广州数学大讲坛第四期

第三十四讲——中山大学罗巍副教授学术报告

题目：Global well-posedness, blow-up phenomenon and sharp ill-posed for the nonlinear shallow water equations

时间：2026年6月7日（星期日）下午15:30—17:00

地点：理学实验楼314

报告人：罗巍 副教授

摘要：This paper investigates the long-time behavior, blow-up phenomena, and sharp ill-posedness for the nonlinear shallow water (NSW) equations. The global existence and well-posedness for the NSW equations in critical spaces have remained open. In this work, we address these issues by establishing, in the case of sufficiently large water depth \( H \), a sharp dichotomy between blow-up and global well-posedness for initial data in a new function space \( E^2(\mathbb{R}) \) associated with the NSW equations. Despite the one-dimensional Euler-type structure of the system, we prove the long-time stability of solutions near the equilibrium. Furthermore, we uncover the sharp ill-posedness of the NSW equations: the Cauchy problem is well-posed in \( E^s(\mathbb{R}) \) for \( s > 3/2 \), while norm inflation occurs in the critical space \( E^s(\mathbb{R}) \) with \( s = 3/2 \).

报告人简介：

罗巍，中山大学杏吧视频 副教授、博士生导师，主要从事非线性偏微分方程与泛函分析研究，聚焦复杂流体（如FENE哑铃模型、聚合物流体）和浅水波系统（如Camassa-Holm方程）的适定性、爆破机制、大时间行为及Liouville型定理。‌‌目前主持国自然面上项目一项，青年基金一项，广东省面上项目一项。目前发表SCI论文20多篇，相关成果发表在ARMA，Adv.Math，ASNS-Pisa, JDE等期刊上。