arXiv subjects

Gyu Eun Lee

Publications and source records attributed to Gyu Eun Lee.

3 recordsLinked to original sources

arXiv2021

Analyticity and infinite breakdown of regularity in mass-subcritical Hartree scattering

We study the asymptotic behavior of solutions to the defocusing mass-subcritical Hartree NLS $iu_t + Δu = F(u) = (|x|^{-γ}*|u|^2)u$ on $\mathbb{R}^d$, $d\geq 2$, $\frac{4}{3} < γ< 2$. We show that the scattering problem associated to this equation is analytically well-posed in the weighted spaces $Σ= H^1\cap\mathcal{F}H^1$ and $\mathcal{F}H^1$. Furthermore, we show that the same problem fails to be analytically well-posed for data in $L^2$. This constitutes an infinite loss of regularity between the scattering problems in weighted spaces and in $L^2$. This further develops an earlier investigation initiated by the author in which a finite breakdown of regularity was proved for the $L^2$ scattering problem for the mass-subcritical NLS with power nonlinearity $F(u) = |u|^pu$.

math.AP↗

arXiv2019

Breakdown of regularity of scattering for mass-subcritical NLS

We study the scattering problem for the nonlinear Schrödinger equation $i\partial_t u + Δu = |u|^p u$ on $\mathbb{R}^d$, $d\geq 1$, with a mass-subcritical nonlinearity above the Strauss exponent. For this equation, it is known that asymptotic completeness in $L^2$ with initial data in $Σ$ holds and the wave operator is well-defined on $Σ$. We show that there exists $0<β<p$ such that the wave operator and the data-to-scattering-state map do not admit extensions to maps $L^2\to L^2$ of class $C^{1+β}$ near the origin. This constitutes a mild form of ill-posedness for the scattering problem in the $L^2$ topology.

math.AP↗

arXiv2019

Local wellposedness for the critical nonlinear Schrödinger equation on $\mathbb{T}^3$

For $p\geq 2$, we prove local wellposedness for the nonlinear Schrödinger equation $(i\partial_t + Δ)u = \pm|u|^pu$ on $\mathbb{T}^3$ with initial data in $H^{s_c}(\mathbb{T}^3)$, where $\mathbb{T}^3$ is a rectangular irrational $3$-torus and $s_c = \frac{3}{2} - \frac{2}{p}$ is the scaling-critical regularity. This extends work of earlier authors on the local Cauchy theory for NLS on $\mathbb{T}^3$ with power nonlinearities where $p$ is an even integer.

math.AP↗