A Hele-Shaw Limit Without Monotonicity

Abstract

We study the incompressible limit of the porous medium equation with a right hand side representing either a source or a sink term, and an injection boundary condition. This model can be seen as a simplified description of non-monotone motions in tumor growth and crowd motion, generalizing the congestion-only motions studied in recent literature (Alexander et al. in Nonlinearity 27(4):823–858, 2014; Perthame et al. in Arch Ration Mech Anal 212(1):93–127, 2014; Kim and Požár in Trans Am Math Soc 370(2):873–909, 2018; Mellet et al. in J Funct Anal 273(10):3061–3093, 2017). We characterize the limit density, which solves a free boundary problem of Hele-Shaw type in terms of the limit pressure. The novel feature of our result lies in the characterization of the limit pressure, which solves an obstacle problem at each time in the evolution.

Appendices

Appendix A. Tumor Growth Model with Nutrient

In [21] (see also [9]), the following model for tumor growth is studied:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \rho _m - \mathrm{div}\,( \rho _m\nabla p_m) = \rho _m G(p_m,c_m) \qquad x\in {\mathbb {R}}^n, \; t\geqq 0\\ \partial _t c_m -\Delta c_m + \rho _m H(c_m) = (c_B -c_m) K(p_m) \\ c_m(x,t)\rightarrow c_B \text{ for } x\rightarrow \infty \end{array}\right. } \end{aligned}$$

(A.1)

where

$$\begin{aligned} p_m = \frac{m}{m-1} \rho _m^{m-1}. \end{aligned}$$

In this system, the evolution of the cell population density $\rho _m\geqq 0$ is coupled to the concentration of nutrients $c_m\geqq 0$ by the cell division rate G(p, c). Importantly, this function satisfies

$$\begin{aligned} \partial _ p G<-\beta <0 \end{aligned}$$

(see [21] for a complete list of the assumptions necessary to get a good existence and uniqueness framework as well as the appropriate estimates to pass to the limit).

It is proved in [21] that $\rho _m(x,t)$, $p_m(x,t)$ and $c_m(x,t)$ converge strongly in $L^1(Q_T)$ (for all $T>0$) to $\rho _\infty ,p_\infty , c_\infty$ in $BV(Q_T)$which solves the system

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \rho _\infty - \mathrm{div}\,( \rho _\infty \nabla p_\infty ) =\rho _\infty G(p_\infty ,c_\infty ) \qquad x\in {\mathbb {R}}^n, \; t\geqq 0\\ \partial _t c_\infty -\Delta c_\infty + \rho _\infty H(c_\infty ) =(c_B -c_\infty ) K(p_\infty ) \\ c_\infty (x,t)\rightarrow c_B \text{ for } x\rightarrow \infty \end{array}\right. } \end{aligned}$$

(A.2)

with the Hele-Shaw relation $p_\infty \in P_\infty (\rho _\infty )$.

Remarkably, the solution of this system is unique, and one would like to interpret the system as a weak form of some geometric Hele-Shaw type free boundary problem. For this one needs to identify the pressure $p_\infty$ as solution of an elliptic equation in $\{\rho _\infty =1\}$.

In [9], it is proved that $p_\infty$ solves the complementarity condition

$$\begin{aligned} p_\infty (\Delta p_\infty + G(p_\infty ,c_\infty ))=0 \quad \text{ in } {\mathcal {D}}'(Q). \end{aligned}$$

This condition says that $p_\infty$ solves an elliptic equation in $\{p_\infty \}$ and is proved by deriving additional estimates on $p_m$.

We will show below that the approach used in this paper can be used to characterize $p_\infty (\cdot ,t)$ as the unique solution of an obstacle problem. First, we summarize the estimates proved in [21]:

Lemma A.1

Under the assumptions listed in [21], the following holds for all $T>0$:

• $\rho _m(t)$ is uniformly compactly supported for $t\in [0,T]$;

• $|\nabla p_m|$ is bounded in $L^2(Q_T)$

• $0\leqq p_m\leqq p_M$, $0\leqq \rho _m\leqq \left( \frac{m-1}{m} p_M\right) ^\frac{1}{m-1}$, $0<c_m<c_B$

• $\rho _m$, $p_m$ and $c_B-c_m$ are bounded in $BV(Q_T)$

• $\rho _m$, $p_m$ and $c_B-c_m$ converge strongly in $L^1$ and almost everywhere to $\rho _\infty$, $p_\infty$ and $c_B-c_\infty$.

