Hele–Shaw Limit for a System of Two Reaction-(Cross-)Diffusion Equations for Living Tissues

Abstract

Multiphase mechanical models are now commonly used to describe living tissues including tumour growth. The specific model we study here consists of two equations of mixed parabolic and hyperbolic type which extend the standard compressible porous medium equation, including cross-reaction terms. We study the incompressible limit, when the pressure becomes stiff, which generates a free boundary problem. We establish the complementarity relation and also a phase-segregation result. Several major mathematical difficulties arise in the two species case. Firstly, the system structure makes comparison principles fail. Secondly, segregation and internal layers limit the regularity available on some quantities to BV. Thirdly, the Aronson–Bénilan estimates cannot be established in our context. We are led, as it is classical, to add correction terms. This procedure requires technical manipulations based on BV estimates only valid in one space dimension. Another novelty is to establish an $L^1$ version in place of the standard upper bound.

Appendix A. Energy

Proposition 5

Let $H_1(p) := \int _0^{p} F(z) \, \mathrm {d}z$ and $H_2(p) := \int _0^{p} G(z) \, \mathrm {d}z$ for $p \geqq 0$. Then, the energy

$$\begin{aligned} {\mathcal {E}}(t) := \int _\Omega \left( \frac{1}{2}\left| \frac{\partial p_{\gamma , \varepsilon }}{\partial x} \right| ^2 - c^{(1)}_{\gamma ,\varepsilon } H_1(p_{\gamma , \varepsilon }) - c^{(2)}_{\gamma ,\varepsilon } H_2(p_{\gamma , \varepsilon })\right) \mathrm {d}x \end{aligned}$$

is such that, for a constant C independent of $\gamma$ and $\varepsilon$,

$$\begin{aligned} {\mathcal {E}}'(t) + \gamma \int _{\Omega } p_{\gamma ,\varepsilon } w_{\gamma ,\varepsilon } ^2 \mathrm {d}x \leqq C. \end{aligned}$$

(44)

Proof

Consider the equation for the pressure (16) and multiply by $- \tfrac{\partial ^2 p_\gamma }{\partial x^2}$. Integration by parts yields

$$\begin{aligned} \frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\int _\Omega \left| \dfrac{\partial p_\gamma }{\partial x}\right| ^2 \mathrm {d}x +\gamma \int _\Omega p_\gamma \left| \dfrac{\partial ^2 p_\gamma }{\partial x^2}\right| ^2 \mathrm {d}x + \gamma \int _\Omega p_\gamma \dfrac{\partial ^2 p_\gamma }{\partial x^2}R \, \mathrm {d}x = 0. \end{aligned}$$

(45)

Moreover, using the equations for $c_\gamma ^{(1)}$ and $c_\gamma ^{(2)}$, we compute

$$\begin{aligned} \begin{aligned}&\frac{\partial \left( c_\gamma ^{(1)}H_1(p_\gamma ) \right) }{\partial t}\\&\quad = H_1(p_\gamma ) \left( \frac{\partial c_\gamma ^{(1)}}{\partial x}\dfrac{\partial p_\gamma }{\partial x}+ c_\gamma ^{(1)}F_1(p_\gamma ) + c_\gamma ^{(2)}G_1(p_\gamma ) \right. \\&\qquad \left. - (c_\gamma ^{(1)})^2F(p_\gamma ) - c_\gamma ^{(1)}\,c_\gamma ^{(2)}G(p_\gamma )\right) \\&\qquad + c_\gamma ^{(1)}F(p_\gamma ) \left[ \left| \dfrac{\partial p_\gamma }{\partial x}\right| ^2 + \gamma p_\gamma w_\gamma \right] , \end{aligned} \end{aligned}$$

(46)

and

$$\begin{aligned} \begin{aligned}&\frac{\partial \left( c_\gamma ^{(2)}H_2(p_\gamma ) \right) }{\partial t}\\&\quad = H_2(p_\gamma ) \left( \frac{\partial c_\gamma ^{(2)}}{\partial x}\dfrac{\partial p_\gamma }{\partial x}+ c_\gamma ^{(1)}F_2(p_\gamma ) + c_\gamma ^{(2)}G_2(p_\gamma )\right. \\&\left. \qquad - (c_\gamma ^{(2)})^2\,G(p_\gamma ) - c_\gamma ^{(1)}\, c_\gamma ^{(2)}F(p_\gamma )\right) \\&\qquad + c_\gamma ^{(2)}G(p_\gamma ) \left[ \left| \dfrac{\partial p_\gamma }{\partial x}\right| ^2 + \gamma p_\gamma w_\gamma \right] . \end{aligned} \end{aligned}$$

(47)

Summing (45), (46) and (47), and using the uniform bounds for $c_\gamma ^{(1)}$, $c_\gamma ^{(2)}$ and the reaction terms, we get

$$\begin{aligned}&\frac{\mathrm {d}}{\mathrm {d}t}\int _\Omega \bigg ( \frac{1}{2} \left| \dfrac{\partial p_\gamma }{\partial x}\right| ^2 - c_\gamma ^{(1)}H_1(p_\gamma ) - c_\gamma ^{(2)}H_2(p_\gamma ) \bigg ) \mathrm {d}x + \gamma \int _{\Omega } p_\gamma w_\gamma ^2 \mathrm {d}x \leqq \\&\quad C \int _{\Omega } \left[ \left| \dfrac{\partial p_\gamma }{\partial x}\right| ^2 + \left| \dfrac{\partial p_\gamma }{\partial x}\right| \left( \left| \frac{\partial c_\gamma ^{(1)}}{\partial x}\right| + \left| \frac{\partial c_\gamma ^{(2)}}{\partial x}\right| \right) \right] \mathrm {d}x. \end{aligned}$$

Theorem 3.1, together with the Hölder inequality and the Sobolev embeddings, yield the desired bound (44). $\quad \square$