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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
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