On the initial-boundary value problem of two-phase incompressible flows with variable density in smooth bounded domain

In the past two decades, the phase field approach was employed by many researchers in various fluid models [2, 3, 9, 16, 17, 18, 20], and have carried out extensive analytical and numerical studies on the two-phase flows. For phase field model of incompressible binary fluids with identical densities, or with small density ratios where Boussinesq approximation can be applied in practice, we refer the readers to [1, 8, 11, 12, 13, 20, 21] and references therein for detailed derivations and mathematical analysis. However, in most cases, the density differences of the components are not negligible, whence studies on the incompressible two-phase fluids with non-matched densities become even more interesting and challenging.

Recently, in [14], by employing the energetic variational approach, a new two-phase incompressible flows with variable density was derived. In fact, before [14], analytical and numerical results for two-phase incompressible flow models that are valid for large density ratios between different species are quite limited [4, 17, 18]. Most studies of the phase-field models for binary fluids have been restricted to the cases with the same density or with small density differences. In the latter case, Boussinesq approximation can be used, where the variable density is replaced by a background constant density value and external gravitational force is added to model the effect of density force [16, 20]. In the model derived in [14], the density ratio between two phases can be quite large and hence the Boussinesq approximation is no more physically valid.

In this paper, we study the hydrodynamics of a diffuse-interface model describing a mixture of two immiscible incompressible fluids in the smooth bounded domain $\Omega\subseteq\mathbb{R}^{3}$ with different densities $\rho_{1}$ and $\rho_{2}$. A phase field $\phi(t,x)$ is introduced to characterize the two fluids such that

$\phi(t,x)=\begin{cases}\quad 1\,,&\textrm{fluid}\ \ 1\ \textrm{with density }\rho_{1}\,,\\ \quad-1\,,&\textrm{fluid}\ \ 2\ \textrm{with density }\rho_{2}\,,\\ \end{cases}$

with a thin, smooth transition region. While the two fluids are mixed, $\phi(t,x)$ will ranged be in $(-1,1)$. More precisely, we study the following Allen-Cahn-Navier-Stokes (briefly, ACNS) system:

$$\left\{\begin{array}[]{l}\rho(\phi)(u_{t}+u\cdot\nabla u)+\nabla p=\nabla\cdot(\mu\nabla u-\lambda\nabla\phi\otimes\nabla\phi)\,,\\[5.69054pt] \qquad\qquad\nabla\cdot u=0\,,\\[2.84526pt] (\dot{\phi}=)\,\phi_{t}+u\cdot\nabla\phi=\gamma(\lambda\Delta\phi-\lambda f^{\prime}(\phi)-\rho^{\prime}(\phi)\frac{\mid u\mid^{2}}{2})\,.\end{array}\right.$$

(1.1)

Here, $\phi:\mathbb{R}^{+}\times\Omega\rightarrow\mathbb{R}$ is the phase field function that labels different species, $u:\mathbb{R}^{+}\times\Omega\rightarrow\mathbb{R}^{3}$ denotes the velocity of the fluid, and $p:\mathbb{R}^{+}\times\Omega\rightarrow\mathbb{R}$ stands for the pressure. $\rho(\cdot)$ is the average density which is a given function of $\phi$. $\mu>0$ is the viscosity, $\lambda>0$ represents the competition between the kinetic energy and the free energy, and $\gamma>0$ comes from microscopic internal damping during the mixing of two immiscible incompressible fluids. $\dot{\phi}$ is the material derivative of $\phi$ with respect to the velocity $u$. The physical relevant energy density functional $f$ that represent the two phases of the mixture usually has a double-well structure. Without loss of generality, in current paper, we assume that

$$f(\phi)=\tfrac{1}{4\varepsilon^{2}}(\phi^{2}-1)^{2}$$

(1.2)

with some small parameter $\varepsilon>0$. For the average density function $\rho(\cdot)$, it is generally assumed that (see [14])

$\displaystyle\rho(\cdot)\in C^{1}(\mathbb{R})\,,\ \rho(-1)=\rho_{1}\,,\ \rho(1)=\rho_{2}\,,\ \rho(s)\in[\rho_{1},\rho_{2}]\ \textrm{for }-1\leq s\leq 1$

