中科院数学与系统科学研究院

数学研究所

偏微分方程研讨班

报告人： Prof. Helena J Nussenzveig Lopes （Universidade Federal do Rio de Janeiro）

题  目：2D Navier-Stokes equations on a bounded domain with holes and Navier friction boundary conditions

时  间：2021.12.02（星期四），21:00-22:00

地  点：Zoom 会议号：910 8088 7251 密码：459913

摘  要：We will discuss the large time behavior of solutions of 2D Navier-Stokes in bounded domains which are not necessarily simply connected, when we impose Navier friction boundary conditions. We establish exponential time decay, for both velocity and vorticity, under various assumptions on the friction coefficient relative to curvature of the boundary, for different types of domains. We also discuss the special role, played by the disk and the annulus, in this analysis. This is joint work with Christophe Lacave, Milton Lopes Filho and Jim Kelliher.

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报告人： Prof. Zhenfu Wang （Beijing International Center for Mathematical Research, Peking University）

题  目：Gaussian fluctuations for interacting  particle systems with singular kernels

时  间：2021.12.03（星期五），10:00-11:00

地  点：数学院南楼N226

摘  要：We consider  the asymptotic behaviour of  the fluctuations for the empirical measures of interacting particle systems with singular kernels. We prove that the sequence of fluctuation processes  converges in distribution to a generalized Ornstein-Uhlenbeck process.  Our result considerably extends classical results  to singular kernels, including the Biot-Savart law.  The result applies to the point vortex model approximating  the 2D incompressible Navier–Stokes equation and the 2D Euler equation.  We also obtain Gaussianity and optimal regularity of the limiting Ornstein-Uhlenbeck process. The method relies on  the martingale approach  and  the  Donsker-Varadhan variational formula, which transfers the uniform estimate to some exponential integrals. Estimation of those exponential integrals follows by  cancellations and combinatorics techniques and is of the type of large deviation principle. Given time, we will also breifly mention our recent result on Large Deviation Principles for 2D Navier-Stokes. This is based on joint work with Xianliang Zhao (AMSS) and Rongchan Zhu (BIT).

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报告人： Prof. Jiajun Tong （Beijing International Center for Mathematical Research, Peking University）

题  目：On the Free Boundary Evolution in a Tumor Growth Model with Nutrient

时  间：2021.12.10（星期五），10:00-11:00

地  点：数学院南楼N226

摘  要：We study a free boundary problem that naturally arises in a 2-D tumor growth model with nutrient. In this model, the tumor cells proliferate by consuming the nutrient locally, and meanwhile they migrate into ambient empty space because of an incompressibility constraint on the cell density. We shall focus on evolution of the tumor boundary, which couples strongly with the dynamics of nutrient in the bulk. We will show, under suitable assumptions, global well-posedness of evolution of the tumor boundary, provided that it is initially close to a circle. We will also establish its long-time convergence to a circular shape as well as a (sharp) decay estimate. Discussions will be made on several related problems.