Furthermore, proceeding as in Lemma 3.7, it is not difficult to show that $\{\rho _m\}_{m\in {\mathbb {N}}}$ is relatively compact in $C^s(0,T;H^{-1}({\mathbb {R}}^n))$ for all $s\in (0,1/2)$ and thus that $\rho _\infty \in C(0,T;H^{-1}({\mathbb {R}}^n))$.

Finally, since $p_\infty$ and $c_B-c_\infty$ are in $BV(Q_T)$, we can define the trace $p^+(\cdot ,t)$ and $c^+(\cdot ,t)$ for all $t>0$ as in (2.10). We can then prove the following result:

Proposition A.2

For all $t>0$, let $E_t$ denote the space

$$\begin{aligned} E_t = \{ v\in H^1({\mathbb {R}}^n)\cap L^1({\mathbb {R}}^n)\, ;\, \; v (x) \geqq 0, \; \langle v, 1-\rho _\infty (t) \rangle _{H^1,H^{-1}} =0 \}. \end{aligned}$$

Then for all $t>0$, the function $x\mapsto p^+(x,t)$ is the unique solution of the minimization problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} p \in E_t \\ \displaystyle \int _{{\mathbb {R}}^n} \frac{1}{2} |\nabla p|^2 -{\mathcal {G}}(p,c^+) \, \mathrm{d}x \leqq \int _{{\mathbb {R}}^n} \frac{1}{2} | \nabla v|^2 -{\mathcal {G}}( v,c^+)\, \mathrm{d}x \qquad \forall v \in E_t \end{array}\right. } \end{aligned}$$

(A.3)

where ${\mathcal {G}}$ is the (concave) function such that $\partial _p {\mathcal {G}}(p,c) =G(p,c)$ and ${\mathcal {G}}(0,c)=0$. Furthermore $p_\infty$ satisfies the complementarity condition

$$\begin{aligned} p_\infty (\Delta p_\infty + G(p_\infty ,c_\infty ) ) =0 \quad \text{ in } {\mathcal {D}}'({\mathbb {R}}^n\times (0,\infty )). \end{aligned}$$

(A.4)

As mentioned in the introduction (see Proposition 2.8), if the complementarity condition (A.4) is known to hold, then one can derive the variational formulation (A.3) from the weak equation (A.2). In particular, this complementarity condition was derived for this particular model in [9] by using a generalized Aronson-Bénilan estimate and the $L^2(W^{1,4})$ estimate on the pressure (but our proof here does not require either of these estimates).

Proof

First we recall the equation for the pressure $p_m$:

$$\begin{aligned} \partial _t p_m = (m-1) p_m( \Delta p_m + G(p_m,c_m)) + |\nabla p_m|^2. \end{aligned}$$

(A.5)

We then proceed as in the proof of Theorem 2.7: Given $t_0>0$ and a function v(x) in $E_{t_0}$, we use the equation for the pressure (A.5) and density (A.2) to write that

$$\begin{aligned}&\int _{{\mathbb {R}}^n} \nabla p_m \cdot \nabla p_m - \rho _m \nabla p_m \cdot \nabla v- {\mathcal {G}}(p_m,c_m) + {\mathcal {G}}(v,c_m)\, \hbox {d}x\\&\quad = -\frac{1}{m-1}\left[ \frac{\text {d}}{\text {d}t}\int _{{\mathbb {R}}^n} p_m\, \hbox {d}x - \int _\Omega |\nabla p_m|^2\, \hbox {d}x\right] + \frac{\text {d}}{\text {d}t} \int _{{\mathbb {R}}^n} v \rho _m\, \hbox {d}x\\&\qquad + \int _{{\mathbb {R}}^n} p_m G(p_m,c_m) - \rho _m v G(p_m,c_m) - {\mathcal {G}}(p_m,c_m) + {\mathcal {G}}(v,c_m)\, \hbox {d}x \end{aligned}$$

in ${\mathcal {D}}'({\mathbb {R}}_+)$. Using the concavity of ${\mathcal {G}}$ to write

$$\begin{aligned} {\mathcal {G}}(v,c_m) - {\mathcal {G}}(p_m,c_m) \leqq G(p_m,c_m)(v-p_m), \end{aligned}$$

we deduce that

$$\begin{aligned}&\int _{{\mathbb {R}}^n} \nabla p_m \cdot \nabla p_m - \rho _m \nabla p_m \cdot \nabla v- {\mathcal {G}}(p_m,c_m) + {\mathcal {G}}(v,c_m)\, \hbox {d}x\\&\quad = -\frac{1}{m-1}\left[ \frac{\text {d}}{\text {d}t}\int _{{\mathbb {R}}^n} p_m \, \hbox {d}x - \int _{{\mathbb {R}}^n}|\nabla p_m|^2\, \hbox {d}x\right] \\&\qquad + \frac{\text {d}}{\text {d}t} \int _{{\mathbb {R}}^n} v \, \rho _m\, \hbox {d}x + \int _{{\mathbb {R}}^n} (1- \rho _m) \, v \, G(p_m,c_m) \, \hbox {d}x. \end{aligned}$$

