Time: March 30 (Wed) 4-5PM
Speaker: Sangdon Jin (Chung-Ang University)
Title: Effective lower-dimensional Gross-Pitaevskii equations
Abstract: We consider the Gross-Pitaevskii equations with external trapping potentials. This model describes the dynamics of a Bose-Einstein condensate(BEC), but its solutions can blow up in finite time due to L^2-supercritical. However, this issue can be overcome in experiments by creating effective lower dimensional systems. In this talk, we study lower-dimensional BECs and their qualitative properties.
Time: April 13 (Wed) 4-5PM
Speaker: Shohei Nakamura (Osaka University)
Title: Regularisation of functional inequalities via Fokker-Planck equation
Abstract: A functional inequality is often used to investigate several properties of PDEs and this is one of the source of its importance. In this talk, we report an idea somehow “reversing” this relation. Namely, we report that some functional inequalities can be improved by applying properties of diffusion equations. Such idea has been already appeared in the work of Bennett-Carbery-Christ-Tao on the theory of Brascamp-Lieb inequality where they employed the standard heat equation. Here we are interested in Nelson's hypercontractivity and Gross’s logarithmic Sobolev inequality and aim to improve the best constant of these inequalities via Fokker—Planck equation. In particular, our result on the logarithmic Sobolev inequality complements a recently obtained result by Eldan-Lehec-Shenfeld concerning a deficit estimate for inputs with small covariance.
Time: April 13 (Wed) 4-5PM
Speaker: Seokchang Hong (Chung-Ang University)
Title: Scattering property for the cubic Dirac equations with Hartree-type nonlinearity
Abstract: Scattering property of the Hartree-type nonlinear Dirac equations with Yukawa potential and Coulomb potential is considered for the scaling critical Sobolev data. Compared to the Yukawa potential, the Coulomb potential possesses singularity at low frequency in the Fourier space. This singularity requires further task on the output frequency localised near origin: $\mu\lesssim1$. With an observation that such a singularity makes a problem when two input frequencies are high in bilinear interactions, the singularity can be managed in bilinear estimates using orthogonal decomposition of cubes of size $\mu$. In this process, we establish small data scattering as well as large data scattering with a certain condition such as bounded dispersive norm or Majorana condition.
Time: May 11 (Wed) 4-5PM
Speaker: Baoping Liu (Peking University)
Title: Wellposedness for the KdV hierarchy
Abstract: The KdV hierarchy is a hierarchy of integrable equations generalizing the KdV equation. Using the modified Muria transform, we first relate it to the Gardner hierarchy, and by exploiting the idea of approximate flow, we show that the whole hierarchy is wellposed for initial data in H^{-1}. This is based on joint work with H.Koch and F. Klaus.
Time: June 8 (Wed) 4-5PM
Speaker: Masayuki Hayashi (Kyoto University)
Title: Instability of degenerate solitons for nonlinear Schrödinger equations of derivative type
Abstract: We consider nonlinear Schrödinger (NLS) equations of derivative type. Inspired from the works on instability theory of the L^2-critical generalized KdV equation, we study the instability of degenerate solitons in a qualitative way, and obtain a large set of initial data yielding the instability. Our arguments give a small step towards understanding the dynamics around algebraic solitons of the derivative NLS equation.