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
时 间：2021.11.24（星期三）
地 点：腾讯会议：890 304 167
n 上午9：00-10：00
报告人：李敬宇 教授 (东北师范大学)
题 目：Asymptotic profiles of the singular Keller-Segel model
摘 要：In this talk we present some results on the characterization of asymptotic profiles, as the time tends to infinity, of solutions to the chemotaxis models with logarithmic sensitivity on the half space. We show that if the density of bacteria is imposed by inward flux boundary condition, then the solution converges to a traveling front that is determined by the flux strength; if the bacteria satisfy no-flux boundary condition and the nutrient satisfies non-homogeneous Dirichlet boundary condition, then the solution converges with algebraic rate to a stationary spike. These two results, respectively, describe the phenomena of invasion of tumor issue and the aggregation of bacteria. The difficulties of the problems are the nonlocal structure of the equation and singularities caused by the vacuum end state of the profiles. The proofs are based on Cole-Hopf transformation, anti-derivative method and weighted energy estimates. We also present the intrinsic relations between this chemotaxis model and the compressible Navier-Stokes equations with density dependent viscosity.
n 上午10：00-11：00
报告人：单敏捷 讲师 (中央民族大学)
题 目：Resonant Decompositions and Global Well-posedness for 2D Zakharov-Kuznetsov Equation in Sobolev spaces of Negative Indices
摘 要：The Cauchy problem for 2D Zakharov-Kuznetsov equation is shown to be global well-posed for the initial date in H^{s} provided s>-\frac{1}{13}. As conservation laws are invalid in Sobolev spaces below L^2, we construct an almost conserved quantity using multilinear correction term following the I-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao. In this paper, we use bilinear Strichartz estimate and the nonlinear Loomis-Whitney inequality to handle the resonant interactions.
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