\eta\Big)\leq1$ for $t > 0$.

Integrating (4.16) from 0 to $t$ leads to

Using (2.11) and (2.17) in (4.17) and setting

we obtain

This is the continuous dependence result for $T$ and $C$. The continuous dependence for $v_i$ follows directly from (4.7).

5.   Conclusions

In this paper, the continuous dependence of the solution is obtained by using the methods of energy estimate and a priori estimates. The main innovation is to deal with the influence of boundary conditions and magnetic field. The structural stability of boundary parameters and magnetic field coefficients is proved.

Acknowledgements

The work was supported national natural Science Foundation of China (Grant No. 11371175), the science foundation of Guangzhou Huashang College (Grant No. 2019HSDS28).

Conflict of interest

The authors declare that they have no competing interests.