(1.3)

with $\rho_{1}<\rho_{2}$ being two positive constants and the following exterior convexity

$\displaystyle s\rho^{\prime}(s)\geq 0\,,\ \textrm{for }|s|>1\,.$

(1.4)

In current paper, we assume that the $\rho(\cdot)\in C^{2}(\mathbb{R})$ is the parabolic average of $\rho_{1}$ and $\rho_{2}$ satisfying (1.3)-(1.4) as follows (see [17]):

$\displaystyle\rho(\phi)=\tfrac{1}{4}\rho_{1}(\phi-1)^{2}+\tfrac{1}{4}\rho_{2}(\phi+1)^{2}\,.$

(1.5)

Let $\mathbf{n}=\mathbf{n}(x)\,,x\in\partial\Omega$ be the outnormal vector of the boudnary $\partial\Omega$. We now impose the boundary conditions:

$$u|_{\partial\Omega}=0\,,\quad\tfrac{\partial}{\partial\mathbf{n}}\phi|_{\partial\Omega}=0,$$

(1.6)

and the initial data:

$$u(0,x)=u^{in}(x)\in\Omega,\quad\phi(0,x)=\phi^{in}(x)\in\mathbb{R}$$

(1.7)

with the compatibility condition $\nabla\cdot u^{in}=0$.

As shown in [14], once $-1\leq\phi^{in}(x)\leq 1$ initially in $\Omega$, the Maxmal Principle of the heat equation implies that $-1\leq\phi(t,x)\leq 1$ for $t\geq 0$ and $x\in\Omega$. We therefore know that the average density $\rho(\phi)$ admits the lower and upper bounds, namely,

$\displaystyle\rho(\phi)=\tfrac{\rho_{1}+\rho_{2}}{4}\Big{(}\phi+\tfrac{\rho_{2}-\rho_{1}}{\rho_{2}+\rho_{1}}\Big{)}^{2}+\tfrac{\rho_{1}\rho_{2}}{\rho_{1}+\rho_{2}}\in[\tfrac{\rho_{1}\rho_{2}}{\rho_{1}+\rho_{2}},\tfrac{2\rho_{1}\rho_{2}+\rho_{2}^{2}}{2(\rho_{1}+\rho_{2})}]$

(1.8)

for all $\phi\in[-1,1]$.

The system (1.1) was derived from employing the energetic variational approach by Jiang-Li-Liu [14], in which they considered the total energy

$\displaystyle E^{total}:=\int_{\Omega}\Big{(}\tfrac{1}{2}\rho(\phi)|u|^{2}+\lambda(\tfrac{1}{2}|\nabla\phi|^{2}+f(\phi))\Big{)}\mathrm{d}x\,.$

It consists of the first part of the macroscopic kinetic energy and the second part of the Helmholtz free energy. They also took the dissipation of the energy as

$\displaystyle\triangle^{dissipative}:=\int_{\Omega}\Big{(}\mu|\nabla u|^{2}+\tfrac{1}{\gamma}|\dot{\phi}|^{2}\Big{)}\mathrm{d}x\,,$

where the first part accounts for the macroscopic dissipation due to viscosity and the second part comes from microscopic internal damping during the mixing. Finally, the so-called energetic variational approach implies the system (1.1). The details can be seen in [14].

In the sequel, we consider the smooth bounded domain $\Omega$ in $\mathbb{R}^{3}$. We first denote by $L^{p}\,(1\leq p\leq\infty)$ by the standard Lebesgue space with norm

$\displaystyle\|f\|_{L^{p}}=\Big{(}\int_{\Omega}|f|^{p}dx\Big{)}^{\frac{1}{p}}\,(1\leq p<\infty)\,,\quad\|f\|_{L^{\infty}}=\underset{x\in\Omega}{\mathrm{ess\ sup}}\,|f(x)|\,.$

