arXiv subjects
Publications and source records attributed to Kailong Yang.
We study long-time Strichartz estimates for the Schr\"{o}dinger equation on waveguide manifolds, and use them to establish upper bounds on the growth of Sobolev norms for the nonlinear Schr\"{o}dinger equation on three-dimensional waveguides.
In this article, we show that the solution to defocusing cubic nonlinear Schr\"odinger equation (NLS) posed on the two-dimensional waveguide \begin{align*} i\partial_tu+\Delta_{\R\times\T}u=|u|^2u \end{align*} is globally well-posed in $H^s(\R\times\T)$ with $s>\frac{1}{2}$. The proof is based on the $I$-method. Inspired by Colliander-Keel-Staffilani-Takaoka-Tao [Discrete Contin. Dyn. Syst. 21 (2008), 665-686], we construct the modified energy to improve the energy increment. The main difficulty lies in controlling the resonant interactions caused by the modified energy. To this end, we establish refined bilinear Strichartz estimates with angular truncation on the rescaled waveguide, thereby generalizing results previously obtained by Takaoka [J. Differ. Equa. 394 (2024), 296-319]. Furthermore, we demonstrate polynomial growth of $H^s$ with $\frac{1}{2} < s < 1$. Our result extends the recent work of Deng-Fan-Yang-Zhao-Zheng [J. Func. Anal. 287 (2024), 110595].
We study local-in-time and global-in-time bilinear Strichartz estimates for the Schr\"odinger equation on waveguides. As applications, we apply those estimates to study global well-posedness of nonlinear Schr\"odinger equations on these waveguides.
We consider the cubic nonlinear Schrödinger equation on product manifolds $\mathbb{R}\times \mathbb{T}$. In this paper, we obtain polynomial bounds on the growth in time of high Sobolev norms of the solutions. The main ingredient of the proof is to establish an iteration bound, which is based on the idea used by Bourgain in \cite{B1}.
In this paper, we prove scattering asymptotics for the 2D (discrete dimension) cubic resonant system. This scattering result was used in Zhao \cite{Z1} as an assumption to obtain the scattering for cubic NLS on $\mathbb{R}^2\times \mathbb{T}^2$ in $H^1$ space. Moreover, the 1D analogue is proved in Yang-Zhao \cite{YZ}. Though the scheme is also tightly based on Dodson \cite{D}, the 2D case is more complicated which causes some new difficulties. One obstacle is the failure of `$l^2$-estimate' for the cubic resonances in 2D (we also discuss it in this paper, which may have its own interests). To fix this problem, we establish weaker estimates and exploit the symmetries of the resonant system to modify the proof of \cite{YZ}. At last, we make a few remarks on the research line of `long time dynamics for NLS on waveguides'.
In this article, we investigate the multilinear distorted multiplier estimate (Coifman-Meyer type theorem) associated with the Schrödinger operator $H=-Δ+ V$ in the framework of the corresponding distorted Fourier transform. Our result is the "distorted" analog of the multilinear Coifman-Meyer multiplier operator theorem in \cite{CM1}, which extends the bilinear estimates of Germain, Hani and Walsh's in \cite{PZS} to the multilinear case for all dimensions. As applications, we give the estimate of Leibniz's law of integer order derivations for the multilinear distorted multiplier for the first time and we obtain small data scattering for a kind of generalized mass-critical NLS with good potential in low dimensions $d=1,2$.
In this article, we will show the global wellposedness and scattering of the cubic defocusing nonlinear Schrödinger equation on waveguide $\mathbb{R}^2\times \mathbb{T}$ in $H^1$. We first establish the linear profile decomposition in $H^{ 1}(\mathbb{R}^2 \times \mathbb{T})$ motivated by the linear profile decomposition of the mass-critical Schrödinger equation in $L^2(\mathbb{R}^2)$. Then by using the solution of the infinite dimensional vector-valued resonant nonlinear Schrödinger system to approximate the nonlinear profile, we can prove scattering in $H^1$ by using the concentration-compactness/rigidity method.
In this article, we consider the infinite dimensional vector-valued resonant nonlinear Schrödinger system, which arises from the study of the asymptotic behavior of the defocusing nonlinear Schrödinger equation on "wave guide" manifolds like $\mathbb{R}^2\times \mathbb{T}$ in [7]. We show global well-posedness and scattering for this system by long time Strichartz estimates and frequency localized interaction Morawetz estimates. As a by-product, our results make the arguments of scattering theory in [7] closed as crucial ingredients for compactness of the critical elements.