arXiv subjects

Ryosuke Hyakuna

Publications and source records attributed to Ryosuke Hyakuna.

5 recordsLinked to original sources

arXiv2026

Fefferman--Stein-type estimates and fractional NLS in Fourier Sobolev spaces

In this paper, we prove a sharp Fefferman--Stein-type estimate for the fractional Schr\"odinger equation, which can be regarded as a generalized Strichartz estimate for data in the Fourier Lebesgue space $\widehat{L^p}$. Then, as an application of the Fefferman--Stein inequality and its off-diagonal generalization, we prove large data local well-posedness and small data global well-posedness results for the one dimensional fractional nonlinear Schr\"odinger equation with pure power nonlinearities in the homonegeneous and inhomogeneous Fourier--Sobolev spaces $\widehat{\dot{H}^s_p},\widehat{H^s_p}$. Solutions are established in $L_x^r(\mathbb{R} ;L^q_t(I))$ spaces in order to overcome the difficulty of a loss of derivatives in the standard Strichartz estimates.

math.AP↗

arXiv2025

Global well-posedness for the 1D cubic nonlinear Schr\"odinger equation in $L^p,\,p>2$

In this paper, we show that the one dimensional cubic nonlinear Schr\"odinger equation is globally well posed in $L^p$ for $2\le p <13/6$. In particular, we prove that the global solution enjoys the persistence property for a twisted variable at any time, which implies the result is a natural exetension of the classical global well-posedness in $L^2$ to $L^p$. The proof exploits the data-decomposition argument originally developed by Vargas-Vega in the functional framework introduced by Zhou.

math.AP↗

arXiv2024

Unconditional well-posedness for the nonlinear Schr\"odinger equation in Bessel potential spaces

The Cauchy problem for the nonlinear Schr\"odinger equation is called unconditionally well posed in a data space $E$ if it is well posed in the usual sense and the solution is unique in the space $C([0,T]; E)$. In this paper, this notion of unconditional well-posedness is redefined so that it covers $L^p$-based Sobolev spaces as data space $E$ and it is equivalent to the usual one when $E$ is an $L^2$-based Sobolev space $H^s$. Next, based on this definition, it is shown that the Cauchy problem for the 1D cubic NLS is unconditionally well posed in Bessel potential spaces $H^s_p$ for $4/3<p\le 2$ under certain regularity assumptions on $s$.

math.AP↗

arXiv2022

Well-posedness for the 1D cubic nonlinear Schrödinger equation in $L^p$, $p>2$

In this paper, local well-posedness is shown for the one dimensional cubic nonlinear Schrödinger equation in $L^p$-spaces for $2<p<4$, which generalizes a classical result for $p=2$ by Y. Tsutsumi and recent work for $1<p<2$ by Y. Zhou. As a consequence, a local theory of solutions is established for a class of data which decay more slowly than square integrable functions. Regularity properties of the local solutions in the $L^p$-based Sobolev spaces and Stricharz spaces are also proved.

math.AP↗

arXiv2018

Well-posedness and decay estimates for 1D nonlinear Schrödinger equations with Cauchy data in $L^p$

In this paper, we establish a standard $L^p$-theory of solutions to one dimensional nonlinear Schrödinger equations with the power like nonlinearity. More precisely, we extend the following three well-known results in the $L^2$ space into $L^p$ setting: 1. Large data local well-posedness for subcritical nonlinearities, 2. Small data global well-posedness for critical nonlinearities, 3. Large data global well-posedness if the subcritical nonlinearity is given by $|u|^{α-1}u$.

math.AP↗