For simplicity, we denote by $\|f\|:=\|f\|_{L^{2}}$. Let $W^{m,p}$ be the standard Sobolev space with norm $\|f\|_{W^{m,p}}^{2}=\sum_{|\alpha|\leq m}\|\partial^{\alpha}f\|^{2}_{L^{p}}$. Here $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})\in\mathbb{N}^{s}$ with $|\alpha|=\alpha_{1}+\alpha_{2}+\alpha_{3}$ and

$\displaystyle\partial^{m}f=\tfrac{\partial^{|\alpha|}f}{\partial x_{1}^{\alpha_{1}}\partial x_{x}^{\alpha_{2}}\partial x_{3}^{\alpha_{3}}}\,.$

As usual, $W^{m,2}$ simply denotes by $H^{m}$ with norm $\|f\|_{m}:=\|f\|_{H^{m}}$. In particular, $\|f\|_{0}=\|f\|$. Moreover, we introduce a weighted $L^{2}$ space by

$\displaystyle\|f\|_{L_{\rho(\phi)}^{2}}=\Big{(}\int_{\Omega}\rho(\phi)|f|^{2}dx\Big{)}^{\frac{1}{2}}\,.$

Remark that the corresponding vector-valued Lebesgue and Sobolev spaces will still be expressed by $L^{p}$, $W^{m,p}$, $H^{m}$ and $L_{\rho(\phi)}^{2}$, etc.

We also employ $A\lesssim B$ to denote by $A\leq CB$ for some harmless constant $C>0$. Moreover, $A\thicksim B$ means that $C_{1}B\leq A\leq C_{2}B$ for two harmless constants $C_{1},C_{2}>0$.

We then introduce the following energy functional $\mathbb{E}_{j}(t)$ and dissipation functional $\mathbb{D}_{j}(t)$ for $j\geq 0$,

	$\displaystyle\mathbb{E}_{j}(t)=$	$\displaystyle\|\partial_{t}^{j}u\|_{L^{2}_{\rho(\phi)}}^{2}+\|\nabla\partial_{t}^{j}u\|^{2}+\|\partial_{t}^{j}\phi\|^{2}_{2}\,,$		(1.9)
	$\displaystyle\mathbb{D}_{j}(t)=$	$\displaystyle\|\nabla\partial_{t}^{j}u\|^{2}_{1}+\|\partial_{t}^{j}u_{t}\|_{L^{2}_{\rho(\phi)}}^{2}+\|\nabla\partial_{t}^{j}\phi\|^{2}_{2}+\|\partial_{t}^{j}\phi_{t}\|^{2}_{1}+\|\partial_{t}^{j}p\|^{2}_{1}\,.$

The next theorem is to prove the global well-posedness of (1.1)-(1.7) near the equilibrium $(0,\pm 1)$. More precisely, let $\phi(t,x)=\pm 1+\varphi(t,x)$. We then know that $(u,p,\varphi)$ satisfies

$$\left\{\begin{array}[]{l}\varrho(\varphi)(u_{t}+u\cdot\nabla u)+\nabla p=\nabla\cdot(\mu\nabla u-\lambda\nabla\varphi\otimes\nabla\varphi)\,,\\[5.69054pt] \qquad\qquad\nabla\cdot u=0\,,\\ \varphi_{t}+u\cdot\nabla\varphi+\tfrac{2\gamma\lambda}{\varepsilon^{2}}\varphi=\gamma(\lambda\Delta\varphi-\lambda h(\varphi)-\varrho^{\prime}(\varphi)\frac{|u|^{2}}{2})\end{array}\right.$$

(1.13)

with the boundary conditions

$$u|_{\partial\Omega}=0,\quad\tfrac{\partial}{\partial\mathbf{n}}\varphi|_{\partial\Omega}=0,$$

(1.14)

and initial data

$\displaystyle u(0,x)=u^{in}(x)\in\mathbb{R}^{3}\,,\quad\varphi(0,x)=\varphi^{in}(x):=\phi^{in}(x)\pm 1\in\mathbb{R}\,,$

(1.15)

which satisfies the compatibility condition $\nabla\cdot u^{in}=0$. Here $h(\varphi)=\tfrac{1}{\varepsilon^{2}}(\varphi^{3}\pm 3\varphi^{2})$ and

