Nonlocal Cahn–Hilliard Equation with Degenerate Mobility: Incompressible Limit and Convergence to Stationary States

Abstract

The link between compressible models of tissue growth and the Hele–Shaw free boundary problem of fluid mechanics has recently attracted a lot of attention. In most of these models, only repulsive forces and advection terms are taken into account. In order to take into account long range interactions, we include a surface tension effect by adding a nonlocal term which leads to the degenerate nonlocal Cahn–Hilliard equation, and study the incompressible limit of the system. The degeneracy and the source term are the main difficulties. Our approach relies on a new $L^{\infty }$ estimate obtained by De Giorgi iterations and on a uniform control of the energy despite the source term. We also prove the long-term convergence to a single constant stationary state of any weak solution using entropy methods, even when a source term is present. Our result shows that the surface tension in the nonlocal (and even local) Cahn–Hilliard equation will not prevent the tumor from completely invading the domain.

Technical Tools

Several tools have been used to carry out some proofs. First, we present a lemma about geometric convergence of numerical sequences, whose proof can be easily obtained by induction (see, e.g., [40, Ch.2, Lemma 5.6]).

Lemma A.1

Let $\{y_n\}_{n\in {\mathbb {N}}\cup \{0\}}\subset {\mathbb {R}}^+$ satisfy the recursive inequality

$$\begin{aligned} y_{n+1}\le Cb^ny_n^{1+\epsilon }, \quad \forall n\ge 0, \qquad \text {and} \qquad y_0\le \theta := C^{-\frac{1}{\epsilon }}b^{-\frac{1}{\epsilon ^2}}, \end{aligned}$$

(A.1)

for some $C>0$, $b>1$ and $\epsilon >0$. Then, $y_n\rightarrow 0$ for $n\rightarrow \infty$ with geometric rate

$$\begin{aligned} y_n\le \theta b^{-\frac{n}{\epsilon }},\qquad \forall n\ge 0. \end{aligned}$$

(A.2)

Next, we state a theorem concerning the absolute continuity of some integrals of convex functions in ${\mathbb {R}}$, whose proof can be found, e.g. in [34, p.101].

Theorem A.2

Let $T>0$ and let $h: {\mathbb {R}}\rightarrow {\mathbb {R}}$ be a convex and lower semicontinuous function. Assume that

• $f\in L^2(0,T;H^1(\Omega ))$ and $\partial _t f\in L^2(0,T; (H^1(\Omega ))')$,

• $g(x,t)\in \partial h(x,t)$ for almost every $(x,t)\in \Omega \times (0,T)$,

• $g\in L^2(0,T;H^1(\Omega ))$.

Then, the function $t\mapsto \int _\Omega h(f(x,t))$ is absolutely continuous on [0, T], and

$$\begin{aligned} \frac{d}{dt} \int _\Omega h(f){\textrm{d}}x =\langle \partial _t f, g\rangle \quad \text {for almost any } t\in (0,T). \end{aligned}$$

We then propose a control on the $H^1(\Omega )$-norm related to the use of $\omega _\varepsilon$.

Lemma A.3

There exists $\varepsilon _0>0$ and a constant C such that for $\varepsilon \in (0, \varepsilon _0)$ and all $f\in L^2(\Omega )$ we have

$$\begin{aligned} \Vert f - {\overline{f}} \Vert _{L^2(\Omega )}^2\le \frac{C}{2\varepsilon ^2} \int _\Omega \int _\Omega \omega _\varepsilon (y)\vert f(x)- f(x-y) \vert ^2{\textrm{d}}x {\textrm{d}}y, \end{aligned}$$

(A.3)

where ${\overline{f}}$ is the average of f over $\Omega$. Similarly, for all $\alpha$, there exists $\varepsilon _0(\alpha )>0$ and constant $C(\alpha )$ such that for all $\varepsilon \in (0, \varepsilon _0)$ and all $f\in H^1(\Omega )$ we have

$$\begin{aligned} \Vert f\Vert _{H^1(\Omega )}^2\le \frac{\alpha }{2\varepsilon ^2} \int _\Omega \int _\Omega \omega _\varepsilon (y)\vert \nabla f(x)-\nabla f(x-y) \vert ^2{\textrm{d}}x {\textrm{d}}y+C(\alpha ) \Vert f \Vert _{L^1(\Omega )}^2. \end{aligned}$$

(A.4)

Proof

The proof is identical to the one in [24, Lemma C.3] by substituting the norm $\Vert ~ \cdot ~\Vert _{L^2({\mathbb {T}}^d)}$ with the norm $\Vert \cdot \Vert _{L^1({\mathbb {T}}^d)}$. Indeed, with the notation of the proof of that Lemma, also $n\Vert g_n\Vert _{L^1({\mathbb {T}}^d)}<1$ implies that the limit function $g=0$, exactly as in the case $L^2({\mathbb {T}}^d)$. $\square$

In conclusion, we recall the Csiszár–Kullback–Pinsker inequality (see, e.g., [7]), which is essential to study the asymptotic behavior of weak solutions.

Lemma A.4

For any non-negative $u\in L^1(\Omega )$

$$\begin{aligned} 4 |\Omega | {\overline{u}} \int _\Omega u\log \left( \dfrac{u}{{\overline{u}}}\right) {\textrm{d}}x \ge \Vert u-{\overline{u}}\Vert ^{2}_{L^{1}(\Omega )}. \end{aligned}$$