Graduate Student Seminar- Stable singularity formations in the nonlinear dispersive equations.

Title: Stable singularity formations in the nonlinear dispersive equations.

Speaker: Kai Yang, GWU

Date and Time: Friday September 22, 1pm-2pm

Place: Rome 206

Abstract: We will consider a couple of basic models used in dispersive (wave-type) differential equations and will address the question of formation of singularities (i.e., solutions that break down in finite time). These solutions are typically referred to blow-up solutions, since they tend to concentrate on some set (e.g. at a point) with amplitude or speed growing unboundedly. We study the case called the L2-critical case, which means that the solutions and the equations preserve the L2-norm, often referred as mass. We introduce the dynamic rescaling method to simulate the rate of blow-up solutions for the L2-critical nonlinear Schrodinger equation (NLS) as well as for the L2-critical generalized Hartree equation (gHartree). We study the solutions initiated from the radial data and consider dimensions from d=4 to d=12 for the NLS equation and from d=3 to d=7 for the gHartree equation. It turns out that the stable singularities have the blowup rate which can be expressed as (T-t)^{-0.5}, with a logarithmic corrections from analyzing the rescaled equations. We also provide a numerically-assisted proof of the spectral property for the NLS equation from d=4 to d=12, which confirms our direct computational results. In the gHartree case, we can only show the spectral property in the 3d radial case, as the spectral property of the nonlocal (convolution) nonlinearity is influenced significantly, and is harder to obtain both analytically and numerically.

Note: The Graduate Student Seminar is mandatory for ALL graduate Students. Attendance will be taken.