Strong nonlinear instability and growth of Sobolev norms near quasiperiodic finite-gap tori of the 2D cubic NLS

Speaker: 

Zaher Hani

Institution: 

University of Michigan

Time: 

Tuesday, November 27, 2018 - 3:00pm

Location: 

RH 306

 We consider a family of quasiperiodic solutions of the nonlinear Schrodinger equation on the 2-torus, namely the family of finite-gap solutions (tori). These solutions are inherited by the 2D equation from its completely integrable 1D counterpart (NLS on the circle) by considering solutions that only depend on one variable. Despite being linearly stable, we prove that these tori (under some genericness conditions) are nonlinearly unstable in the following strong sense: there exists solutions that start very close to those tori in certain Sobolev spaces, but eventually become larger than any given factor at later times. This is the first instance where (unstable) long-time nonlinear dynamics near (linearly stable) quasiperiodic tori is studied and constructed. (joint work with M. Guardia (UPC, Barcelona), E. Haus (University of Naples), M. Procesi (Roma Tre), and A. Maspero (SISSA))

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.

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