主讲人：王晓明 东方理工大学教授
时间：2025年10月9日15：00
地点：徐汇校区三号楼332室
举办单位：数理学院
主讲人介绍：王晓明教授本科及硕士阶段就读于复旦大学数学系，后赴美在印第安纳大学伯明顿分校获得应用数学博士学位，并在纽约大学库朗数学科学研究所完成博士后研究。2024年加入东方理工大学，担任数学学科创校讲席教授。在此之前，他曾在多所高校担任终身教职，包括密苏里科技大学首任 Havener 讲席系主任、南方科技大学讲席教授、复旦大学特聘教授、佛罗里达州立大学教授。
王晓明教授长期致力于现代应用数学前沿研究，主要方向包括流体动力学、地下水流动、地球物理流体力学、湍流与气候变化等。他善于融合偏微分方程、动力系统、随机分析、数值方法、科学计算及机器学习等多种数学工具，致力于在严谨数学理论与复杂物理系统之间架起桥梁，推动理论突破与交叉创新。他已在《Communications on Pure and Applied Mathematics》（CPAM）等国际一流期刊发表学术论文一百余篇，并由剑桥大学出版社出版学术专著一本。
内容介绍：Convection in porous media plays a central role in geophysical fluid dynamics, geothermal energy, carbon sequestration, and other climate-related processes. Layered porous structures often arise naturally or through design, leading to systems with abrupt material transitions. In such cases, the Darcy–Boussinesq equations give rise to a nonlinear transmission problem, raising a fundamental question: what interfacial conditions are appropriate?
In this talk, I address this issue by viewing the sharp interface model as the limit of a more physically realistic diffuse-interface formulation, where properties vary smoothly across layers. Assuming constant porosity, we prove that as the transition-layer thickness vanishes, solutions of the diffuse model converge to those of the sharp interface system over finite time intervals for suitable data. The analysis highlights velocity boundary layer formation and requires delicate elliptic and parabolic estimates with nearly discontinuous coefficients. Beyond finite time, we show that both sharp and diffuse models admit global attractors, and these attractors converge as the transition layers shrink.
This work provides a rigorous foundation for the sharp interface approximation, linking it to more realistic diffuse-interface models. I will also discuss implications for long-time dynamics and outline numerical methods adapted to layered porous structures.
This is joint work with Hongjie Dong (Brown University) and Kaijian Sha (EIT).