We can now proceed as in the proof of Theorem 2.7: Integrating this equality with respect to $t\in [t_0,t_0+\delta )$ and using the weak $L^2$ convergence of $\nabla p_m$ and $\rho _m \nabla p_m$ to $\nabla p$, we get

$$\begin{aligned}&\int _{t_0}^{t_0+\delta } \int _{{\mathbb {R}}^n} |\nabla p_\infty |^2 -\nabla p_\infty \cdot \nabla v- {\mathcal {G}}(p_\infty ,c_\infty ) +{\mathcal {G}}(v,c_\infty )\, \hbox {d}x\, \hbox {d}t\\&\quad \leqq \int _{{\mathbb {R}}^n} v(x) [\rho _\infty (x,t_0 +\delta ) - \rho _\infty (x,t_0)] \, \hbox {d}x+ \int _{t_0}^{t_0+\delta } \int _{{\mathbb {R}}^n} (1- \rho _\infty ) v G(p_\infty ,c_\infty ) \, \hbox {d}x\, \hbox {d}t\\&\quad \leqq \Vert G(p_\infty ,c_\infty )\Vert _{L^\infty } \int _{t_0}^{t_0+\delta } \int _{{\mathbb {R}}^n} v(1- \rho _\infty )\, \hbox {d}x\, \hbox {d}t \end{aligned}$$

(where we used the fact that $v(x) \rho _\infty (x,t_0) = v(x)$ and $v(x) \rho _\infty (x,t) \leqq v(x)$ for all t)

Finally, dividing by $\delta$ and using Young’s inequality, we rewrite the inequality as

$$\begin{aligned}&\frac{1}{\delta }\int _{t_0}^{t_0+\delta } \int _{{\mathbb {R}}^n}\frac{1}{2} |\nabla p_\infty |^2 - {\mathcal {G}}(p_\infty ,c_\infty ) \, \hbox {d}x\\&\quad \leqq \frac{1}{\delta }\int _{t_0}^{t_0+\delta } \int _{{\mathbb {R}}^n} \frac{1}{2} |\nabla v|^2 - {\mathcal {G}}(v,c_\infty )\, \hbox {d}x\, \hbox {d}t + \frac{C}{\delta }\int _{t_0}^{t_0+\delta } \int _{{\mathbb {R}}^n} v (1- \rho _\infty ) \, \hbox {d}x\, \hbox {d}t \\&\quad \leqq \int _{{\mathbb {R}}^n} \frac{1}{2} |\nabla v|^2 -\frac{1}{\delta }\int _{t_0}^{t_0+\delta }{\mathcal {G}}(v,c_\infty )\, \hbox {d}t\, \hbox {d}x + \frac{C}{\delta }\int _{t_0}^{t_0+\delta } \langle v, 1 -\rho _\infty \rangle _{H^1,H^{-1}} \, \hbox {d}x\, \hbox {d}t. \end{aligned}$$

The continuity of $t\mapsto \langle v, 1- \rho _\infty \rangle _{H^1,H^{-1}}$ and the fact that $v\in E_t$ implies that the last term converges to zero as $\delta \rightarrow 0$. We can now conclude as in the proof of Theorem 2.7.

Finally, given a test function $\varphi \in {\mathcal {D}} ({\mathbb {R}}^n\times (0,\infty ))$, we take $v = p_\infty + \varepsilon (p_\infty \varphi ) =p_\infty (1+\varepsilon \varphi )$ in (A.3), with $|\varepsilon |$ small enough so that $1+\varepsilon \varphi \geqq 0$. Passing to the limit $\varepsilon \rightarrow 0^-$ and $\varepsilon \rightarrow 0^+$ yields

$$\begin{aligned} \int _{{\mathbb {R}}^n} \nabla p_\infty \cdot \nabla (p_\infty \varphi ) -G(p_\infty ,c_\infty ) p_\infty \varphi \, \hbox {d}x =0 \end{aligned}$$

and (A.4) follows. $\square$

Appendix B. The Complementarity Condition

Proof of Proposition 2.8

We note that $\partial _t \rho = \Delta p + \lambda \rho \in L^2(0,T;H^{-1} (\Omega ))$. Given $u\in E_t$, we have $p-u \in L^2(0,T;H^1_0(\Omega ))$ and so we can write (in ${\mathcal {D}}'({\mathbb {R}}_+)$)

$$\begin{aligned} \langle \partial _t \rho , (p-u)\rangle _{H^{-1}, H^1_0} =\langle \Delta p + \lambda \rho , p-u\rangle _{H^{-1}, H^1_0}= - \int _\Omega \nabla p \cdot \nabla (p-u) -\lambda \rho (p-u)\, \hbox {d}x . \end{aligned}$$