$\displaystyle\varrho(\varphi):=\rho(\varphi\pm 1)=\left\{\begin{array}[]{l}\tfrac{\rho_{1}}{4}\varphi^{2}+\tfrac{\rho_{2}}{4}(\varphi+2)^{2}\quad\textrm{for }\varphi+1\,,\\[5.69054pt] \tfrac{\rho_{1}}{4}(\varphi-2)^{2}+\tfrac{\rho_{2}}{4}\varphi^{2}\quad\textrm{for }\varphi-1\,.\end{array}\right.$

(1.16)

We now introuce the global energy functionals and dissipations as follows: for $j\geq 0$,

$$\begin{split}\mathds{E}_{j}(t)=&\|\partial_{t}^{j}u\|_{L^{2}_{\varrho(\varphi)}}^{2}+\|\nabla\partial_{t}^{j}u\|^{2}+\|\partial_{t}^{j}\varphi\|^{2}_{2}\,,\\ \mathds{D}_{j}(t)=&\|\nabla\partial_{t}^{j}u\|^{2}_{1}+\|\partial_{t}^{j}u_{t}\|_{L^{2}_{\varrho(\varphi)}}^{2}+\|\partial_{t}^{j}\varphi\|^{2}_{3}+\|\partial_{t}^{j}\varphi_{t}\|^{2}_{1}+\|\partial_{t}^{j}p\|^{2}_{1}\,.\end{split}$$

(1.17)

The first goal of this paper is to prove the local well-posedness of the initial-boundary value problem (1.1)-(1.7) over the smooth bounded domain $\Omega$ in the functions spaces that the spatial variables with regularity $H^{s}$ for large index $s$, i.e., Theorem 1.1. In many known literatures that considered the well-posedness of various models, only the period domain or whole space was considered in the spaces with spatial regularity $H^{s}$ for large $s$. Oppositely, if the bounded domains were focused, the aims were to prove the existence of weak solutions or strong solutions (in $H^{2}$ space, for example), in which cases only the information of boundary values was required, rather than that of higher order spartial derivatives of the boundary values.

In the smooth bounded domain $\Omega$, if one investigates the well-posedness of the ACNS model in the functions spaces with $H^{s}$-regularity of the spatial variables, the main difficulties come from dealing with the boundary values of the higher order spatial derivatives. Generally speaking, although the boundary value of a function is finite, the higher order derivatives may be infinite. For example, the function $f(x)=\sqrt{x}$ for $x\geq 0$ is continuous up to the boundary $x=0$ but $f^{\prime}(0)=\infty$. Note that it is impossible to control the boundary values of higher order spatial derivatives only employing the usual $H^{s}$-theory over the period domain or whole space. For instance,

$\displaystyle-\int_{\Omega}\Delta\partial^{m}u\cdot\partial^{m}udx=-\int_{\partial\Omega}\tfrac{\partial}{\partial\mathbf{n}}\partial^{m}u\cdot\partial^{m}udS+\int_{\Omega}|\nabla\partial^{m}u|^{2}dx\,,$

the boundary integral $\int_{\partial\Omega}\tfrac{\partial}{\partial\mathbf{n}}\partial^{m}u\cdot\partial^{m}udS$ cannot be controlled by the “good” quantity $\int_{\Omega}|\nabla\partial^{m}u|^{2}dx$ combining with the Trace Theorem.

In order to overcome the difficulty, we employ the Agmon-Douglis-Nirenberg (briefly, ADN) theory associated with the general elliptic system in [6], which was reviewed in Section 2 below. The main ideas are as follows. The ACNS system (1.1) with boundary conditions (1.6) can be rewritten as the abstract forms

$$\left\{\begin{aligned} -\mu\Delta u+\nabla p=\mathcal{U}(u_{t},u,\phi)\,,\quad\nabla\cdot u=0\,,\\ \Delta\phi=\Theta(\phi_{t},u,\phi)\,,\\ u|_{\partial\Omega}=0\,,\quad\tfrac{\partial\phi}{\partial\mathbf{n}}|_{\partial\Omega}=0\,.\end{aligned}\right.$$

By the ADN theory, $\|(u,\phi)\|_{s+2}$ can be bounded by $\|(u,\phi)_{t}\|_{s}$, namely, the second order spatial derivatives of $(u,\phi)$ can be transformed to the first order time derivatives of $(u,\phi)$. We therefore mutate the higher order spatial derivatives problem to the higher order time derivatives problem.

