复杂流体中几类非线性发展方程组的适定性与渐近性态

The project investigates the well-posedness and asymptotic behavior (also including the associated infinite dimensional dynamical systems) for the following important nonlinear evolution equations arising from the material sciences, fluid mechanics and biology: (A) hydrodynamic systems for liquid crystal flow of nematic as well as smectic type; (B) diffuse-interface model for the two-phase fluids with Marangoni effects; (C) phase-field-fluid system modeling the interactions between vesicle membrane and fluid; (D) chemotaxis-fluid system modeling cellular swimming in fluid drops, and so on. All these problems have important physical backgrounds such that they originate from very active research areas in the recent years. Besides, they are very useful in applications. These problems have attracted a lot of interests from many mathematicians so far. However, the theoretical study of the associated nonlinear evolution equations is still far from complete. Therefore, it is worth performing a deep detailed study on these problems. The aim of our project is to make significant advances both in the qualitative description of the physical phenomena and in the analytical study of the related mathematical models. We note that the complexities due to the highly nonlinear structures of the problems bring a lot of new challenges in mathematical analysis. In order to overcome the difficulties, we have to develop new techniques and methods in partial differential equations. Our research will make important contributions to the development of the theory of nonlinear evolution equations and it will also be helpful to related numerical analysis and simulations.

本项目拟研究源于材料科学、流体力学及生物学等自然学科的几类重要非线性发展方程组的适定性及其整体解的大时间渐近性态（含相关无穷维动力系统的研究）。具体问题包括：（A）关于向列相以及近晶相液晶的流体力学方程组；（B）关于伴有Marangoni效应的两相流的相场-流体方程组；（C）关于生物膜泡形变的相场-流体方程组；（D）生物趋化-流体力学方程组，等等。这些问题均源自于当前国际上十分活跃的研究领域，具有重要的物理背景和应用价值，近年来受到国内外同行的高度关注。然而，其相关理论分析还远未成熟，因此对这些问题展开深入细致的研究十分必要。为解决这些非线性发展方程组的复杂数学结构带来的挑战，我们需要发展新的数学方法、进行偏微分方程技巧的创新，这对完善非线性发展方程的理论和应用研究具有重要意义。

本项目对源于材料科学、流体力学及生物学等的几类重要非线性发展方程组的适定性及其整体解的大时间渐近性态开展研究。具体问题包括：液晶流体动力学方程组（Ericsken-Leslie 方程组、Landau–De Gennes Q-张量模型），两相流方程组（伴有 Marangoni 效应的两相流、Cahn-Hilliard-Darcy 方程组，Cahn-Hilliard-Stokes-Darcy 方程组），关于生物膜泡形变的相场-流体方程组，生物趋化-流体力学方程组，修正晶体相场模型，等等。为解决这些非线性发展方程组的复杂数学结构带来的挑战，我们发展了新的数学方法、进行偏微分方程技巧的创新，这对完善非线性发展方程的理论和应用研究具有重要意义。在本项目资助下，我们在高水平数学期刊上发表了学术论文12篇，包括 ARMA, SIMA, JDE, M3AS, EJAM 等。

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