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Shinya Kinoshita

Publications and source records attributed to Shinya Kinoshita.

At least 19 recordsLinked to original sources
arXiv2025

Maximal estimates for orthonormal systems of wave equations

This paper investigates maximal estimates of the wave operators for orthonormal families of initial data. We extend the classical maximal estimates for the wave operator by making partial progress on maximal estimates for orthonormal systems in low dimensions. Our novel approach is based on a geometric analysis of the kernel of wave operators within the framework of Schatten $2$ estimates. In particular, we exploit Wolff's geometric lemma on the intersection patterns of thickened spheres.

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arXiv2025

Global well-posedness and scattering for the 2D modified Zakharov-Kuznetsov equation

We consider the Cauchy problem associated with the modified Zakharov-Kuznetsov equation over $\mathbb{R}^2$. Taking into consideration the associated dispersive effects, we introduce, for $s,a\ge 0$, a two-parameter space $H^{s,a}(\mathbb{R}^2)$, which scales as the classic $H^s$ spaces. In this new class, we prove local well-posedness for $s+a\ge 1/4$, $0<a<1/4$, and global well-posedness and scattering for small data in the case $s=0, \ a=1/4$. These results are shown to be sharp in the sense of $C^3$-flows.

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arXiv2025

Well-posedness and ill-posedness for a system of periodic quadratic derivative nonlinear Schrödinger equations

We consider the Cauchy problem of a system of quadratic derivative nonlinear Schrödinger equations which was introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction. For the nonperiodic setting, the authors proved some well-posedness results, which contain the scaling critical case for $d\geq 2$. In the present paper, we prove the well-posedness of this system for the periodic setting. In particular, well-posedness is proved at the scaling critical regularity for $d\geq 3$ under some conditions for the coefficients of the Laplacian. We also prove some ill-posedness results. As long as we use an iteration argument, our well-posedness results are optimal except for some critical cases.

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arXiv2024

Improved well-posedness for quasilinear and sharp local well-posedness for semilinear KP-I equations

We show new well-posedness results in anisotropic Sobolev spaces for dispersion-generalized KP-I equations with increased dispersion compared to the KP-I equation. We obtain the sharp dispersion rate, below which generalized KP-I equations on $\mathbb{R}^2$ and on $\mathbb{R} \times \mathbb{T}$ exhibit quasilinear behavior. In the quasilinear regime, we show improved well-posedness results relying on short-time Fourier restriction. In the semilinear regime, we show sharp well-posedness with analytic data-to-solution mapping. On $\mathbb{R}^2$ we cover the full subcritical range, whereas on $\mathbb{R} \times \mathbb{T}$ the sharp well-posedness is strictly subcritical. Nonlinear Loomis-Whitney inequalities are one ingredient. These are presently proved for Borel measures with growth condition reflecting the different geometries of the plane $\mathbb{R}^2$, the cylinder $\mathbb{R} \times \mathbb{T}$, and the torus $\mathbb{T}^2$. Finally, we point out that on tori $\mathbb{T}^2_γ$, KP-I equations are never semilinear.

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arXiv2023

Decoupling inequality for paraboloid under shell type restriction and its application to the periodic Zakharov system

In this paper, we establish local well-posedness for the Zakharov system on $\mathbb{T}^d$, $d\ge3$ in a low regularity setting. Our result improves the work of Kishimoto. Moreover, the result is sharp up to $\varepsilon$-loss of regularity when $d=3$ and $d\ge5$ as long as one utilizes the iteration argument. We introduce ideas from recent developments of the Fourier restriction theory. The key element in the proof of our well-posedness result is a new trilinear discrete Fourier restriction estimate involving paraboloid and cone. We prove this trilinear estimate by improving Bourgain--Demeter's range of exponent for the linear decoupling inequality for paraboloid under the constraint that the input space-time function $f$ satisfies ${\rm supp}\, \hat{f} \subset \{ (ξ,τ) \in \mathbb{R}^{d+1}: 1- \frac1N \le |ξ| \le 1 + \frac1N,\; |τ- |ξ|^2| \le \frac1{N^{2}} \} $ for large $N\ge1$.

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arXiv2023

Boundary Strichartz estimates and pointwise convergence for orthonormal systems

We consider maximal estimates associated with fermionic systems. First we establish maximal estimates with respect to the spatial variable. These estimates are certain boundary cases of the many-body Strichartz estimates pioneered by Frank, Lewin, Lieb and Seiringer. We also prove new maximal-in-time estimates, thereby significantly extending work of Lee, Nakamura and the first author on Carleson's pointwise convergence problem for fermionic systems.

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arXiv2023

A note on Strichartz estimates for the wave equation with orthonormal initial data

This note is concerned with Strichartz estimates for the wave equation and orthonormal families of initial data. We provide a survey of the known results and present what seems to be a reasonable conjecture regarding the cases which have been left open. We also provide some new results in the maximal-in-space boundary cases.

