论文标题
适用于Navier-Stokes-Cahn-Hilliard系统，用于与表面活性剂的不可压缩的两相流
Well-posedness for a Navier-Stokes-Cahn-Hilliard System for Incompressible Two-phase Flows with Surfactant
论文作者
论文摘要
我们研究了一个漫射界面模型，该模型描述了用表面活性剂不可压缩的两相粘性流的动力学。所得的部分微分方程系统由六阶Cahn-hilliard方程组成，该方程与二元流体混合物的局部浓度差，并与四阶Cahn-Hilliard方程相连，用于表面活性剂的局部浓度。前者具有平稳的潜力，而后者具有单一的潜力。这两个方程都与用于（体积平均）流体速度的Navier-Stokes系统相结合。进化系统具有合适的初始条件，速度场的无滑动边界条件以及相位功能以及化学电位的均匀的Neumann边界条件。我们首先证明了一个全球弱解决方案的存在，该解决方案在两个维度上是唯一的。对初始数据的更牢固的规律性假设使我们能够证明在两个（分别三个）维度中存在独特的全局（局部）强解决方案。在两个维度的情况下，我们可以根据总能量控制的规范得出一个连续的依赖估计。然后，我们以$ t> 0 $建立了全球弱解决方案的瞬时正则属性。特别是，我们表明表面活性剂浓度在积极的时间后与纯州$ 0 $和$ 1 $保持一致。
We investigate a diffuse-interface model that describes the dynamics of incompressible two-phase viscous flows with surfactant. The resulting system of partial differential equations consists of a sixth-order Cahn-Hilliard equation for the difference of local concentrations of the binary fluid mixture coupled with a fourth-order Cahn-Hilliard equation for the local concentration of the surfactant. The former has a smooth potential, while the latter has a singular potential. Both equations are coupled with a Navier-Stokes system for the (volume averaged) fluid velocity. The evolution system is endowed with suitable initial conditions, a no-slip boundary condition for the velocity field and homogeneous Neumann boundary conditions for the phase functions as well as for the chemical potentials. We first prove the existence of a global weak solution, which turns out to be unique in two dimensions. Stronger regularity assumptions on the initial data allow us to prove the existence of a unique global (resp. local) strong solution in two (resp. three) dimensions. In the two dimensional case, we can derive a continuous dependence estimate with respect to the norms controlled by the total energy. Then we establish instantaneous regularization properties of global weak solutions for $t>0$. In particular, we show that the surfactant concentration stays uniformly away from the pure states $0$ and $1$ after some positive time.
