Nonlinear Analysis Seminar ---- Fall 2020 (Organizers: Yanyan Li, Dennis Kriventsov)

  • The seminar is run remotely on Zoom mostly on Wednesdays (9:30am--10:30am).

    Registration Link   Meeting ID: 992 8122 5008

     

     

  • October 21, 2020, Wed, 9:30--10:30am

    Speaker: Yannick Sire, Johns Hopkins University

    Title: Blow-up solutions via parabolic gluing

    Abstract: We will present some recent results on the construction of blow-up solutions for critical parabolic problems of geometric flavor. Initiated in the recent years, the inner/outer parabolic gluing is a very versatile parabolic version of the well-known Lyapunov-Schmidt reduction in elliptic PDE theory. The method allows to prove rigorously some formal matching asymptotics (if any available) for several PDEs arising in porous media, geometric flows, etc….I will give an overview of the strategy and will present several applications to (variations of) the harmonic map flow, Yamabe flow and Yang-Mills flow. I will also present some open questions.

     

  • October 28, 2020, Wed, 9:30--10:30am

    Speaker: Zhifei Zhang, Peking University

    Title: Linear stability of pipe Poiseuille flow at high Reynolds number regime

    Abstract: The linear stability of pipe Poiseuille flow is a long standing problem since Reynolds experiment in 1883. Joint with Qi Chen and Dongyi Wei, we solve this problem at high Reynolds regime. We first introduce a new formulation for the linearized 3-D Navier-Stokes equations around this flow. Then we establish the resolvent estimates of this new system under favorable artificial boundary conditions. Finally, we solve the original system by constructing a boundary layer corrector.

     

  • November 4, 2020, Wed, 9:30--10:30am

    Speaker: Daniela De Silva, Barnard College

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  • November 11, 2020, Wed, 9:30--10:30am

    Speaker: Fanghua Lin, Courant Institute, NYU

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  • November 18, 2020, Wed, 9:30--10:30am

    Speaker: Alessio Figalli, ETH

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  • November 25, 2020, Wed, 9:30--10:30am

    Speaker: Shaoming Guo, University of Wisconsin

    Title: Fourier restrictions and decouplings for surfaces of co-dimension two

    Abstract: I will report some recent process on Fourier restriction estimates and decoupling inequalities for quadratic surfaces of co-dimension two.

     

  • December 2, 2020, Wed, 9:30--10:30am

    Speaker: Maria Esteban, Universite Paris-Dauphine

    Title: Flows and symmetry properties of positive solutions of nonlinear elliptic PDEs

    Abstract: In this talk will be presented various results concerning the symmetry and symmetry breaking properties of positive solutions of nonlinear elliptic PDEs. In some cases related to the study of Caffarelli-Kohn-Nirenberg inequalities the use of flows can provide optimal results concerning symmetry or symmetry breaking. In other cases, where magnetic Laplacians are concerned, the situation becomes more complicated, and one cannot always prove optimal results. Various examples will be discussed during the talk.

     

  • December 9, 2020, Wed, 9:30--10:30am

    Speaker: Angela Pistoia, Universita "La Sapienza"

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  • September 16, 2020, Wed, 9:30--10:30am

    Speaker: Luis Caffarelli, University of Texas at Austin

    Title: Propagation on a domain, driven by expansion on its boundary

    Abstract: Diverse models of propagation of a species that travels along a line and diffuses sideways were introduced among others by Berestycki, Coulon, Roquejoffre, Rossi I will describe work in collaboration with Roquejoffre and Tomasetti in this area of research.

     

  • September 23, 2020, Wed, 9:30--10:30am

    Speaker: Hao Jia, University of Minnesota

    Title: Long time dynamics of 2d Euler and nonlinear inviscid damping

    Abstract: In this talk, we will discuss some joint work with Alexandru Ionescu on the nonlinear inviscid damping near point vortex and monotone shear flows in a finite channel. We will put these results in the context of long time behavior of 2d Euler equations and indicate further important open problems in the field.

     

  • September 30, 2020, Wed, 9:30--10:30am

    Speaker: Yao Yao, Georgia Institute of Technology

    Title: Two results on the interaction energy

    Abstract: For any nonnegative density f and radially decreasing interaction potential W, the celebrated Riesz rearrangement inequality shows the interaction energy E[f] = \int f(x)f(y)W(x-y) dxdy satisfies E[f] \le E[f^*], where f^* is the radially decreasing rearrangement of f. It is a natural question to look for a quantitative version of this inequality: if its two sides almost agree, how close must f be to a translation of f^*? Previously the stability estimate was only known for characteristic functions. I will discuss a recent work with Xukai Yan, where we found a simple proof of stability estimates for general densities. I will also discuss another work with Matias Delgadino and Xukai Yan, where we constructed an interpolation curve between any two radially decreasing densities with the same mass, and show that the interaction energy is convex along this interpolation. As an application, this leads to uniqueness of steady states in aggregation-diffusion equations with any attractive interaction potential for diffusion power m\ge 2, where the threshold is sharp.

     

  • October 7, 2020, Wed, 9:30--10:30am

    Speaker: Alexandru Ionescu, Princeton University

    Title: On the global regularity for the Einstein-Klein-Gordon coupled system

    Abstract: In joint work with Benoit Pausader we consider the Einstein field equations of General Relativity for self-gravitating massive scalar fields (the Einstein-Klein-Gordon system). Our main results concern the global regularity, modified scattering, and precise asymptotic analysis of solutions of this system with initial data in a small neighborhood of the Minkowski space-time.

     

  • October 14, 2020, Wed, 9:30--10:30am

    Speaker: Weiyong He, University of Oregon

    Title: Harmonic and biharmonic almost complex structures

    Abstract: Harmonic almost complex structures was introduced by C. Wood in 1990s. We study the existence and regularity of weak harmonic almost complex structure from analytic point of view, as a tensor-valued version of harmonic maps. We also introduce the notion of biharmonic almost complex structures, in particular in dimension four. We prove that biharmonic almost complex structures are smooth, and there always exist an energy-minimizing biharmonic almost complex structure. Moreover, given a homotopy class, we also show existence results. We conjecture that the homotopy class of energy-minimizing biharmonic almost complex structures do not depend on a generic background metric.