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08-12微分方程系列报告【Charles Collot】
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Title:Recent advances in the construction of singular solutions to evolution partial differential equations 
Speaker:Charles Collot  (Courant Institute, New York University)
Time:2019年8月12日           下午  14:30-15:30
Room:东区管理科研楼  数学科学学院1418室

Abstract:Singularity formation in partial differential equations of evolution has been an important research field over the last forty years. This talk will focus on semi-linear equations for scalar fields, or equations from fluid dynamics. For semi-linear equations, singularity formation in the radial case is now well understood in several cases. It already displays a rich variety of phenomena: true or degenerate self-similarity, stable blowup regimes coexisting with instable ones,  depending on the so-called criticality of the equation. I will discuss recent progresses made on understanding nonradial dynamics and in particular anisotropic singularities: focusing on a work of Merle, Rapha\"el and myself, and on a work of del Pino, Musso and Wei. In fluid mechanics the situation remains poorly understood. However, there has been some very interesting recent work regarding the incompressible Euler equations: existence of $C^{1+\alpha}$ singular solutions by Elgindi, and numerical results supporting blowup in a domain by Hou and Luo. One  dimensional toy models also have been investigated intensively. I will discuss similarities and differences between the various approaches, mentioning which open problems still remain.


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