Mathematical Analysis of a Diffuse Interface Model for Multi-phase Flows of Incompressible Viscous Fluids with Different Densities

Abstract

We analyze a diffuse interface model for multi-phase flows of N incompressible, viscous Newtonian fluids with different densities. In the case of a bounded and sufficiently smooth domain existence of weak solutions in two and three space dimensions and a singular free energy density is shown. Moreover, in two space dimensions global existence for sufficiently regular initial data is proven. In three space dimension, existence of strong solutions locally in time is shown as well as regularization for large times in the absence of exterior forces. Moreover, in both two and three dimensions, convergence to stationary solutions as time tends to infinity is proved.

Appendix

1.1 Bihari’s Lemma

Lemma 10.1

([5]) Let u and f be non-negative continuous functions defined on $[0,T']$, where $T'\ge T$ and $T>0$ is such that

$$\begin{aligned} G(\alpha )+\int _0^t\,f(s)\,ds\in {\mathfrak {D}}(G^{-1}),\qquad \forall \, t \in [0,T], \end{aligned}$$

where $\alpha$ is a non-negative constant, the function G is defined by

$$\begin{aligned} G(x)=\int _{x_0}^x \frac{dy}{w(y)},\qquad \text {for } x \ge 0, \end{aligned}$$

for a fixed $x_0>0$, and $G^{-1}$ denotes its inverse. Moreover, w is a continuous non-decreasing function defined on $[0,\infty )$, with $w(u)>0$ on $(0,\infty )$. If u satisfies the following integral inequality

$$\begin{aligned} u(t)\le \alpha + \int _0^t f(s)\,w(u(s))\,ds,\qquad \forall t\in [0,T'], \end{aligned}$$

then

$$\begin{aligned} u(t)\le G^{-1}\left( G(\alpha )+\int _0^t\,f(s) \, ds\right) ,\qquad \forall t\in [0,T]. \end{aligned}$$