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On a diffuse interface model for a two-phase flow of compressible viscous fluids. (English) Zbl 1144.35041
The authors consider a model of binary mixture for two non-miscible compressible fluids, where the difference in the concentrations between the two fluids plays the role of an order parameter. The resulting model, consisting in a compressible barotropic Navier-Stokes system coupled to a convection-diffusion model of Cahn-Hilliard type in a bounded domain $$\Omega\subset {\mathbb R}^3$$, is the following: $\partial_t\rho+ \text{div}\;\rho\vec{v}=0,$ $\rho\partial_t \vec{v}+ \rho\vec{v}\cdot \nabla \vec{v} = -\nabla p+\text{div}\;{\mathbb S}-\text{div}\;\left(\nabla c\otimes \nabla c-\frac{1}{2}|\nabla c|^2\;{\mathbb I}\right),$ $\rho \partial_t c+ \rho \vec{v}\cdot \nabla c=\Delta \mu,$ $\rho \mu=\rho \partial_c f-\Delta c,$ where $$\rho$$ is the density, $$\vec{v}$$ is the velocity, $$c$$ is the mass concentration difference of the two components and $$\mu$$ is the chemical potential.
In this system:
$${\mathbb S}=\nu(c)\left(\nabla \vec{v}+\nabla {\vec{v}}^T -\frac{2}{3}\;\;\text{div}\;\vec{v}\;{\mathbb I}\right) +\eta(c)\;\text{div}\;\vec{v}\;{\mathbb I}$$ is the viscous stress tensor with $$\nu(c)>0$$ and $$\eta(c)\geq 0$$; the free energy $$f$$, given as a sum of elastic and mixture contributions $$f(\rho,c)=f_e(\rho) + H(c)\log \rho+G(c)$$, is related to the pressure $$p(\rho,c)=\rho^2 \partial_{\rho}f$$ with $$f_e(\rho)=\int_1^{\rho} \frac{p_e(z)}{z^2}\;dz$$, where $$p_e\in C[0,\infty)\cap C^1(0,\infty)$$.
Finally, the following behaviours are assumed for the continuously differentiable functions $$\nu$$, $$\eta$$, $$p_e$$, $$H$$ and $$G$$ $0<\underline{n}\leq \nu(c)\leq \overline{n}, \;\;\;0\leq \eta(c)\leq \overline{e}.$ $p_e(0)=0,\;\;\;\underline{p}_1 \rho^{\gamma-1}-\underline{p}_2\leq p'_e(\rho)\leq \overline{p}(1+\rho^{\gamma-1})\;\;\text{for}\;\gamma>3/2,$ $H(c)\leq \overline{H},\;\;\;-\underline{h}\leq H'(c),\;\;\;\underline{g}_1 c-\underline{g}_2\leq G'(c)\leq \overline{g}(1+c).$ Adapting the methods of E. Feireisl [Dynamics of viscous compressible fluids. (Oxford: Oxford University Press) (2004; Zbl 1080.76001)], the authors prove the global existence of a variational solution for the previous system, under the boundary conditions $\left. \vec{v}\right|_{\partial\Omega}=\vec{0},\;\;\left. \nabla c\cdot\vec{n}\right|_{\partial\Omega}=0,\;\;\left. \nabla \mu\cdot\vec{n}\right|_{\partial\Omega}=0,$ where $$\partial\Omega$$ is a $$C^{2+\nu}$$ surface with $$\nu>0$$, and $$\vec{n}$$ is its external normal, and supposing that the initial conditions $$\rho_0$$, $$\vec{m}_0$$ and $$c_0$$ satisfy $\rho_0\geq 0,\;\;\;\rho_0\in L^{\gamma}(\Omega),\;\;\;\frac{\vec{m}_0^2}{\rho_0}\in L^1(\Omega),\;\;\;c_0\in H^1(\Omega),$ for a $$\gamma>3/2$$.
The proof relies on the construction of an implicit time discretization scheme for a regularized system with artificial pressure depending on two positive numbers $$\varepsilon$$ and $$\delta$$, for which global existence of a solution may be proved using the Faedo-Galerkin method and the Leray-Schauder theorem. Note that, contrary to the pure barotropic model, a new argument has to be introduced in order to control $$c$$ and $$\nabla c$$ which may strongly oscillate with respect to time, moreover specific estimates are required concerning the Cahn-Hilliard part of the system. Then energy-entropy estimates and refined pressure estimates allow to get strong compactness of the concentration gradients and then pointwise convergence for the regularized density. After passing successively to the limit $$\varepsilon\rightarrow 0$$ and $$\delta\rightarrow 0$$ a solution of the original problem is obtained.

##### MSC:
 35Q30 Navier-Stokes equations 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 76T99 Multiphase and multicomponent flows
##### Keywords:
diffuse interface; two-phase flow; compressible; viscous
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