由Cahn-Hilliard模型与Hele-Shaw模型和Brinkman模型耦合而成的CHHS模型和CHB模型，已广泛应用于Hele-Shaw细胞中二元流体旋节分解和二元流体的变相分离运动，能精准刻画实体肿瘤的生长过程，是偏微分方程领域的又一研究热点。.本课题主要研究可压CHHS模型强解的适定性、正则性、长时间行为和CHB模型在高维空间中解的全局适定性。1、利用能量估计法和插值不等式研究可压CHHS模型二维全局解，三维局部解的适定性，利用高阶先验估计和特征函数法研究解的正则性；2、针对化学势函数为一般凸函数时研究可压CHHS模型强解的适定性、正则性和长时间行为；3、利用能量临界性、内积法、Lojasiwica-Simon不等式研究CHB模型解的全局适定性，全局吸引子的存在性及与CHHS模型解之间的关系。.本课题的完成将进一步丰富上述两类模型在医学、生物学的应用。

The CHHS model and CHB model are the type of partial differential equations coupled by the Cahn-Hilliard model, the Hele-Shaw model, and the Brinkman model. They have been widely applied to the binary fluid spinodal decomposition and the separation of binary fluid in Hele-Shaw cells, with particular application in the study of morphogenesis of solid tumor growth. The study of morphogenesis of tumor growth is one of research hotspots in the field of partial differential equations.. This project will study the well-posedness, regularity and large-time behavior of a strong solution for a class of compressible CHHS models and the global well-posedness of CHB models related to tumor growth. Firstly, the energy estimation method and a series of inequality can be used to study CHHS models. When the initial conditions have finite energy, the existence of two-dimensional global solutions and the existence and uniqueness of three-dimensional local solutions are studied. Then Combining high-order prior estimation with eigenfunction method, the com pressible CHHS model can be used to solve high-order spatial regularity and Gevrey spatial regularity. Secondly, we will study the well-posed ness, regularity and large-time behavior of the strong solution of the compressible CHHS model for the general convex function in the chemical potential F. Thirdly, By using of the energy criticality，interior integral method and Lojasiewica-Simon inequality, the global well-posedness of the solution and the existence of a global attractor for CHB models are researched, then the relation of solution between the CHHS model and CHB model is also analyzied.. The completion of this topic will address some qualitative theoretical studies of the solution of CHHS and CHB models, and provide theoretical basis and numerical analysis for promoting the application of CHHS and CHB models in the field of tumor research.