Convergence to equilibrium of global weak solutions for a Cahn–Hilliard–Navier–Stokes vesicle model

Abstract

In this paper, we consider a model describing the dynamics of vesicle membranes within an incompressible viscous fluid in 3D domains. The system consists of the Navier–Stokes equations, with an extra stress tensor depending on the membrane, coupled with a Cahn–Hilliard phase-field equation associated with a bending energy plus a penalization related to the area conservation (volume is exactly conserved). This problem has a dissipative in time free energy which leads, in particular, to prove the existence of global in time weak solutions. We analyze the large-time behavior of the weak solutions. By using a modified Lojasiewicz–Simon’s result, we prove the convergence as time goes to infinity of each (whole) trajectory to a single equilibrium. Finally, the convergence of the trajectory of the phase is improved by imposing more regularity on the domain and initial phase.

Appendix

Proof of Lemma 3.1

Let us remember that $F(\phi )=\displaystyle \frac{1}{4}(\phi ^2-1)^2$, $F'(\phi )=(\phi ^2-1)\phi$, $F''(\phi )=3\phi ^2-1$, $F'''(\phi )=6\phi$ and ${\mathcal {A}}(\phi )=\displaystyle \int _\Omega \left( \displaystyle \frac{\varepsilon }{2}\vert \nabla \phi \vert ^2+\displaystyle \frac{1}{\varepsilon }F(\phi )\right) \, {\text {d}}{{\varvec{x}}}$. We prove the first inequality for $j=0$:

$$\begin{aligned} \vert F(\phi _i)\vert _p= \displaystyle \frac{1}{4}\vert \phi ^4+2\phi ^2-1\vert _p \le C(\vert \phi \vert _\infty ^4+2\vert \phi \vert _\infty ^2+1) \le C(\Vert \phi \Vert _2^4+2\Vert \phi \Vert _2^2+1)\le C(K), \end{aligned}$$

The cases $j=1,2,3$ can be proved in an analogous way. We prove now the second inequality for $j=0$:

$$\begin{aligned} \begin{array}{l} \vert F(\phi _1)-F(\phi _2)\vert _p\le \vert F'(\theta \phi _1+(1-\theta )\phi _2)\vert _\infty \vert \phi _1-\phi _2\vert _p \le C(K)\vert \phi _1-\phi _2\vert _p. \end{array} \end{aligned}$$

The cases $j=1,2,3$ can be proved in an analogous way. Finally, the third one is as follows:

$$\begin{aligned} \begin{array}{c} \displaystyle \vert {\mathcal {A}}(\phi _1)-{\mathcal {A}}(\phi _2)\vert \le \displaystyle \frac{1}{2}\vert |\nabla \phi _1|_2^2-|\nabla \phi _2|_2^2\vert + \displaystyle \int \limits _\Omega |F(\phi _1) - F(\phi _2)| \\ \displaystyle \le \displaystyle \frac{1}{2}\vert \nabla (\phi _1+\phi _2)\vert _2\vert \nabla (\phi _1-\phi _2)\vert _2 +C(K)\vert \phi _1-\phi _2\vert _1 \le C(K)\Vert \phi _1-\phi _2\Vert _1. \end{array} \end{aligned}$$

$\square$

Proof of Lemma 3.2

From (2.11), we have that

$$\begin{aligned} \begin{array}{l} \vert G(\phi _1)-G(\phi _2)\vert _2\le \displaystyle \frac{2}{\varepsilon }\vert F''(\phi _1)\Delta \phi _1-F''(\phi _2)\Delta \phi _2\vert _2 \\ \qquad +\,\displaystyle \frac{1}{\varepsilon }\vert F'''(\phi _1)\vert \nabla \phi _1\vert ^2-F'''(\phi _2)\vert \nabla \phi _2\vert ^2\vert _2 +\displaystyle \frac{1}{\varepsilon ^3}\vert F'(\phi _1)F''(\phi _1)-F'(\phi _2)F''(\phi _2)\vert _2 \\ \qquad +\,\varepsilon M\vert ({\mathcal {A}}(\phi _1)-\alpha )\Delta \phi _1-({\mathcal {A}}(\phi _2)-\alpha )\Delta \phi _2\vert _2 \\ \qquad +\,\displaystyle \frac{M}{\varepsilon } \vert ({\mathcal {A}}(\phi _1)-\alpha )F'(\phi _1)-({\mathcal {A}}(\phi _2)-\alpha )F'(\phi _2)\vert _2 :=\sum _{i=1}^5 I_i. \end{array} \end{aligned}$$

