UC San Diego Analysis Seminar


Time Location Organizers
Tuesdays at 11am Zoom (contact organizers for details) Ioan Bejenaru and Mohandas Pillai

Fall 2022

Date Speaker Title + Abstract
October 18

Sebastian Herr

Bielefeld University

Global wellposedness of the Zakharov System below the ground state

The Zakharov system is a quadratically coupled system of a Schrödinger and a wave equation, which is related to the focussing cubic Schrödinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that it is globally well-posed in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schrödinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate. This is joint work with Timothy Candy and Kenji Nakanishi.

November 4

4 PM

APM 6402

Note the unusual day, time, location

Dallas Albritton


Title and Abstract
November 15

Kihyun Kim


Rigidity of long-term dynamics for the self-dual Chern-Simons-Schrödinger equation within equivariance

We consider the long time dynamics for the self-dual Chern-Simons-Schrödinger equation (CSS) within equivariant symmetry. Being a gauged 2D cubic nonlinear Schrödinger equation (NLS), (CSS) is L2-critical and has pseudoconformal invariance and solitons. However, there are two distinguished features of (CSS), the self-duality and non-locality, which make the long time dynamics of (CSS) surprisingly rigid. For instance, (i) any finite energy solutions to (CSS) decomposes into at most one(!) modulated soliton and a radiation. Moreover, (ii) in the high equivariance case (i.e., the equivariance index ≥ 1), any smooth finite-time blow-up solutions even have a universal blow-up speed, namely, the pseudoconformal one. We explore this rigid dynamics using modulation analysis, combined with the self-duality and non-locality of the problem.

November 17

3-4 PM

APM 7218

Note the unusual day, time, location

Jan Slovak

Masaryk U

Nearly invariant calculus for a few CR (and all parabolic) geometries


November 29

Manuel Del Pino

University of Bath


A classical problem that traces back to Helmholtz and Kirchhoff is the understanding of the dynamics of solutions to the Euler equations of an inviscid incompressible fluid when the vorticity of the solution is initially concentrated near isolated points in 2d or vortex lines in 3d. We discuss some recent results on these solutions' existence and asymptotic behavior. We describe, with precise asymptotics, interacting vortices, and traveling helices, and extension of these results for the 2d generalized SQG. This is research in collaboration with J. Dávila, A. Fernández, M. Musso, and J. Wei.

December 6

Benoit Pausader

Brown University

Stability of a point charge for the repulsive Vlasov-Poisson system

We consider solutions of the repulsive Vlasov-Poisson system which are a combination of a point charge and a small density with respect to Liouville measure (a ``cloud''), and we show that these solutions exist globally, that the electric field decays at an optimal rate and that the particle distribution converges along a modified scattering dynamics. This follows by a Lagrangian study of the linearized equation, which is integrated by means of an asymptotic action-angle coordinate transformation, and an Eulerian study of the nonlinear dynamic which exhibits the ``mixing'' mechanism responsible for the asymptotic behavior. This is joint work with Klaus Widmayer (U. Zurich) and Jiaqi Yang (ICERM).

December 6

3-4 PM

APM 5402

Note the unusual time, location

Nordine Mir

Texas A&M University at Qatar

Finite jet determination of CR maps into real-algebraic sets

We present recent results about finite jet determination of CR maps of positive codimension from real-analytic CR manifolds into real-algebraic subsets in complex space, or more generally Nash subsets. One instance of such results is the unique jet determination of germs of CR maps from minimal real-analytic CR submanifolds in $\C^N$ into Nash subsets in $\C^{N'}$ of D'Angelo finite type, for arbitrary $N,N'\geq 2$. This is joint work with B. Lamel.

Winter 2023

Date Speaker Title + Abstract
February 7

Tadele Mengesha

University of Tennessee, Knoxville

Spring 2023

Date Speaker Title + Abstract

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