The key point to dominate the higher order time derivatives is that the boundary condition (1.6), i.e., $u|_{\partial\Omega}=0$ and $\tfrac{\partial\phi}{\partial\mathbf{n}}|_{\partial\Omega}=0$ can imply the boundary conditions (2.16) of the higher order time derivatives, i.e., $\partial_{t}^{k}u|_{\partial\Omega}=0$ and $\tfrac{\partial}{\partial\mathbf{n}}\partial_{t}^{k}\phi|_{\partial\Omega}=0$ for any $k\geq 0$. The sketch of the proofs as follows.

1. (1)

   In Section 2 below, we first review the general ADN theory, and reduce the special forms of estimates in Lemma 2.1 and Lemma 2.2 required in this paper.

2. (2)

   In Subsection 3.2, we derive the closed $H^{2}$-estimates for the ACNS system (1.1). Here the boundary conditions $u|_{\partial\Omega}=0$ and $\tfrac{\partial\phi}{\partial\mathbf{n}}|_{\partial\Omega}=0$ in (1.6) are required.

3. (3)

   In Subsection 3.3, we derive the closed $H^{2}$-estimates for the ACNS system (1.1) after acting the higher order time derivatives operator $\partial_{t}^{k}$ for $k\geq 1$. Here the boundary conditions $\partial_{t}^{k}u|_{\partial\Omega}=0$ and $\tfrac{\partial}{\partial\mathbf{n}}\partial_{t}^{k}\phi|_{\partial\Omega}=0$ in (2.16) below are required.

4. (4)

   Based on closed $H^{2}$-bounds of the various orders of the time derivatives in Step (2) and (3), we apply the ADN theory in Lemma 2.1 and Lemma 2.2 to dominate the higher order time-spatial mixed derivatives, see Lemma 3.5. Here the core is to control the quantities $\bm{\Phi}_{\ell,s}$ and $\mathbf{U}_{\ell,s}$ as in Corollary 3.1.

The second goal of this paper is to investigate the global stability and long time decay of the ACNS system (1.1) near the equilibrium $(0,\pm 1)$, hence, Theorem 1.2. In this case, under the purturbation $\phi(t,x)=\varphi(t,x)\pm 1$, we reduce the equivalent system (1.13). The way to deal with the information of the boundary values is totally the same as that in proving the locall well-posedness to (1.1). The key ingredients to verify the global stability is to seek a new dissipation or damping structure on the unknown $\varphi(t,x)$. Fortunately, by the perturbation $\phi=\varphi\pm 1$, the derivative $f^{\prime}(\phi)=f^{\prime}(\varphi\pm 1)$ of the physical relevant energy density $f(\phi)$ in the $\phi$-equation of (1.1) will generate an additional damping effective $\tfrac{2}{\varepsilon^{2}}\varphi$, which gaurantees us to prove the global existence of the ACNS system near the equilibrium $(0,\pm 1)$.

On the other hand, due to the boundary conditions $\partial_{t}^{j}u|_{\partial\Omega}=0$ for $j\geq 0$, there holds the Poincaré inequality $\|\partial_{t}^{j}u\|\lesssim\|\nabla\partial_{t}^{j}u\|$. We thereby imply that

$\displaystyle\mathds{E}_{j}(t)\lesssim\mathds{D}_{j}(t)\quad(\forall\,j\geq 0)\,,$

where $\mathds{E}_{j}(t)$ and $\mathds{D}_{j}(t)$ are defined in (1.17). Then by the arguments in the end of Section 5, we can justify the global solution near $(0,\pm 1)$ admits the exponetial decay $e^{-c_{\#}t}$ for some constant $c_{\#}>0$.

