中科院数学与系统科学研究院
数学研究所
学术报告
偏微分方程研讨班
报告人：Charles Collot (CY Cergy Paris Université)
题 目：Description of singularity formation for Burgers, Prandtl and Keller-Segel equations：Singularity formation for the Burgers equation (1)
时 间：2023.07.10（星期一）10:00-11:00
地 点：思源楼S813
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题 目：Description of singularity formation for Burgers, Prandtl and Keller-Segel equations：Singularity formation for the inviscid Prandtl equations (2)
时 间：2023.07.11（星期二）10:00-11:00
地 点：思源楼S813
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题 目：Description of singularity formation for Burgers, Prandtl and Keller-Segel equations：Preliminaries on the parabolic-elliptic Keller-Segel system (3)
时 间：2023.07.12（星期三）10:00-11:00
地 点：思源楼S817
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题 目：Description of singularity formation for Burgers, Prandtl and Keller-Segel equations：Stable self-similar singularity for the supercritical Keller-Segel system(4)
时 间：2023.07.13（星期四）10:00-11:00
地 点：思源楼S813
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题 目：Description of singularity formation for Burgers, Prandtl and Keller-Segel equations： Self-similarity of the second kind for the critical Keller-Segel system (5)
时 间：2023.07.14（星期五）10:00-11:00
地 点：思源楼S813
摘 要：These lectures will concern the qualitative study of some evolution partial differential equations modelling advection, diffusion and reaction. The Burgers and Prandtl equations describe fluid flows, and the Keller-Segel system bacteria motion. We will focus on singularity formation, and will describe precisely mathematically the associated concentration patterns. A common mathematical phenomenon will be observed: the solution is to leading order self-similar. We will first describe how self-similarity occurs for shocks in transport equations, with an approach that differs from the usual geometric one of John, Alinhac, Christodoulou and others. This will be achieved via the use of representation formulas given by characteristics based on joint works from the author with Ghoul, Ibrahim, Lin and Masmoudi. The stability of self-similar singularity formation will then be addressed for the Keller-Segel system. This will rely on a spectral approach, and will be based on recent works of Glogic and Schorkhuber, and from the author with Ghoul, Masmoudi, Merle, Nguyen and Raphael.
Detailed content:
1. Singularity formation for the Burgers equation.
We will first recall the method of characteristics. Then, we will present backward self-similar solutions and will classify them. Finally, we shall show how any singular solution starting from an analytic initial data resolves into a sum of such self-similar solutions near the singular points.
2. Singularity formation for the inviscid Prandtl equations.
We will first present the problem of the inviscid limit of the Navier-Stokes equations. Then, we will present the characteristics for this incompressible flow. We will study the backward self-similar equation using a modifier Crocco transform. Finally, we shall show that generically, singular solutions are described by the Van Dommelen and Shen self-similar profile.
3. Preliminaries on the parabolic-elliptic Keller-Segel system.
We will recall the Cauchy theory in Lebesgue spaces for this system, adapting the works of Weissler, and Brezis and Cazenave. Then, parabolic regularizing effects will be studied. Finally, we shall show that in the radially symmetric case, the partial mass transformation reduces the equations to local ones.
4. Stable self-similar singularity for the supercritical Keller-Segel system.
An explicit backward self-similar solution exists in dimensions three and higher ; it was found by Constantin et al. Schorkhuber and Glocic obtained its stability by Schwartz class perturbations. We will present some tools for studying such stability problem: renormalization techniques and spectral analysis.
5. Self-similarity of the second kind for the critical Keller-Segel system.
In two dimensions, singularity formation involves self-similarity of the second kind (also referred to as type II). An example was constructed in the seminal work by Herrero and Velazquez, and further studied by Raphael-Schweyer. We will explain how spectral analysis can also address the study of its stability, but via the understanding of a singularly perturbed operator.
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