Weak solutions to the stationary Cahn-Hillard/Navier-Stokes equations for compressible fluids

The Cahn-Hilliard/Navier-Stokes system is one of the important diffuse interface models (cf.[3, 8, 17]) describing the evolution of mixing fluids. The mixture is assumed to be macroscopically immiscible, with a partial mixing in a small interfacial region where the sharp interface is regularized by the Cahn-Hilliard type diffusion in terms of the mass concentration difference. Roughly speaking, the Cahn-Hilliard equation is used for modeling the loss of mixture homogeneity and the formation of pure phase regions, while the Navier-Stokes equations describe the hydrodynamics of the mixture that is influenced by the order parameter, due to the surface tension and its variations, through an extra capillarity force term.

In this paper, we are interested in the following stationary Cahn-Hilliard/Navier-Stokes system for the mixture of compressible fluid flows in a three-dimensional bounded domain $\Omega\subset\mathbb{R}^{3}$:

where $\rho$ denotes the total density, $u$ the mean velocity field, $c$ the mass concentration difference of the two components, $\mu$ the chemical potential, and $g$ the external force; the tensor

is the Navier-Stokes stress tensor, where $\mathbb{I}$ is the $3\times 3$ identity matrix, $\lambda_{1},\,\lambda_{2}$ are constant such that

the tensor

is the capillary stress tensor; and

is the pressure with the free energy density (cf. [1, 17])

where $\gamma>1$ is the adiabatic exponent, and $H_{i}\,(i=1,2)$ are two given functions. The corresponding evolutionary diffuse interface model was derived in [1, Section 2.2] where the existence of weak solutions was obtained for $\gamma>\frac{3}{2}$. We refer the readers to [3, 8, 17, 1, 16] for more discussions on the physics and models of mixing fluids with diffuse interfaces.

We briefly review some related results in literature. For the stationary Navier-Stokes equations of compressible flows, the existence of weak solutions was studied in Lions [15] with $\gamma>\frac{5}{3}$, Novotný-Strašcraba [20] with $\gamma>\frac{3}{2}$, Frehse-Steinhauer-Weigant [12] with $\gamma>\frac{4}{3}$, Plotnikov-Weigant[24] with $\gamma>1$, as well as in Jiang-Zhou [14] and Bresch-Burtea [7] for periodic domains. For the stationary Cahn-Hilliard/Navier-Stokes equations of incompressible flows, the existence of weak solutions was obtained in Biswas-Dharmatti-Mahendranath-Mohan [5], Ko-Pustejovska-Suli [22], and Ko-Suli [23]. For the compressible Cahn-Hilliard/Navier-Stokes equations, Liang-Wang in [16] proved the existence of weak solutions in case of the adiabatic exponent $\gamma>2.$ See [15, 20, 10, 12, 24, 14, 19, 18, 5, 22, 23, 16, 7, 6, 9] and their references for more results.

In this paper, we shall continue our study on the existence of weak solutions, and improve our previous result obtained in [16] for $\gamma>2$ to the case of $\gamma>\frac{4}{3}$ for the stationary equations (1.1) subject to the following boundary conditions:

and the additional conditions:

with two given constants $m_{1}$ and $m_{2}$, where $n$ is the normal vector of $\partial\Omega$.

Before stating our main results, we introduce some notation that will be used throughout this paper. For two given matrices $\mathbb{A}=(a_{ij})_{3\times 3}$ and $\mathbb{B}=(b_{ij})_{3\times 3}$, we denote their scalar product by $\mathbb{A}:\mathbb{B}=\sum_{i,j=1}^{3}a_{ij}b_{ij}$. For two vectors $a,\,b\in\mathbb{R}^{3}$, denote $a\otimes b=(a_{i}b_{j})_{3\times 3}.$ We use $\int f=\int_{\Omega}f(x)dx$ for simplicity. For any $p\in[1,\infty]$ and integer $k\geq 0,$ $W^{k,p}(\Omega)$ is the standard Sobolev space (cf. [2]), and

where $(f)_{\Omega}=\frac{1}{|\Omega|}\int f$ is the average of $f$ over $\Omega$.