In the next section, we give some preliminaries, in particular review the general ADN theory. In Section 3, we derive three types of the a priori estimates of the system (1.1): 1) $H^{2}$-estimates; 2) $H^{2}$-estimates of the higher order time derivatives; 3) The estimates for the higher order time-space mixed derivatives. In Section 4, based on the a priori estimates, we prove the local well-posedness by employing the iteration methods. In Section 5, we prove the global classical solution for the system (1.13)-(1.7) near the equilibrium $(0,\pm 1)$. Moreover, the time exponetial decay $e^{-c_{\#}t}$ is also obtained. In Appendix A, we give the proof of Lemma 3.4.

In order to deal with the high order spatial derivatives of the solutions $(u,\phi)$, we shall employ the well-known Agmon-Douglis-Nirenberg (briefly, ADN) theory [6]. For convenience of readers, we sketch the theory here. More precisely, they studied the general linear elliptic system on the bounded smooth domain $\Omega\subseteq\mathbb{R}^{n}$ with the following forms:

$$\left\{\begin{aligned} \sum_{j=1}^{N}l_{ij}(x,\partial)u_{j}(x)=F_{i}(x)\ (i=1,\cdots,N)\textrm{ on }\Omega\,,\\ \sum_{j=1}^{N}B_{hj}(x,\partial)u_{j}(x)=\phi_{h}(x)\ (h=1,\cdots,m)\textrm{ on }\partial\Omega\,,\end{aligned}\right.$$

(2.1)

where the $l_{ij}(x,\partial)$, linear differential operators, are polynomials in $\partial$ with coefficients depending on $x\in\Omega$. The orders of these operators are be assumed to depend on two groups of integer weights, $s_{1},\cdots,s_{N}$ and $t_{1},\cdots,t_{N}$, attached to the equations and to the unknowns, respectively, $s_{i}$ corresponding to the $i$-th equation and $t_{j}$ to the $j$-th dependent unknown $u_{j}$. The manner of the dependence is represented by the inequality

$\displaystyle\deg l_{ij}(x,\Xi)\leq s_{i}+t_{j}\,,\ i,j=1,\cdots,N\,,$

(2.2)

where “$\deg$” refers of course to the degree in $\Xi$, and $s_{i}\leq 0$. Moreover, $l_{ij}=0$ if $s_{i}+t_{j}<0$. The ellipticity of (2.1) is characterized by

$\displaystyle L(x,\Xi):=\det(l_{ij}^{\prime}(x,\Xi))\neq 0\quad\textrm{ for real }\Xi\neq 0\,,$

(2.3)

where $l_{ij}^{\prime}(x,\Xi)$ consists of the terms in $l_{ij}(x,\Xi)$ which are just of the order $s_{i}+t_{j}$. Furthermore, the following supplementary condition on $L$ should be imposed:

1. (SC)

   $L(x,\Xi)$ is of even degree $2m$ (with respect to $\Xi$). For every pair of linearly independent real vectors $\Xi,\Xi^{\prime}$, the polynomial $L(x,\Xi+\tau\Xi^{\prime})$ in the complex variable $\tau$ has exactly $m$ roots $\tau_{k}^{+}(x,\Xi)$ ($k=1,\cdots,m$) with positive imaginary part, i.e., $\mathrm{Im}\tau_{k}^{+}(x,\Xi)>0$.

Uniform ellipticity will be required in the sense that there is a positive constant $A$ such that

$\displaystyle A^{-1}|\Xi|^{2m}\leq|L(x,\Xi)|\leq A|\Xi|^{2m}$

(2.4)

for every real vector $\Xi=(\xi_{1},\cdots,\xi_{n})$ and for every point $x$ in the closure of the domain $\Sigma$, where $m=\tfrac{1}{2}\deg(L(x,\Xi))>0$.

