2022 Fall (Abstracts)

Time: September 28 (Wed) 4-5PM

Speaker: Yehun Kwon (Changwon National University)

Title: Carleman inequalities and unique continuation for the polyharmonic operators

Abstract: We obtain a complete characterization of $L^p-L^q$ Carleman estimates with weight $e^{v\cdot x}$ for the polyharmonic operators. Our result extends the Carleman inequalities for the Laplacian due to Kenig--Ruiz--Sogge. Consequently, we obtain new unique continuation properties of higher order Schr\"odinger equations relaxing the integrability assumption on the solution spaces.

Time: October 12 (Wed) 4-5PM

Speaker: Seunghyeok Kim (Hanyang University)

Title: Infinite-time blowing-up solutions to small perturbations of the Yamabe flow

Abstract: In this talk, we will examine a PDE aspect of the Yamabe flow as an energy-critical parabolic equation of the fast-diffusion type.

It is well-known that under the validity of the positive mass theorem, the Yamabe flow on a smooth closed Riemannian manifold $M$ exists

for all time $t$ and uniformly converges to a solution to the Yamabe problem on $M$ as $t \to \infty$. We will observe that such results no longer hold if some arbitrarily small and smooth perturbation is imposed on it, by constructing solutions to the perturbed flow that blow up at multiple points on $M$ in the infinite time. We are also concerned about the stability of the blow-up phenomena under a negativity condition on the Ricci curvature at blow-up points.

This is joint work with Monica Musso (University of Bath, UK).

Time: November 9 (Wed) 9-10AM

Speaker: Fabio Pusateri (University of Toronto)

Title: Some recent results on PDEs with potentials and the stability of solitons

Abstract: This talk will be an overview of some recent results on nonlinear evolution equations with large potentials, and applications to the stability of kinks, as well as to the phenomenon of “radiation damping”. Our approach to this class of problems is based on the use of the distorted Fourier transform and the development of multilinear harmonic analysis in this setting. This talk is based on joint works with P. Germain (NYU), A. Soffer (Rutgers), T. Leger (Princeton), Gong Chen (Georgia Tech) and Adilbek Kairzhan (U of Toronto).

Time: December 14 (Wed) 4-5PM

Speaker: Hyunseok Kim (Sogang University)

Title: On global weak solutions of a Stokes-Magneto system with fractional diffusions

Abstract: We study a Stokes-Magneto system in $\mathbb{R}^d$ ($d\geq 2)$ with fractional diffusions $\Lambda^{2\alpha}\mathbf{u}$ and $\Lambda^{2\beta} \mathbf{b}$ for the velocity $\mathbf{u}$ and the magnetic field $\mathbf{b}$, respectively. Here $\alpha,\beta$ are positive constants and $\Lambda^s = (-\Delta)^{s/2}$ is the fractional Laplacian of order $s\in \mathbb{R}$. Global existence of weak solutions is shown for initial data in $L_2 (\mathbb{R}^d )$ when $\alpha$, $\beta$ satisfy $1/2<\alpha<(d+1)/2$, $\beta >0$, and $\min\{\alpha+\beta,2\alpha+\beta-1\}>d/2$. Moreover, weak solutions are unique if $\beta \geq 1$ and $\min \{\alpha+\beta,2\alpha+\beta-1\}\geq d/2+1$, in addition. This talk is based on a paper with Hyunwoo Kwon at Brown University.