报告题目:Semi-similar Prandtl Flows near a Stagnation Point: Dynamical Selection and Nonlinear Stability
报告人：陶涛(山东大学)
报告时间: 2026年10月29日08：40-09：40
报告地点： 伍卓群楼第一报告厅
校内联系方式：郭斌bguo@jlu.edu.cn
报告摘要：This talk studies time-dependent semi-similar Prandtl flows with outer flow $U(x)=x^m$, where $m\geq1$. The normal coordinate is scaled by the local boundary-layer thickness, and time by the associated diffusion scale. Every time-monotone solution in the Oleinik class extends globally in time. The Falkner--Skan boundary layer globally attracts every such unsteady semi-similar profile. The convergence rate is exponential for $m=1$ and algebraic for $m>1$. We also prove nonlinear stability of each admissible time-dependent semi-similar trajectory under small compatible perturbations that break the semi-similar ansatz. Together, these results rigorously justify the time-dependent semi-similar reduction and establish its solutions as a dynamically meaningful class of Prandtl flows.
For semi-similar flows, the Crocco transformation reduces the problem to a one-dimensional degenerate parabolic equation. Combined with weighted barriers, comparison principles propagate time monotonicity and lead to global continuation. Zero-order convergence and a one-sided gradient estimate then produce uniform strict concavity after a transient period; this concavity supplies the damping needed for derivative decay. The same damping mechanism extends to the full degenerate hyperbolic--parabolic perturbation equation, allowing us to close the nonlinear stability estimates..
报告人简介：陶涛,山东大学教授，主要从事不可压缩流体方程组的研究，特别是Convex Integration在不可压缩流体方程组中的应用(构造奇异耗散弱解)以及不可压缩Navier-Stokes方程的粘性消失极限,研究成果发表在Comm. Math.Phys.，Advances in Mathematics, J.Math.Pures Appl.，J.Funct.Anal.，SIAM J. Math. Anal.，Calc.Var.Partial Differential Equations等国际期刊上。