(8.1)

By taking into account Lemma 3.1, we can estimate each term $I_i$. For example,

$$\begin{aligned} I_1= & {} \displaystyle \frac{2}{\varepsilon }\vert F''(\phi _1)\Delta \phi _1-F''(\phi _2)\Delta \phi _2\vert _2 \\\le & {} C( \vert (F''(\phi _1)-F''(\phi _2))\Delta \phi _1\vert _2+\vert F''(\phi _2)(\Delta \phi _1-\Delta \phi _2)\vert _2) \\\le & {} C( \vert \Delta \phi _1\vert _2\vert F''(\phi _1)-F''(\phi _2)\vert _\infty + \vert F''(\phi _2)\vert _\infty \vert \Delta \phi _1-\Delta \phi _2\vert _2) \\\le & {} C(\Vert \phi _1\Vert _2 C(K)\vert \phi _1-\phi _2\vert _\infty + C(K)\Vert \phi _1- \phi _2\Vert _2) \\\le & {} C(K)\Vert \phi _1-\phi _2\Vert _2. \\ I_2= & {} \displaystyle \frac{1}{\varepsilon }\vert F'''(\phi _1)\vert \nabla \phi _1\vert ^2-F'''(\phi _2)\vert \nabla \phi _2\vert ^2\vert _2 \\\le & {} C (\vert (F'''(\phi _1)- F'''(\phi _2))\vert \nabla \phi _1\vert ^2\vert _2+\vert F'''(\phi _2)(\vert \nabla \phi _1\vert ^2-\vert \nabla \phi _2\vert ^2)\vert _2) \\\le & {} C (\vert \nabla \phi _1\vert ^2_6\vert F'''(\phi _1)- F'''(\phi _2)\vert _6+\vert F'''(\phi _2)\vert _6\vert \nabla \phi _1+\nabla \phi _2\vert _6\vert \nabla \phi _1-\nabla \phi _2\vert _6) \\\le & {} C (\Vert \phi _1\Vert ^2_2C(K)\vert \phi _1- \phi _2\vert _6+C(K)\Vert \phi _1+\phi _2\Vert _2\Vert \phi _1-\phi _2\Vert _2) \\\le & {} C(K)\Vert \phi _1-\phi _2\Vert _2. \end{aligned}$$

Observe that

$$\begin{aligned} \vert {\mathcal {A}}(\phi _2)-\alpha \vert \le C( \vert \nabla \phi _2\vert _2^2+\vert \phi _2^2-1\vert _2^2) \le C(\Vert \phi _2\Vert _1^2+\Vert \phi _2\Vert _2^4+1)\le C(K), \end{aligned}$$

therefore,

$$\begin{aligned} I_4= & {} \varepsilon M\vert ({\mathcal {A}}(\phi _1)-\alpha )\Delta \phi _1-({\mathcal {A}}(\phi _2)-\alpha )\Delta \phi _2\vert _2 \\\le & {} C(\vert {\mathcal {A}}(\phi _1)-{\mathcal {A}}(\phi _2)\vert \, \vert \Delta \phi _1\vert _2+\vert {\mathcal {A}}(\phi _2)-\alpha \vert \, \vert \Delta (\phi _1-\phi _2 )\vert _2) \\\le & {} C(K)(\Vert \phi _1-\phi _2\Vert _1+\Vert \phi _1-\phi _2\Vert _2) \le C(K)\Vert \phi _1-\phi _2\Vert _2, \end{aligned}$$

$\square$