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arXiv2022

Sharp well-posedness for the Cauchy problem of the two dimensional quadratic nonlinear Schrödinger equation with angular regularity

This paper is concerned with the Cauchy problem of the quadratic nonlinear Schrödinger equation in $\mathbb{R} \times \mathbb{R}^2$ with the nonlinearity $η|u|^2$ where $η\in \mathbb{C} \setminus \{0\}$ and low regularity initial data. If $s < -1/4$, the ill-posedness result in the Sobolev space $H^{s}(\mathbb{R}^2)$ is known. We will prove the well-posedness in $H^s(\mathbb{R}^2)$ for $-1/2 < s < -1/4$ by assuming some angular regularity on initial data. The key tools are the modified Fourier restriction norm and the convolution estimate on thickened hypersurfaces.

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arXiv2021

Local well-posedness of a system describing laser-plasma interactions

A degenerate Zakharov system arises as a model for the description of laser-plasma interactions. It is a coupled system of a Schrödinger and a wave equation with a non-dispersive direction. In this paper, a new local well-posedness result for rough initial data is established. The proof is based on an efficient use of local smoothing and maximal function norms.

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arXiv2020

Global Well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation in 2D

This paper is concerned with the Cauchy problem of the $2$D Zakharov-Kuznetsov equation. We prove bilinear estimates which imply local in time well-posedness in the Sobolev space $H^s({\mathbb{R}}^2)$ for $s > -1/4$, and these are optimal up to the endpoint. We utilize the nonlinear version of the classical Loomis-Whitney inequality and develop an almost orthogonal decomposition of the set of resonant frequencies. As a corollary, we obtain global well-posedness in $L^2({\mathbb{R}}^2)$.

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arXiv2020

The Zakharov-Kuznetsov equation in high dimensions: Small initial data of critical regularity

The Zakharov-Kuznetsov equation in spatial dimension $d\geq 5$ is considered. The Cauchy problem is shown to be globally well-posed for small initial data in critical spaces and it is proved that solutions scatter to free solutions as $t \to \pm \infty$. The proof is based on i) novel endpoint non-isotropic Strichartz estimates which are derived from the $(d-1)$-dimensional Schrödinger equation, ii) transversal bilinear restriction estimates, and iii) an interpolation argument in critical function spaces. Under an additional radiality assumption, a similar result is obtained in dimension $d=4$.

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arXiv2020

Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in 2D

This paper is concerned with the Cauchy problem of $2$D Klein-Gordon-Zakharov system with very low regularity initial data. We prove the bilinear estimates which are crucial to get the local in time well-posedness. The estimates are established by the Fourier restriction norm method. We utilize the bilinear Strichartz estimates and the nonlinear version of the classical Loomis-Whitney inequality which was applied to Zakharov system.

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arXiv2019

Well-posedness for the Cauchy problem of the modified Zakharov-Kuznetsov equation

This paper is concerned with the Cauchy problem of the modified Zakharov-Kuznetsov equation on $\mathbb{R}^d$. If $d=2$, we prove the sharp estimate which implies local in time well-posedness in the Sobolev space $H^s(\mathbb{R}^2)$ for $s \geq 1/4$. If $d \geq 3$, by employing $U^p$ and $V^p$ spaces, we establish the small data global well-posedness in the scaling critical Sobolev space $H^{s_c}(\mathbb{R}^d)$ where $s_c = d/2-1$.

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arXiv2018

Well-posedness for KdV-type equations with quadratic nonlinearity

We consider the Cauchy problem of the KdV-type equation \[ \partial_t u + \frac{1}{3} \partial_x^3 u = c_1 u \partial_x^2u + c_2 (\partial_x u)^2, \quad u(0)=u_0. \] Pilod (2008) showed that the flow map of this Cauchy problem fails to be twice differentiable in the Sobolev space $H^s(\mathbb{R})$ for any $s \in \mathbb{R}$ if $c_1 \neq 0$. By using a gauge transformation, we point out that the contraction mapping theorem is applicable to the Cauchy problem if the initial data are in $H^2(\mathbb{R})$ with bounded primitives. Moreover, we prove that the Cauchy problem is locally well-posed in $H^1(\mathbb{R})$ with bounded primitives.

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arXiv2018

Well-posedness for a system of quadratic derivative nonlinear Schrödinger equations with radial initial data

In the present paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schrödinger equations. This system was introduced by M. Colin and T. Colin (2004). The first and second authors obtained some well-posedness results in the Sobolev space $H^{s}(\mathbb{R}^d)$. We improve these results for conditional radial initial data by rewriting the system radial form.

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arXiv2018

Sharp bilinear estimates and its application to a system of quadratic derivative nonlinear Schrödinger equations

In the present paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schrödinger equations for the spatial dimension $d=2$ and $3$. This system was introduced by M. Colin and T. Colin (2004). The first author obtained some well-posedness results in the Sobolev space $H^{s}$. But under some condition for the coefficient of Laplacian, this result is not optimal. We improve the bilinear estimate by using the nonlinear version of the classical Loomis-Whitney inequality, and prove the well-posedness in $H^s$ for $s\ge 1/2$ if $d=2$, and $s>1/2$ if $d=3$.

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