Yongguen Cho | Searcharxiv

arXiv2020

On the focusing energy-critical inhomogeneous NLS: weighted space approach

This paper is concerned with the global well-posedness and finite time blowup problem for the 3D focusing energy-critical inhomogeneous NLS. In the previous results \cite{chkl2, chkl3} the authors considered the same problems with the spatial inhomogeneity coefficient $g$ such that $g(x) \sim |x|^{-b}$ for $0 \le b < \frac43$. Here we extend the inhomogeneous index $b$ up to $\frac32$. For this purpose, we improve the local theory and develop a new profile decomposition based on weighted space.

math.AP↗

arXiv2020

Almost critical regularity of non-abelian Chern-Simons-Higgs system in the Lorenz gauge

In this paper we consider a Cauchy problem on the self-dual relativistic non-abelian Chern-Simons-Higgs model, which is the system of equations of $\mathfrak{su}(n)\, (n \ge 2)$-valued matter field $ϕ$ and gauge field $A$. Based on the frequency localization as well as the null structure we show the local well-posedness in Sobolev space $H^{s+\frac12} \times H^s$ for $s>\frac14$. We also prove that the solution flow map $(ϕ(0), A(0)) \mapsto (ϕ(t), A(t))$ fails to be $C^2$ at the origin of $H^s \times H^σ$ when $σ< \frac14$ regardless of $s \in \mathbb R$. This means the regularity $H^s$, $s>\frac14$ is almost critical.

math.AP↗