中科院数学与系统科学研究院
数学研究所
偏微分方程研讨会
报告人：Matthew Rosenzweig ( MIT)
题 目：Global solutions of aggregation equations and other flows with random diffusion
时 间：2021.10.28（星期四），21:00-21:50
地 点：Zoom Meeting ID: 982 4274 5864 Passcode: 1m3Lkm
摘 要：Aggregation equations, such as the parabolic-elliptic Patlak-Keller-Segel model, are known to have an optimal threshold for global existence vs. finite-time blow-up. In particular, if the diffusion is absent, then all smooth solutions with finite second moment can exist only locally in time. Nevertheless, one can ask whether global existence can be restored by adding a suitable noise to the equation, so that the dynamics are now stochastic. In this talk, we investigate whether random diffusion can restore global existence for a large class of active scalar equations in arbitrary dimension with possibly singular velocity fields. This class includes Hamiltonian flows, such as the SQG equation and its generalizations, and gradient flows, such as those arising in aggregation models. For this class, we show global existence of solutions in Gevrey-type Fourier-Lebesgue spaces with quantifiable high probability.
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报告人：Jeremy Louis Marzuola ( University of North Carolina at Chapel Hill)
题 目：On 4th order nonlinear thin-film like PDEs describing crystal surface evolution
时 间：2021.11.04（星期四），21:00-21:50
地 点：Zoom Meeting ID: 970 8664 6295 Passcode: M35hJ9
摘 要：We discuss recent results with a number of collaborators on PDEs relating to the relaxation of a crystal surface. After a brief overview of the motivating microscopic process that leads to the models, we will present results on the well-posedness of these models in various settings. Towards the end, we will show numerical evidence to motivate a number of open questions about this family of models.
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