As in [16], we define the weak solution as follows.

We now state our main result.

The main contribution of this paper is to develop new ideas to improve the existence result of [16] from the adiabatic exponent $\gamma>2$ in [16] to a wider range $\gamma>\frac{4}{3}$. Our approach is mainly motivated by the papers [14, 24] where the authors studied the existence of weak solutions to the stationary Navier-Stokes equations of compressible fluids. In order to prove the Theorem 1.1, we start with the approximate solution sequence $(\rho_{\delta},u_{\delta},\mu_{\delta},c_{\delta})$ stated in Proposition 2.1 in Section 2, and use the weighted total energy as in [14, 24] together with new techniques to handle the capillary stress to establish the uniform in $\delta$ bound on $(\rho_{\delta},u_{\delta},\mu_{\delta},c_{\delta})$ in (3.1). Then, we shall be able to take the limit as $\delta\rightarrow 0$ and complete the proof of Theorem 1.1 by means of the weak convergence arguments in [16]. More precisely, our proof includes the following key ingredients and new ideas:

1. (1)

   In light of [14, 24], for any given $x^{*}\in\overline{\Omega}$ we estimate the weighted total energy

   $$\int_{\Omega}\frac{\left(\delta\rho^{4}+P+\rho|u|^{2}\right)(x)}{|x-x^{*}|^{\alpha}}dx,$$

   instead of

   $$\int_{\Omega}\frac{\left(\delta\rho^{4}+P\right)(x)}{|x-x^{*}|^{\alpha}}dx,$$

   where the advantage is that the involved kinetic energy

   $$\int_{\Omega}\frac{\rho|u|^{2}(x)}{|x-x^{*}|^{\alpha}}dx$$

   helps us relax the restriction on $\gamma$.

2. (2)

   In order to analyze the weighted total energy we need to overcome the new difficulties caused by the capillary stress $\mathbb{S}_{c}$ in (1.4), besides the Navier-Stokes stress tensor $\mathbb{S}_{ns}$. In particular, we are required to control $\|\rho\mu\|_{L^{\frac{3}{2}}}^{2}$ appearing in (3.10) and (3.11). For this purpose we make the following estimate

   $\displaystyle\begin{aligned} &\int_{\Omega\cap B_{r_{0}}(x^{*})}\frac{\left(\delta\rho^{4}+P+\rho|u|^{2}\right)(x)}{|x-x^{*}|^{\alpha^{2}}}dx\\ &\leq Cr_{0}^{\alpha(1-\alpha)}\int_{\Omega\cap B_{r_{0}}(x^{*})}\frac{\left(\delta\rho^{4}+P+\rho|u|^{2}\right)(x)}{|x-x^{*}|^{\alpha}}dx,\end{aligned}$

   where $r_{0}>0$ is small and $\alpha\in(0,1)$. By virtue of the Finite Coverage Theorem, $\overline{\Omega}$ can be covered by a finite number of balls of radius $r_{0}$ centered at $x_{1}^{*},...,x_{K}^{*}$, then

   $\displaystyle\begin{aligned} &\sup_{x^{*}\in\overline{\Omega}}\int_{\Omega}\frac{\left(\delta\rho^{4}+P+\rho|u|^{2}\right)(x)}{|x-x^{*}|^{\alpha^{2}}}dx\\ &\leq\max_{1\leq k\leq K}\int_{\Omega\cap B_{r_{0}}(x_{k}^{*})}\frac{\left(\delta\rho^{4}+P+\rho|u|^{2}\right)(x)}{|x-x^{*}|^{\alpha^{2}}}dx\\ &\leq Cr_{0}^{\alpha(1-\alpha)}\|\rho\mu\|_{L^{\frac{3}{2}}}^{2}+\cdots.\end{aligned}$

