The $L^{2}$ weak sequential convergence of radial mass critical NLS solutions with mass above the ground state

TL;DR

This paper investigates the weak convergence behavior of radial mass critical NLS solutions with mass slightly above the ground state, revealing convergence to the ground state along a sequence and exploring mass concentration phenomena.

Contribution

It demonstrates the existence of a sequence where solutions weakly converge to the ground state and provides partial results on mass concentration for minimal mass blow-up solutions.

Findings

Existence of a sequence where solutions weakly converge to the ground state.

Partial results on mass concentration of minimal mass blow-up solutions.

Insights into the asymptotic behavior of solutions above the ground state mass.

Abstract

We study the non-scattering $L^{2}$ solution $u$ to the radial mass critical nonlinear Schr\"odinger equation with mass just above the ground state, and show that there exists a time sequence $\{t_{n}\}_{n}$, such that $u(t_{n})$ weakly converges to the ground state $Q$ up to scaling and phase transformation. We also give some partial results on the mass concentration of the minimal mass blow up solution.