Stationary Solutions for a Navier-Stokes/Cahn-Hilliard System with Singular Free Energies

Abstract

We consider a stationary Navier-Stokes/Cahn-Hilliard type system. The system describes a so-called diffuse interface model for the two-phase flow of two macroscopically immiscible incompressible viscous fluids in the case of matched densities, also known as Model H. We prove existence of weak solutions for the stationary system for general exterior forces and singular free energies, which ensure that the order parameter stays in the physical reasonable interval. To this end we reduce the system to an abstract differential inclusion and apply the theory of multi-valued pseudo-monotone operators.

Dedicated to Yoshihiro Shibata on the occasion of his 60th birthday

Appendix

Lemma 5.1

Let X be a real, reflexive Banach space and $\tilde{\mathcal{A}}: X \times X \rightarrow X'$ such that for all u ∈ X:

• 1. $\tilde{\mathcal{A}} (u,.): X \rightarrow X'$ is monotone and hemi-continuous.

• 2. $\tilde{\mathcal{A}} (.,u): X \rightarrow X'$ is completely continuous.

Then the operator $\mathcal{A}: X \rightarrow X'$ defined by $\mathcal{A}(u) = \tilde{\mathcal{A}} (u,u)$ is pseudo-monotone.

Proof

Let $(u_{k})_{k\in \mathbb{N}} \subseteq X$ be such that $u_{k} \rightharpoonup u$ in X and

$$\displaystyle{\mathop{\lim \sup }\limits_{k \rightarrow \infty }\langle \mathcal{A}(u_{k}),u_{k} - u\rangle _{X',X} \leq 0.}$$

We have to show that

$$\displaystyle{\mathop{\lim \inf }\limits_{k \rightarrow \infty }\langle \mathcal{A}(u_{k}),u_{k} - w\rangle _{X',X}\ \geq \ \langle \mathcal{A}(u),u - w\rangle _{X',X}}$$

for all w ∈ X. Since $\tilde{\mathcal{A}} (u,.)$ is monotone,

$$\displaystyle\begin{array}{rcl} \langle \tilde{\mathcal{A}}(u_{k},u_{k}) - \tilde{\mathcal{A}} (u_{k},w_{t}),u_{k} - w_{t}\rangle _{X',X} \geq 0,& & {}\\ \end{array}$$

where we choose w _t : = (1 − t)u + tv for 0 < t < 1 and v ∈ X arbitrary.

Now we use that u _k − w _t  = (u _k − u) + t(u − v) and thus we get two terms. For the first one we use

$$\displaystyle\begin{array}{rcl} \mathop{\lim \inf }\limits_{k \rightarrow \infty }\langle \tilde{\mathcal{A}} (u_{k},u_{k}) - \tilde{\mathcal{A}} (u_{k},w_{t}),u_{k} - u\rangle _{X',X}& =& \mathop{\lim \inf }\limits_{k \rightarrow \infty }\langle \mathcal{A}(u_{k}),u_{k} - u\rangle _{X',X}, {}\\ \end{array}$$

where we used that $\tilde{\mathcal{A}} (\cdot,w_{t})$ is completely continuous for every 0 < t < 1.

This means that for the remaining terms it holds

$$\displaystyle\begin{array}{rcl} \mathop{\lim \inf }\limits_{k \rightarrow \infty }\left (\langle \tilde{\mathcal{A}} (u_{k},u_{k}) - \tilde{\mathcal{A}} (u_{k},w_{t}),t(u - v)\rangle _{X',X} +\langle \mathcal{A}(u_{k}),u_{k} - u\rangle _{X',X}\right ) \geq \ 0.& & {}\\ \end{array}$$

Since $\mathop{\lim \sup }\limits_{k \rightarrow \infty }\langle \mathcal{A}(u_{k}),u_{k} - u\rangle _{X',X} \leq 0$ and 0 < t < 1, we can conclude

$$\displaystyle\begin{array}{rcl} \mathop{\lim \inf }\limits_{k \rightarrow \infty }\left (\langle \tilde{\mathcal{A}} (u_{k},u_{k}) - \tilde{\mathcal{A}} (u_{k},w_{t}),u - v\rangle _{X',X} +\langle \mathcal{A}(u_{k}),u_{k} - u\rangle _{X',X}\right ) \geq \ 0.& & {}\\ \end{array}$$

This yields

$$\displaystyle\begin{array}{rcl} \mathop{\lim \inf }\limits_{k \rightarrow \infty }\left (\langle \mathcal{A}(u_{k}),u_{k} - v\rangle _{X',X} -\langle \tilde{\mathcal{A}} (u_{k},w_{t}),u - v\rangle _{X',X}\right ) \geq 0& & {}\\ \end{array}$$

Using that $\tilde{\mathcal{A}} (u_{k},\cdot )$ is hemi-continuous for every $k \in \mathbb{N}$ and $\tilde{\mathcal{A}} (\cdot,u)$ is completely continuous for every u ∈ X yields the lemma. □