标题：The Cahn-Hilliard Equation with Dynamic Boundary Conditions: Model Derivation and Mathematical Analysis

地点：数学与计算机科学学院4号楼229室

时间：2018年1月9日上午9：30-10：30

摘要：The Cahn-Hilliard equation is a fundamental model that describes phase separation processes of binary mixtures. In order to account for possible short-range interactions of the material with the solid wall, various dynamic boundary conditions have been proposed in the literature. In this talk, we introduce a new class of dynamic boundary conditions for the Cahn-Hilliard equation in a rather general setting. The derivation is based on an energetic variational approach that combines the least action principle and Onsager's principle of maximum energy dissipation. Then under suitable assumptions, we prove the existence and uniqueness of global weak/strong solutions to the initial boundary problem with or without surface diffusion. Furthermore, we establish the uniqueness of asymptotic limit as time goes to infinity and characterize the stability of local energy minimizers for the system.

题目：Analysis of the Cahn-Hilliard-Hele-Shaw System with Singular Potential

地点：数学与计算机科学学院4号楼229室

时间：2018年1月9日下午14：30-15：30

摘要：In this talk, we discuss the Cahn-Hilliard-Hele-Shaw system with a physically relevant singular potential (i.e., of logarithmic type) that models an incompressible binary fluid confined in a Hele-Shaw cell. We first prove the existence of global weak solutions with finite energy. Then in dimension two, we show the uniqueness weak solutions. Besides, we prove that any weak solution satisfies the strict separation property, which further leads to the regularity of weak solutions. When the spatial dimension is three, we prove the existence of a unique global strong solution, provided that the initial datum is regular enough and sufficiently close to any local minimizer of the free energy. This also yields the local Lyapunov stability of the local minimizer itself.

吴昊，复旦大学数学科学学院教授，2003年毕业于复旦大学获理学学士学位，2007年毕业于复旦大学获理学博士学位。研究领域为非线性发展方程以及复杂流体的数学理论。2015年获中国工业与应用数学学会优秀青年学者奖，2016年入选上海市青年拔尖人才。目前已在《Arch. Rational Mech. Anal.》，《SIAM J. Math. Anal.》，《Ann. Inst. H. Poincare Anal. Non Lineaire》，《Calc. Var. Partial Differential Equations》，《J. Math. Fluid Mech.》，《J. Differential Equations》，《Math. Models Methods Appl. Sci.》等高水平杂志发表论文44多篇。