   Next, we assume the following a priori bound

   $$\textbf{M}=\max\{1,\|\rho\|_{L^{2}}\}<\infty$$

   (1.11)

   that is uniform in $\delta>0.$ If we select $r_{0}=r_{0}(\alpha,\textbf{M})$ small enough such that

   $\displaystyle r_{0}^{\alpha(1-\alpha)}\|\rho\mu\|_{L^{\frac{3}{2}}}^{2}\leq Cr_{0}^{\alpha(1-\alpha)}\textbf{M}^{\frac{8}{3}}(\|\nabla\mu\|_{L^{2}}^{2}+1)\leq C(\|\nabla\mu\|_{L^{2}}^{2}+1),$

   we are able to derive the following estimate

   $\displaystyle\begin{aligned} \int_{\Omega}\frac{\left(\delta\rho^{4}+P+\rho|u|^{2}\right)(x)}{|x-x^{*}|^{\alpha^{2}}}dx\leq C+C\|\nabla\mu\|_{L^{2}}^{2}+\cdots.\end{aligned}$

3. (3)

   With the above two key steps, we can show that there is a constant $C$ that does not rely on M, such that

   $\displaystyle\|\rho^{\gamma}\|_{L^{s}}\leq C+C\|\rho\mu\|_{L^{\frac{3}{2}}}^{2}\leq C+C\|\rho\|_{L^{2}}^{\frac{4}{3}}\leq C+\frac{1}{2}\|\rho^{\gamma}\|_{L^{s}},$

   as long as $\gamma s>2$. This yields $\|\rho^{\gamma}\|_{L^{s}}\leq 2C$, and then we have the estimate $\|\rho\|_{L^{2}}\leq C_{0}$ for some positive constant $C_{0}$ independent of M. By choosing the a priori bound $\textbf{M}=2C_{0}$, one can close the a priori assumption (1.11) and prove the existence of weak solutions in the Theorem 1.1.

The rest of the paper is organized as follows. In Section 2, we present the approximate solutions constructed in [16] and provide some preliminary lemmas. In Section 3, we prove the Theorem 1.1.

We start with the following approximate solutions constructed in [16].

The next lemma gives an embedding from $H^{1}$ to $L^{2}$ in a three-dimensional bounded domain, via the Green representation formula.

Finally, we present the properties of the Bogovskii operator whose proof is available in [11, 21].

For the approximate solution $(\rho_{\delta},u_{\delta},\mu_{\delta},c_{\delta})$ given in Proposition 2.1, if we can show that there is a constant $C$ uniform in $\delta$ such that

then, from (3.1) we are able to control the possible oscillation of density and the nonlinearity in the free energy density (1.6), and hence we can take the limit as $\delta\rightarrow 0$ to prove that the approximate solution $(\rho_{\delta},u_{\delta},\mu_{\delta},c_{\delta})$ converges weakly to some limit function which satisfies (1.1)-(1.8) in the sense of Definition 1.1. This convergence proof relies heavily on the compactness arguments in [10, 16, 15, 21] and the details can be found in [16]. Therefore, it suffices to prove the following proposition in order to complete the proof of Theorem 1.1.

For the sake of simplicity of notation, in the proof of the Proposition 3.1 below we will drop the subscript in $(\rho_{\delta},u_{\delta},\mu_{\delta},c_{\delta})$ and denote it by $(\rho,u,\mu,c)$.

Next, we shall deduce some weighted estimates on the pressure and kinetic energy together, i.e., the weighted total energy motivated by [14, 24]. As in [12], we introduce

where the function $\phi(x)\in C^{2}(\overline{\Omega})$ can be regarded as the distance function when $x\in\Omega$ is close to the boundary, smoothly extended to the whole domain $\Omega.$ In particular,

where the constants $k_{i}>0$, $i=1,2$, are given. See for example [26, Exercise 1.15] for details.