Date/Time : 27 Dec 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Jingbo Wan, Columbia University

In this talk, we discuss the rigid behaviour of contracting maps between positively curved manifolds via the method of geometric heat flows. We establish rigidity results for distance and area non-increasing maps between spheres, complex projective spaces and other positively curved manifolds under various curvature conditions. The methods rely on energy monotonicity for harmonic map heat flow, and the Strong Maximum Principle for Mean Curvature Flow or the coupled Mean Curvature Flow-Ricci Flow system. These results demonstrate and extend an interesting application of geometric heat flows to the homotopy classes of maps, pioneered by Prof. Mao-Pei Tsui and Prof. Mu-Tao Wang in 2004. This is based on joint works with Man-Chun Lee and Luen-Fai Tam.

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Date/Time : 11 Dec 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Jiaqi Liu, University of Southern California

The Guderley problem describes the behavior of a strong shock wave propagating radially in an ideal gas. A spherical shock converges radially inwards to the spatial origin, strengthening as it collapses. At the collapse point, the shock’s strength becomes infinite, leading to the formation of a new outgoing shock wave of finite strength, which then propagates outwards to infinity. In this talk, I will present recent work on the rigorous construction of the Guderley solution for $\gamma \in (1,3]$. These solutions are analytic away from the shock interfaces and the blow-up point. The proof relies on continuity arguments, nonlinear invariances, and barrier functions.

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Date/Time : 21 Nov 2024 15:00
Venue : S17 #04-04 (SR3)
Speaker : Gunther Uhlmann, University of Washington

We will consider the question on whether we can determine the structure of space time by making measurements near the worldline of an observer. We will consider both active and passive measurements. For the case of passive measurements one measures the fronts of light sources near the observer. For the case of active measurements we couple Einstein equations with matter or electromagnetic fields and formulate the question of determining the structure of space time as the problem of recovering the metric from observations of waves near the observer, The method applies to several other inverse problems for nonlinear equations, for example, the Westervelt equation arising in nonlinear acoustics.

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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 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 : 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 : 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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