In the boundary value of (2.1), $m$ is exactly $\tfrac{1}{2}\deg(L(x,\Xi))$. The linear boundary operator $B_{hj}(x,\partial)$ are of complex coefficients depending on $x$. The orders of $B_{hj}(x,\partial)$, like those of the operators $l_{ij}(x,\partial)$, depend on two groups of integer weights, in this case the group $t_{1},\cdots,t_{N}$ already attached to the dependent unknowns and a new group $r_{1},\cdots,r_{m}$ of which $r_{h}$ pertains to the $h$-th boundary condition, $h=1,\cdots,m$. The exact dependence is expressed by the inequality

$\displaystyle\deg B_{hj}(x,\Xi)\leq r_{h}+t_{j}\,,$

(2.5)

and $B_{hj}=0$ when $r_{h}+t_{j}<0$. Moreover, the following complementing boundary condition on the boundary operator should also be imposed:

1. (CBC)

   For any $x\in\partial\Omega$ and any real, non-zero vector $\Xi$ tangent to $\partial\Omega$ at $x$, let us regard $M^{+}(x,\Xi,\tau)=\prod_{h=1}^{m}(\tau-\tau_{h}^{+}(x,\Xi))$ and the elements of the matrix

   $\displaystyle\sum_{j=1}^{N}B_{hj}^{\prime}(x,\Xi+\tau\mathbf{n})L^{jk}(x,\Xi+\tau\mathbf{n})$

   (2.6)

   as polynomial in the indeterminate $\tau$. The rows of the latter matrix are required to be linearly independent modulo $M^{+}(x,\Xi,\tau)$, i.e.,

   $\displaystyle\sum_{h=1}^{m}C_{h}\sum_{j=1}^{N}B^{\prime}_{hj}L^{jk}\equiv 0\ (\mathrm{mod}M^{+})\,,$

   only if the constant $C_{h}$ are all zero. Here $\mathbf{n}$ is the normal to $\partial\Omega$ at $x$, $B^{\prime}_{hj}(x,\Xi)$ consists of the terms in $B_{hj}(x,\Xi)$ which are just of the order $r_{h}+t_{j}$, and $(L^{j}k(x,\Xi+\tau\mathbf{n}))$ denotes the matrix adjoint to $(l^{\prime}_{ij}(x,\Xi+\tau\mathbf{n}))$.

Specifically, the operators $l_{ij}(x,\partial)$ and $B_{hj}(x,\partial)$ are of the forms

$\displaystyle l_{ij}(x,\partial)=\sum_{|\alpha|=0}^{s_{i}+t_{j}}a_{ij,\alpha}(x)\partial^{\alpha}\,,\quad B_{hj}(x,\partial)=\sum_{|\beta|=0}^{r_{h}+t_{j}}b_{hj,\beta}(x)\partial^{\beta}\,,$

(2.7)

where $\alpha$ and $\beta$ denote multi-indices indicative of the precise differentiation involved. Then the following results hold.

Remark that if the solution to (2.1) is unique, the term $\sum_{j=1}^{N}\|u_{j}\|_{L^{p}(\Omega)}$ on the right can be omitted.

By employing the ADN theory in Proposition 2.1, Temam [19] proved the following conclusion.

Next, the usual $L^{p}$-theory of elliptic equation can be reduced by the ADN theory in Proposition 2.1. One takes the following Laplace equation into consideration:

	$\displaystyle\Delta\phi=\tfrac{\partial^{2}}{\partial x_{1}^{2}}\phi+\cdots+\tfrac{\partial^{2}}{\partial x_{n}^{2}}\phi=$	$\displaystyle h\quad\textrm{ in }\Omega\,,$		(2.12)
	$\displaystyle\tfrac{\partial\phi}{\partial\mathbf{n}}=$	$\displaystyle g\quad\textrm{ in }\partial\Omega\,.$

By letting $\phi_{i}=\tfrac{\partial}{\partial x_{i}}\phi$ with $i=1,\cdots,n$, and $\phi_{n+1}=\phi$, one gains a first order differential system

$$\left\{\begin{aligned} &-\phi_{1}+\tfrac{\partial}{\partial x_{1}}\phi_{n+1}=0\,,\\ &\qquad\cdots\cdots\\ &-\phi_{n}+\tfrac{\partial}{\partial x_{n}}\phi_{n+1}=0\,,\\ &\tfrac{\partial}{\partial x_{1}}\phi_{1}+\cdots+\tfrac{\partial}{\partial x_{n}}\phi_{n}=h\end{aligned}\right.$$

