Time: April 14 (Fri) 2:30-3:30 PM
Speaker: Beomjong Kwak(KAIST)
Title: Critical local well-posedness of the nonlinear Schrödinger equation on torus T^d
Abstract: In this talk, we study the local well-posedness theory of nonlinear Schrödinger equations on torus at the critical regularity. This talk focuses on NLS with nonalgebraic nonlinearities, which were less understood compared to algebraic ones. For those, we suggest a bilinear Strichartz estimate and design a new function space adapted to our approach. This talk is based on joint works with Soonsik Kwon.
Time: May 12 (Fri) 2:30-3:30 PM
Speaker: Chenjie Fan(Chinese Academy of Sciences)
Title: Scattering for mass critical nls with a small multiplicative noise
Abstract: We present our proof of scattering for mass critical NLS with a small multiplicative noise. As a byproduct or as a toy model of our analysis, we also obtain the associated global in time Strichartz type estimates for the related linear model. Though some stochastic background is necesary in the proof, this is essentially a determinitic PDE talk. Based on joint work with Weijun Xu and Zehua Zhao.sed on joint work with Weijun Xu and Zehua Zhao.
Time: May 26 (Fri) 2:30-3:30 PM
Speaker: Jaemin Park(University of Basel)
Title: Construction of quasi-periodic solutions to active scalar equations
Abstract: Tautologically, a smooth, steady fluid remains smooth for all time since it does not evolve non-trivially in time. A natural question is whether a small perturbation of such a smooth steady state can also remain smooth for all time. Especially, when the well-posedness of the governing equation is in question, the investigation of initial data near stable steady states can give insight into potential global-in-time solutions. In this talk, we will discuss the construction of global solutions to the generalized surface quasi-geostrophic equations (gSQG) by means of the KAM theory. This is a joint work with Javier Gomez-Serrano and Alexandru Ionescu.
Time: June 9 (Fri) 2:30-3:30 PM
Speaker: Yifei Wu(Tianjin University)
Title: A modified splitting method for the cubic nonlinear Schrodinger equation
Abstract: As a classical time-stepping method, it is well-known that the Strang splitting method reaches the first-order accuracy by losing two spatial derivatives. In this talk, we propose a modified splitting method for the 1D cubic nonlinear Schr\"odinger equation. Suitably choosing the filters, it is shown rigorously that it reaches the first-order accuracy by only losing $\frac32$-spatial derivatives. The result is better than the expected one for the standard (filtered) Strang splitting methods. Moreover, the mass is conserved. The key idea is based on the observation that the low frequency and high frequency components of solutions are almost separated (up to some smooth components). Then the algorithm is constructed by tracking the solution behavior at the low and high frequency components separately.
Time: June 23 (Fri) 2:30-3:30 PM
Speaker: Timothy Candy(University of Otago)
Title: The energy critical Zakharov equation below the ground state
Abstract: We review recent work on the Zakharov system in the energy critical dimension d=4 with energy below the ground state. In the radial setting, it is known that below the ground state, solutions exists globally in time and scatter. In the non-radial setting, solutions below the ground state exist globally in time. Scattering in the non-radial setting is currently an open question. The key input in the non-radial setting is a certain bilinear restriction estimate for inhomogeneous solutions to the Schrodinger equation. We review these estimates, and explain how they can be applied to the Zakaharov system. Part of this work is joint with S. Herr and K. Nakanishi.