(B.1)

Next, proceeding as in the beginning of the proof of Lemma 8.1 (using the comparison principle for the limiting problem, Proposition 5.1), we can show that $\rho =1$ in $U\times {\mathbb {R}}_+$ for some neighborhood U of K and that $\mathrm{supp}\,p$ is bounded in $\Omega \times [0,T]$. In particular, $\partial _t \rho$ vanishes in $U\times {\mathbb {R}}_+$. Taking a smooth function $\phi (x)$ which is equal to 1 in $\mathrm{supp}\,p\setminus (U\times [0,T])$ and vanishes on $\partial K$, we can write

$$\begin{aligned} \langle \partial _t \rho , (p-u)\rangle _{H^{-1}, H^1_0}&= \langle \partial _t \rho , (p-u)\phi \rangle _{H^{-1}, H^1_0}\\&= \langle \Delta p + \lambda \rho , p \phi \rangle _{H^{-1}, H^1_0 } -\langle \partial _t \rho , u\phi \rangle _{H^{-1}, H^1_0} \\&= \langle p( \Delta p + \lambda \rho ) , \phi \rangle _{{\mathcal {D}}',{\mathcal {D}}} -\langle \partial _t \rho , u\phi \rangle _{H^{-1}, H^1_0} \\&= \langle p( \Delta p + \lambda \rho ) , \phi \rangle _{{\mathcal {D}}',{\mathcal {D}}} -\frac{\text {d}}{\text {d}t} \int _\Omega \rho u \phi \, \hbox {d}x\\&= - \frac{\text {d}}{\text {d}t} \int _\Omega \rho u \phi \, \hbox {d}x \qquad \text{ in }\ {\mathcal {D}}'({\mathbb {R}}_+), \end{aligned}$$

where we used the fact that $\langle p( \Delta p + \lambda \rho ) , \phi \rangle _{{\mathcal {D}}',{\mathcal {D}}}=0$ (this is the complementarity condition). Using (B.1), we deduce

$$\begin{aligned} \int _\Omega \nabla p \cdot \nabla (p-u) - \lambda \rho (p-u)\, \hbox {d}x = \frac{\text {d}}{\text {d}t} \int _\Omega \rho u \phi \, \hbox {d}x \qquad \text{ in }\ {\mathcal {D}}'({\mathbb {R}}_+). \end{aligned}$$

Using the fact that $\rho (x,t)p(x,t) = p(x,t)$, we deduce that

$$\begin{aligned}&\int _\Omega \nabla p \cdot \nabla (p-u)- \lambda (p-u)\, \hbox {d}x \, \hbox {d}t\\&\quad =\int _\Omega \nabla p \cdot \nabla (p-u) - \lambda \rho (p-u)\, \hbox {d}x \, \hbox {d}t +\int _\Omega \lambda (1-\rho ) u\, \hbox {d}x \, \hbox {d}t\\&\quad \leqq \frac{\text {d}}{\text {d}t} \int _\Omega \rho u \phi \, \hbox {d}x +\Lambda \int _\Omega (1-\rho ) u\, \hbox {d}x \, \hbox {d}t\\ \end{aligned}$$

Integrating with respect to $t\in [t_0,t_o+\delta ]$, we get

$$\begin{aligned}&\int _{t_0}^{t_0+\delta } \int _\Omega \nabla p \cdot \nabla (p-u) -\lambda (p-u)\, \hbox {d}x \, \hbox {d}t \\&\quad \leqq \int _\Omega (\rho (t_0+\delta ) -\rho (t_0)) u\phi \, \hbox {d}x+ \Lambda \int _{t_0}^{t_0+\delta } \int _\Omega (1-\rho ) u\, \hbox {d}x \, \hbox {d}t\\&\quad \leqq \int _\Omega (\rho (t_0+\delta )-1) u\phi \, \hbox {d}x +\Lambda \int _{t_0}^{t_0+\delta } \int _\Omega (1-\rho ) u\, \hbox {d}x \, \hbox {d}t\\&\quad \leqq \Lambda \int _{t_0}^{t_0+\delta } \int _\Omega (1-\rho ) u\, \hbox {d}x \, \hbox {d}t \end{aligned}$$

and the result now follows by proceeding as in the proof of Theorem 2.7. $\square$