Speaker: Fangchi Yan
Virginia Tech
Will give a PDE, Complex Analysis and Differential Geometry Seminar entitled:
The Schrödinger equation with cubic nonlinearities on the half-line in low regularity spaces
Abstract: The initial-boundary value problem for the Schr\"odinger equation with cubic nonlinearities of the form $u^{3-k}\bar{u}^{k}$ is studied on the half-line. Using the Fokas solution formula for the corresponding linear forced problem linear estimates are derived with data in Sobolev spaces and forcing in Bourgain solution spaces. Then, using these linear estimates and the trilinear estimates indicated by the forcing it is shown that the iteration map defined by the Fokas solution formula is a contraction in appropriate solution spaces. Thus, local well-posedness is proved for Sobolev exponents $s\ge 0$ when $k=0,1,2$, and for $s> -1/3$ when $k=3$. The methodology used is analogous to the one used for the corresponding initial value problems that is based on the Fourier transform for solving the forced linear problem.
Date: 05-14-2024
Time: 11:00 am
Location: 258 Hurley Bldg