- Series
- PDE Seminar
- Time
- Tuesday, January 16, 2024 - 2:00pm for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Boyang Su – University of Chicago – boyang@math.uchicago.edu – https://mathematics.uchicago.edu/people/profile/boyang-su/
- Organizer
- Gong Chen
The existence of global solutions for the Schrödinger equation
i\partial_t u + \Delta u = P_d(u),
with nonlinearity $P_d$ homogeneous of degree $d$, has been extensively studied. Most results focus on the case with gauge invariant nonlinearity, where the solution satisfies several conservation laws. However, the problem becomes more complicated as we consider a general nonlinearity $P_d$. So far, global well-posedness for small data is known for $d$ strictly greater than the Strauss exponent. In dimension $3$, this Strauss exponent is $2$, making NLS with quadratic nonlinearity an interesting topic.
In this talk, I will present a result that shows the global existence and scattering for systems of quadratic NLS for small, localized data. To tackle the challenge presented by the $u\Bar{u}$-type nonlinearity, we require an $\epsilon$ regularization for the terms of this type in the system.