Spectral transitions for Schr\"odinger operators with decaying potentials and Laplacians on asymptotically flat (hyperbolic) manifolds

Speaker: 

Wencai Liu

Institution: 

UCI

Time: 

Friday, October 26, 2018 - 3:00pm

Location: 

RH 440R

We apply piecewise constructions and gluing  techniques  to  construct

asymptotically flat (hyperbolic) manifolds such that associated

Laplacians have dense embedded eigenvalues or singular continuous

spectra.  The method also allows us to provide various examples of

operators with embedded singular spectra. With additional perturbation theory,    several sharp

spectral transitions (even criteria) for singular spectra are obtained.

In this talk, I will focus on two models-Laplacians on manifolds and Stark operators.

Instability of the Couette flow in low regularity spaces

Speaker: 

Yu Deng

Institution: 

USC

Time: 

Friday, November 2, 2018 - 3:00pm

Host: 

Location: 

RH 440R

 In an exciting paper, J. Bedrossian and N. Masmoudi established the stability of the 2D Couette flow in Gevrey spaces of index greater than 1/2. I will talk about recent joint work with N. Masmoudi, which proves, in the opposite direction, the instability of the Couette flow in Gevrey spaces of index smaller than 1/2. This confirms, to a large extent, that the transient growth predicted heuristically in earlier works does exist and has the expected strength. The proof is based on the fremawork of the stability result, with a few crucial new observations. I will also discuss related works regarding Landau damping, and possible extensions to infinite time.

Asymptotic self-similarity of entire solutions for nonlinear elliptic equations

Speaker: 

Soo-Hyun Bae

Institution: 

Hanbat National University (Daejeon, Korea)

Time: 

Friday, October 12, 2018 - 3:00pm

Location: 

RH 440R

I consider solutions with asymptotic self-similarity. The behavior shows an invariance which comes naturally from nonlinearity. The basic model is Lane-Emden equation. Solution structures depend on the dimension as well as the exponent describing the nonlinearity. More generally, I will explain the corresponding result for quasilinear equations in the radial setting.

The incompressible limit of a tumor growth model

Speaker: 

Olga Turanova

Institution: 

UCLA

Time: 

Tuesday, April 24, 2018 - 3:00pm

Host: 

Location: 

RH306

This talk concerns a PDE system that models tumor growth. We show that a novel free boundary problem arises via the incompressible limit of this model. We take a viscosity solutions approach; however, since the system lacks maximum principle, there are interesting challenges to overcome. This is joint work with Inwon Kim.

The stability of full dimensional KAM tori for nonlinear Schrödinger equation

Speaker: 

Hongzi Cong

Institution: 

UCI and Dalian University of Technology (China)

Time: 

Thursday, January 25, 2018 - 3:00pm

Host: 

Location: 

510N

In this talk, we will show  that the full dimensional invariant tori obtained by Bourgain [J. Funct. Anal., 229 (2005), no. 1, 62–94] is stable in a very long time for 1D nonlinear Schrödinger equation with periodic boundary conditions.

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