The APDE seminar on Monday, 4/24, will be given by Perry Kleinhenz (MSU) in-person in Evans 732, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto (fpasqualotto@berkeley.edu) or Mengxuan Yang (mxyang@math.berkeley.edu).

Title: Energy decay for the damped wave equation

Abstract: The damped wave equation describes the motion of a vibrating system exposed to a damping force. For the standard damped wave equation, exponential energy decay is equivalent to the Geometric Control Condition (GCC). The GCC requires every geodesic to meet the positive set of the damping coefficient in finite time.  A natural generalization is to allow the damping coefficient to depend on time, as well as position. I will give an overview of the classical results and discuss how a time dependent generalization of the GCC implies exponential energy decay. I will also mention some results for unbounded damping when the GCC is not satisfied. 

The APDE seminar on Monday, 4/17, will be given by Kihyun Kim in-person in Evans 732, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto (fpasqualotto@berkeley.edu) or Mengxuan Yang (mxyang@math.berkeley.edu).

Title: Rigidity of long-term dynamics for the self-dual Chern-Simons-Schrödinger equation within equivariance

Abstract: We consider the long time dynamics for the self-dual Chern-Simons-Schrödinger equation (CSS) within equivariant symmetry. Being a gauged 2D cubic nonlinear Schrödinger equation (NLS), (CSS) is L2-critical and has pseudoconformal invariance and solitons. However, there are two distinguished features of (CSS), the self-duality and non-locality, which make the long time dynamics of (CSS) surprisingly rigid. For instance, (i) any finite energy spatially decaying solutions to (CSS) decompose into at most one(!) modulated soliton and a radiation. Moreover, (ii) in the high equivariance case (i.e., the equivariance index ≥ 1), any smooth finite-time blow-up solutions even have a universal blow-up speed, namely, the pseudoconformal one. We explore this rigid dynamics using modulation analysis, combined with the self-duality and non-locality of the problem.

The APDE seminar on Monday, 4/10, will be given by Jian Wang (UNC) in-person in Evans 732, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto (fpasqualotto@berkeley.edu) or Mengxuan Yang (mxyang@math.berkeley.edu).

Title: Damping for Fractional Wave Equations

Abstract: Motivated by highly successful numerical methods for damping the surface water wave equations proposed by Clamond et al. (2005), we study the following leading order linear model for damped gravity water waves
\[ \partial_t^2 U + |D| U + \chi \partial_tU = 0 \]
We show that the energy of the solution has polynomial decay by proving a resolvent estimate. Joint work with Thomas Alazard and Jeremy L. Marzuola.