(2.13)

on $\Omega$ with the boundary condition

$\displaystyle\sum_{j=1}^{n}\mathbf{n}_{j}\phi_{j}=g\quad\textrm{on }\partial\Omega\,,$

(2.14)

where $\mathbf{n}_{j}$ denotes the $j$-th component of the normal vector $\mathbf{n}$ to $\partial\Omega$. When the weights

$\displaystyle t_{1}=\cdots=t_{n}=1\,,\ t_{n+1}=2$

are assigned to $\phi_{1},\cdots,\phi_{n},\phi_{n+1}$, respectively, and

$\displaystyle s_{1}=\cdots=s_{n}=-1\,,\ s_{n+1}=0$

to the first, $\cdots$, the $n$-th, and the $(n+1)$-th equations, respectively, the characteristic determinant of (2.13) thereby is

$$L(x,\Xi)=\left|\begin{array}[]{ccccc}-1&0&\cdots&0&\xi_{1}\\ 0&-1&\cdots&0&\xi_{2}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&\cdots&-1&\xi_{n}\\ \xi_{1}&\xi_{2}&\cdots&\xi_{n}&0\end{array}\right|=(-1)^{n}(\xi_{1}^{2}+\cdots\xi_{n}^{2})=(-1)^{n}|\Xi|^{2}\,,$$

whose degree is $2$, i.e., $m=\tfrac{1}{2}\deg L(x,\Xi)=1$. The system (2.13) is thus elliptic. Moreover, the equations (2.13) only need one boundary condition (2.14). It is obvious that the (SC) condition on $L(x,\Xi)$ holds, and the root $\tau^{+}(\Xi,\Xi^{\prime})$ of $L(x,\Xi+\tau\Xi^{\prime})=0$ with positive imaginary is

$\displaystyle\tau^{+}(\Xi,\Xi^{\prime})=\tfrac{-\Xi\cdot\Xi^{\prime}+\sqrt{-1}\sqrt{|\Xi|^{2}|\Xi^{\prime}|^{2}-|\Xi\cdot\Xi^{\prime}|^{2}}}{|\Xi^{\prime}|^{2}}\,.$

Moreover, in (2.14), the weight $r_{1}=-1$ is assigned to the only boundary condition. It is easy to verify that

$\displaystyle M^{+}(x,\Xi,\tau)=\tau-\tau^{+}(\Xi,\mathbf{n})=\tau-\sqrt{-1}|\Xi|\,.$

Observe that $B_{1j}=\mathbf{n}_{j}$ for $1\leq j\leq n$ and $B_{1,n+1}=0$. We have $B_{1j}=B_{1j}^{\prime}$. A direct calculation shows that the matrix $(L^{jk}(x,\Xi))_{1\leq j,k\leq n+1}$ is

$$|\Xi|^{-2}\left(\begin{array}[]{ccccc}\xi_{1}^{2}-|\Xi|^{2}&\xi_{1}\xi_{2}&\cdots&\xi_{1}\xi_{n}&\xi_{1}\\ \xi_{2}\xi_{1}&\xi_{2}^{2}-|\Xi|^{2}&\cdots&\xi_{2}\xi_{n}&\xi_{2}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \xi_{n}\xi_{1}&\xi_{n}\xi_{2}&\cdots&\xi_{n}^{2}-|\Xi|^{2}&\xi_{n}\\ \xi_{1}&\xi_{2}&\cdots&\xi_{n}&1\end{array}\right)\,.$$

By $\mathbf{n}\cdot\Xi=0$ and $|\mathbf{n}|=1$, the vector with elements $C_{1}\sum_{j=1}^{n+1}B^{\prime}_{1j}L^{jk}(x,\Xi+\tau\mathbf{n})$ is then equal to

$\displaystyle C_{1}(\tau\Xi-|\Xi|^{2}\mathbf{n},0)\,,$

which is zero modulo $M^{+}$ if and only if $C_{1}=0$. Therefore, the complementing boundary condition (CBC) holds.