Date/Time : 23 Oct 2024 15:00
Venue : S17 #05-11 (SR5)
Speaker : Zhou Tao, National University of Singapore

While finite-time blowup solutions have been studied in depth for the Keller-Segel equation, a fundamental model describing chemotaxis, the existence of finite-time blowup solutions to chemotaxis-fluid models remains largely unexplored. To fill this gap in the literature, we use a quantitative method to directly construct a smooth finite-time blowup solution for the Keller-Segel-Navier-Stokes system with buoyancy in 3D. The heart of the proof is to establish the non-radial finite-codimensional stability of an explicit self-similar blowup solution to 3D Keller-Segel equation with the abstract semigroup tool from [Merle-Raphaël-Rodnianski-Szeftel, 2022], which partially generalizes the radial stability result [Glogić-Schörkhuber, 2024] to the non-radial setting. Additionally, we introduce a robust localization argument to find blowup solutions with non-negative density and finite mass. This is talk is based on the joint work with Zexing Li.

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Date/Time : 16 Oct 2024 15:00
Venue : S17 #05-11 (SR5)
Speaker : Raphael Winter, Cardiff University

We present new results on the kinetic Balescu-Lenard and Landau equations, which describe the collisional relaxation of plasma. The first result establishes global existence of solutions to the Balescu-Lenard equation for initial data close to equilibrium, while the second result shows a new blow-down mechanism for the Landau equation for a family of super-critical initial data. We also discuss the breakthrough result of Guillen and Silvestre who identify the Fisher information as a Lyapunov functional of the spatially homogeneous Landau equation. Joint work with Mitia Duerinckx and Maria Gualdani.

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Date/Time : 16 Oct 2024 14:00
Venue : S17 #05-11 (SR5)
Speaker : Havva Yoldas, Delft University of Technology

After a short introduction on kinetic equations and classical L^2/H^1 hypocoercivity techniques due to Dolbeault, Mouhot, Schmeiser AMS 2015, I will talk about Harris-type theorems that is an alternative method for obtaining quantitative convergence rates. I will discuss some applications of these theorems summarising some recent results obtained jointly with Jo Evans (Warwick) on the run and tumble equations that is a kinetic-transport equation modelling the bacterial movement under the effect of a chemoattractant.

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Date/Time : 11 Sep 2024 15:00
Venue : S17 #05-11 (SR5)
Speaker : Yuxuan Wang, Tsinghua University

First examined by Riemann in 1860 in one-dimensional space, the Riemann problem has since been extensively studied and advanced by numerous scholars. Our talk will investigate the interaction of two approaching shocks in the context of spherical symmetry in three-dimensional space, presuming the fluid to be ideally barotropic. We will outline the proof of the local existence, uniqueness, and higher regularity of the solution to the shock interaction problem.

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Date/Time : 04 Sep 2024 15:00
Venue : S17 #05-11 (SR5)
Speaker : Meiyue Jiang, Peking University

In this talk we will first present an elementary treatment of the Brunn-Minkowski inequalities in dimension 2, which include the classical Brunn-Minkowski inequality, the $L^p$ and the log Brunn-Minkowski inequalities due to Lutwak and his collaborators, then we discuss a Brunn-Minkowski type inequality for the functional associated with the dual $L^p$−Minkowski problem. If time permits, a similar inequality for higher dimensional PDEs will be proposed.

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Date/Time : 23 Aug 2024 16:00
Venue : S17 #04-05 (SR2)
Speaker : Yixuan Wang, Caltech

We provide a general framework that could be potentially amenable to the stability of a large class of problems with type I singularities in a weighted Sobolev space and use nonlinear heat and complex Ginzburg-Landau equations as examples, which are joint works with Tom Hou, Van Tien Nguyen, and Jiajie Chen. A generalized dynamic rescaling formulation is introduced, with modulation parameters capturing the spatial translation and rotation symmetries of the equation and novel additional modulation parameters perturbing the scaling symmetry. This new formulation provides enough degrees of freedom to impose vanishing conditions on the rescaled solution, completely eliminating the unstable and neutrally stable modes of the linearized operator around the blowup profile. We then establish the full stability of the blowup using weighted energy estimates, without relying on a topological argument or a spectrum analysis. The log correction for the blowup rate is automatically inferred by refined estimates of the modulation parameters.

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Date/Time : 21 Aug 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Van Tien Nguyen, National Taiwan University

Advanced mathematical techniques have been developed in the last two decades for precise construction of blowup solutions within the class of finite energy. However, the question of a complete classification of blowup dynamics remains open. In this talk, we consider the energy-critical nonlinear heat equation in $\mathbb{R}^3$  as a model example to illustrate our technique to archive such a blowup classification. It consists of two parts: 1) a review of different techniques for constructing finite-energy blowup solutions, and 2) a rigidity result of the finite-energy blowup.  

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Date/Time : 14 Aug 2024 10:00
Venue : S17 #06-11 (SR6)
Speaker : Jinxin Xue, Tsinghua University

The mean curvature flow is to evolve a hypersurface in Euclidean space using the mean curvatures at each point as the velocity field. The flow has good smoothing property, but also develops singularities. The singularities are modeled on an object called shrinkers, which give homothetic solutions to the flows. As there are infinitely many shrinkers that seem impossible to classify, it is natural to explore the idea of generic mean curvature flows that is to introduce a generic perturbation of the initial conditions. In this talk, we shall explain our work on this topic, including perturbing away nonspherical and noncylindrical shrinkers, and generic isolatedness of cylindrical singularities. The talk is based on a series of works jointly with Ao Sun.

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Date/Time : 01 Aug 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Armin Schikorra, University of Pittsburgh

n-Laplace systems with antisymmetric potential are known to govern geometric equations such as n-harmonic maps between manifolds and generalized prescribed H-surface equations. Due to the nonlinearity of the leading order n-Laplace and the criticality of the equation they are very difficult to treat.

I will discuss some progress we obtained, combining stability methods by Iwaniec and nonlinear potential theory for vectorial equations by Kuusi-Mingione.

Joint work with Dorian Martino

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