arXiv subjects

Qingquan Deng

Publications and source records attributed to Qingquan Deng.

6 recordsLinked to original sources

arXiv2021

Large global solutions for nonlinear Schrödinger equations I, mass-subcritical cases

In this paper, we consider the nonlinear Schrödinger equation, $$ i\partial_{t}u+Δu= μ|u|^p u, \quad (t,x)\in \mathbb{R}^{d+1}, $$ with $μ=\pm1, p>0$. In this work, we consider the mass-subcritical cases, that is, $p\in (0,\frac4d)$. We prove that under some restrictions on $d,p$, any radial initial data in the critical space $\dot H^{s_c}(\mathbb{R}^d)$ with compact support, implies global well-posedness.

math.AP↗

arXiv2021

Large global solutions for nonlinear Schrödinger equations II, mass-supercritical, energy-subcritical cases

In this paper, we consider the defocusing mass-supercritical, energy-subcritical nonlinear Schrödinger equation, $$ i\partial_{t}u+Δu= |u|^p u, \quad (t,x)\in \mathbb R^{d+1}, $$ with $p\in (\frac4d,\frac4{d-2})$. We prove that under some restrictions on $d,p$, any radial function in the rough space $H^{s_0}(\mathbb R^d),\textit{for some } s_0<s_c$ with the support away from the origin, there exists an incoming/outgoing decomposition, such that the initial data in the outgoing part leads to the global well-posedness and scattering forward in time; while the initial data in the incoming part leads to the global well-posedness and scattering backward in time. The proof is based on Phase-Space analysis of the nonlinear dynamics.

math.AP↗

arXiv2019

Large global solutions for nonlinear Schrödinger equations III, energy-supercritical cases

In this work, we mainly focus on the energy-supercritical nonlinear Schrödinger equation, $$ i\partial_{t}u+Δu= μ|u|^p u, \quad (t,x)\in \mathbb{R}^{d+1}, $$ with $μ=\pm1$ and $p>\frac4{d-2}$. %In this work, we consider the energy-supercritical cases, that is, $p\in (\frac4{d-2},+\infty)$. We prove that for radial initial data with high frequency, if it is outgoing (or incoming) and in rough space $H^{s_1}(\mathbb{R}^d)$ $(s_1<s_c)$ or its Fourier transform belongs to $W^{s_2,1}(\mathbb{R}^d)$ $(s_2<s_c)$, the corresponding solution is global and scatters forward (or backward) in time. We also construct a class of large global and scattering solutions starting with many bubbles, which are mingled with in the physical space and separate in the frequency space. The analogous results are also valid for the energy-subcritical cases.

math.AP↗

arXiv2017

Endpoint Strichartz Estimates for Charge Transfer Hamiltonians

We prove the optimal endpoint Strichartz estimates for Schrödinger equation with charge transfer potentials and a general source term in $\mathbb{R}^n$ for $n\geq3$. The proof is based on using the projection on the scattering states defined implicitly in Rodnianski, Schlag and Soffer \cite{RSS1}, and asymptotic completeness for such systems. The method works as well in the matrix non-selfadjoint case.

math.AP↗

arXiv2017

Soliton-potential interactions for nonlinear Schrödinger equation in $\mathbb{R}^3$

In this work we mainly consider the dynamics and scattering of a narrow soliton of NLS equation with a potential in $\mathbb{R}^3$, where the asymptotic state of the system can be far from the initial state in parameter space. Specifically, if we let a narrow soliton state with initial velocity $\upsilon_{0}$ to interact with an extra potential $V(x)$, then the velocity $\upsilon_{+}$ of outgoing solitary wave in infinite time will in general be very different from $\upsilon_{0}$. In contrast to our present work, previous works proved that the soliton is asymptotically stable under the assumption that $\upsilon_{+}$ stays close to $\upsilon_{0}$ in a certain manner.

math.AP↗

arXiv2016

Riesz transforms associated with higher-order Schrödinger type operators

In this paper, let $L=L_{0}+V$ be a Schrödinger type operator where $L_{0}$ is higher order elliptic operator with complex coefficients in divergence form and $V$ is signed measurable function, under the strongly subcritical assumption on $V$, the authors study the $L^{q}$ boundedness of Riesz transforms $\nabla^{m}L^{-1/2}$ for $q\leq 2$ and obtain a sharp result. Furthermore, the authors impose extra regularity assumptions on $V$ to obtain the $L^{q}$ boundedness of Riesz transforms $\nabla^{m}L^{-1/2}$ for $q>2$. As an application, the main results can be applied to the operator $L=(-Δ)^{m}-γ|x|^{-2m}$ for suitable $γ$

math.CA↗