We derive, through the deterministic homogenization theory in thin domains, a new model consisting of Hele–Shaw equation with memory coupled with the convective Cahn–Hilliard equation. The obtained system, which models in particular tumor growth, is then analyzed and we prove its well-posedness in dimension 2. To achieve our goal, we develop and use the new concept of sigma-convergence in thin heterogeneous media, and we prove some regularity results for the upscaled model.

中文翻译：

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薄异质层中流动的非局域 Hele-Shaw-Cahn-Hilliard 系统的推导和分析

通过薄域中的确定性均质化理论，我们推导了一个由带记忆的 Hele-Shaw 方程与对流 Cahn-Hilliard 方程组成的新模型。然后对所获得的系统进行分析，该系统模拟了特定的肿瘤生长，并证明了其在 2 维上的适定性。为了实现我们的目标，我们开发并使用了薄异质介质中西格玛收敛的新概念，并证明了一些放大模型的规律性结果。