Abstract Models of tissue growth are now well established, in particular, in relation to their applications to cancer. They describe the dynamics of cells subject to motion resulting from a pressure gradient generated by the death and birth of cells, itself controlled primarily by pressure through contact inhibition. In the compressible regime, pressure results from the cell densities and when two different populations of cells are considered, a specific difficulty arises; the equation for each cell density carries a hyperbolic character, and the equation for the total cell density has a degenerate parabolic property. For that reason, few a priori estimates are available and discontinuities may occur. Therefore the existence of solutions is a difficult problem. Here, we establish the existence of weak solutions to the model with two cell populations which react similarly to the pressure in terms of their motion but undergo different growth/death rates. In opposition to the method used in the recent paper of Carrillo et al., our strategy is to ignore compactness of the cell densities and to prove strong compactness of the pressure gradient. For that, we propose a new version of Aronson–Bénilan estimate, working in L2 rather than We improve known results in three directions; we obtain new estimates, we treat dimensions higher than 1 and we deal with singularities resulting from vacuum.

中文翻译：

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组织生长的两种双曲线-抛物线模型

摘要 组织生长的模型现在已经很好地建立起来，特别是与它们在癌症中的应用有关。他们描述了受细胞死亡和出生产生的压力梯度引起的运动的细胞动力学，其本身主要由压力通过接触抑制控制。在可压缩状态下，压力来自细胞密度，当考虑两种不同的细胞群体时，就会出现一个特定的困难；每个单元密度的方程具有双曲线特征，总单元密度的方程具有退化抛物线特性。出于这个原因，很少有先验估计可用，并且可能会出现不连续性。因此，解的存在性是一个难题。这里，我们建立了具有两个细胞群的模型的弱解的存在，这两个细胞群在它们的运动方面对压力的反应相似，但经历了不同的生长/死亡率。与 Carrillo 等人最近的论文中使用的方法相反，我们的策略是忽略单元密度的紧凑性并证明压力梯度的强紧凑性。为此，我们提出了一个新版本的 Aronson-Bénilan 估计，在 L2 中工作而不是我们在三个方向上改进已知结果；我们获得新的估计，我们处理高于 1 的维度，我们处理由真空引起的奇点。我们的策略是忽略单元密度的紧凑性并证明压力梯度的紧密性。为此，我们提出了一个新版本的 Aronson-Bénilan 估计，在 L2 中工作而不是我们在三个方向上改进已知结果；我们获得新的估计，我们处理高于 1 的维度，我们处理由真空引起的奇点。我们的策略是忽略单元密度的紧凑性并证明压力梯度的紧密性。为此，我们提出了一个新版本的 Aronson-Bénilan 估计，在 L2 中工作而不是我们在三个方向上改进已知结果；我们获得新的估计，我们处理高于 1 的维度，我们处理由真空引起的奇点。