The APDE seminar on Monday, 10/4, will be given by Sanchit Chaturvedi (Stanford University) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (firstname.lastname@example.org)
Title: Stability of vacuum for the non-cut-off Boltzmann equation with moderately soft potentials.
Abstract: The vector field method developed by Klainerman has been widely successful in the study of wave equations and general relativity. Recently, the vector field approach has been adapted to understand the dispersion due to the transport operator in both collisionless and collisional kinetic models. As a proof of concept, I will discuss the stability of vacuum for Boltzmann equation with moderately soft potentials. The nonlocality of the Boltzmann operator poses a lot of difficulty and forces us to use a purely energy based approach. This is in contrast to the paper by Luk (Stability of vacuum for the Landau equation with moderately soft potentials) on Landau equation in a similar setting where a maximum principle is both proved and needed.
The APDE seminar on Monday, 9/27, will be given by Steve Zelditch (Northwestern University) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (email@example.com)
Title: Riemannian 3-manifolds whose eigenfunctions have just
two nodal domains.
Abstract: In 1977, Hans Lewy published an article constructing a
series $\phi_N$ of spherical harmonics of degree $N$ whose nodal (zero) sets cut the 2-sphere into just 2 nodal domains. It is an ingenious construction. Recently, Junehyuk Jung and I showed that there is a simple and canonical way to construct an infinite dimensional family of Riemannian metrics on certain 3 manifolds, all of whose eigenfunctions have this property (except for a certain trivial sequence). The construction generalizes to all odd dimensions. This unexpected behavior of 3D eigenfunctions is heuristically related to numerical computations of nodal sets of 3D spherical harmonics, where only one connected component is visible (the `giant component’). Sarnak conjectured that its genus is maximal among degree N polynomials . Jung and I showed that our result holds for random “equivariant” 3D spherical harmonics and showed that in a certain range the genus has maximal order of growth $N^3$. The purpose of my talk is to explain these phenomena of 3D nodal sets, which have no analogue for the much more studied 2D case
The APDE seminar on Monday, 9/20, will be given by Kenji Nakanishi (Kyoto University) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (firstname.lastname@example.org)
Title: Global dynamics around 2-solitons for the nonlinear damped Klein-Gordon equation
Abstract: This is joint work with Kenjiro Ishizuka (Kyoto). We study global behavior of solutions for the nonlinear Klein-Gordon equation with a damping and a focusing nonlinearity on the Euclidean space. Recently, Cote, Martel and Yuan proved the soliton resolution conjecture for this equation completely in the one-dimensional case: every global solution in the energy space is asymptotic to superposition of solitons. Since the solitons are unstable, a natural question is which initial data evolve into each of the asymptotic forms. We consider the simplest setting in general space dimensions: the global behavior of solutions starting near a superposition of two ground states.
The main result is a complete classification of those solutions into 5 types of global behavior. Two of them are asymptotic to the positive ground state and the negative one respectively. They form two manifolds of codimension-1 that are joined at their boundary, which is the manifold of solutions asymptotic to superposition of two solitons. The connected union of those three manifolds separates the other solutions into the open set of global decaying solutions and that of blow-up. The manifold of 2-solitons was constructed by Cote, Martel, Yuan and Zhao. To get the classification, the main difficulty is in controlling the direction of instability attached to the two soliton components, because the soliton interactions are not integrable in time, breaking the simple linearized approximation. It is resolved by showing that the non-integrable interactions do not essentially affect the direction of instability, using the reflection symmetry of the equation and the 2-solitons.
I will also talk about a much harder difficulty in the 3-soliton case,
which may be called soliton merger.
The APDE seminar on Monday, 9/13, will be given by Viorel Barbu (AI. I. Cuza) online via Zoom from 9.10am to 10.00am PST (note the time change). To participate, email Sung-Jin Oh (email@example.com)
Title: Nonlinear Fokker-Planck equations and trend to equilibrium in
We survey a few recent results on the H-theorem for nonlinear
Fokker-Planck equations and the existence of compact attractors in $L^1$
for the solutions.