arXiv subjects

Nobu Kishimoto

Publications and source records attributed to Nobu Kishimoto.

At least 19 recordsLinked to original sources

arXiv2025

Global well-posedness for intermediate NLS with nonvanishing conditions at infinity

The intermediate nonlinear Schr\"odinger equation (INLS) describes the dynamics of the envelope of weakly nonlinear internal waves in a stratified fluid of finite depth. While the INLS equation is known to admit dark soliton solutions, these solutions possess nonvanishing boundary conditions at spatial infinity and therefore fall outside the scope of existing well-posedness frameworks. This paper establishes the local and global well-posedness of a generalized INLS equation in Zhidkov-type spaces tailored to these nonvanishing boundary conditions. Furthermore, we rigorously justify the deep-water limit, proving that solutions of the generalized INLS converge to those of the generalized Calogero-Moser (CM) derivative NLS equation in Zhidkov-type spaces. Our well-posedness theory relies on the modified energy method combined with frequency envelopes, marking the first application of these techniques to Zhidkov-type spaces.

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arXiv2025

Unconditional uniqueness for the derivative nonlinear Schr\"{o}dinger equation by normal form approach

We prove uniqueness of solutions to the Cauchy problem for the derivative nonlinear Schr\"odinger equation in $L^\infty_tH^{1/2}_x$. Our proof is based on the method of normal form reduction (NFR), which has been employed to obtain the uniqueness in $C_tH^s_x$, $s>1/2$. To overcome logarithmic divergences at the $H^{1/2}$ regularity, we exploit the $B^{0+}_{\infty,1}$ control of solutions provided by a refined Strichartz estimate. Our NFR argument consists of two stages: we first use NFR finitely many times to derive an intermediate equation in which the main cubic nonlinearity is restricted to a certain type of frequency interaction; we then apply the infinite NFR scheme to the intermediate equation. Moreover, we modify the usual NFR argument relying on continuity in time of solutions so that the uniqueness in the class $L^\infty_tH^{1/2}_x$ can be obtained directly.

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arXiv2025

Global solution and asymptotic behavior for the kinetic derivative NLS on $\mathbb R$

In this paper we investigate the global well-posedness and long-term behavior of solutions to the kinetic derivative nonlinear Schr\"odinger equation (KDNLS) on the real line. The equation incorporates both local cubic nonlinearities with derivative terms and a non-local term arising from the Hilbert transform, modeling interactions in plasma physics. We establish global existence for small initial data in the weighted Sobolev space $H^2 \cap H^{1,1}$ and optimal time decay effect. Using energy methods and a frequency-localized gauge transformation, we overcome the difficulties posed by the non-local nonlinearities and provide a rigorous analysis of the asymptotic behavior. Our results also describe modified scattering phenomena with a suitable phase modification, showing that the solutions exhibit a precise asymptotic profile as $t \to \infty$.

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arXiv2025

Local and global well-posedness for the kinetic derivative NLS on $\mathbb{R}$

We investigate the local and global well-posedness of the kinetic derivative nonlinear Schr\"odinger equation (KDNLS) on $\mathbb{R}$, described by \[ i\partial_t u + \partial_x^2 u = i\alpha \partial_x (|u|^2 u) + i\beta \partial_x (H(|u|^2) u), \] where $\alpha, \beta \in \mathbb{R}$, and $H$ represents the Hilbert transformation. For KDNLS, the $L^2$ norm of a solution is decreasing (resp. increasing, conserved) when $\beta$ is negative (resp. positive, zero). Focusing on the Sobolev spaces $H^2$ and $H^2 \cap H^{1,1}$, we establish local well-posedness via the energy method combined with gauge transformations to address resonant interactions in both cases of negative and positive $\beta$. For the dissipative case $\beta < 0$, we further demonstrate global well-posedness by deriving an a priori bound in $H^2$.

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arXiv2023

Gauge transformation for the kinetic derivative nonlinear Schrödinger equation on the torus

We consider the kinetic derivative nonlinear Schrödinger equation, which is a one-dimensional nonlinear Schrödinger equation with a cubic derivative nonlinear term containing the Hilbert transformation. In our previous work, we proved small-data global well-posedness of the Cauchy problem on the torus in Sobolev space $H^s$ for $s>1/2$ by combining the Fourier restriction norm method with the parabolic smoothing effect, which is available in the periodic setting. In this article, we improve the regularity range to $s>1/4$ for the global well-posedness by constructing an effective gauge transformation. Moreover, we remove the smallness assumption by making use of the dissipative nature of the equation.

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arXiv2023

Low regularity a priori estimate for KDNLS via the short-time Fourier restriction method

In this article, we consider the kinetic derivative nonlinear Schrödinger equation (KDNLS), which is a one-dimensional nonlinear Schrödinger equation with a cubic derivative nonlinear term containing the Hilbert transformation. For the Cauchy problem both on the real line and on the circle, we apply the short-time Fourier restriction method to establish a priori estimate for small and smooth solutions in Sobolev spaces $H^s$ with $s>1/4$.

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arXiv2022

Unconditional uniqueness for the periodic Benjamin-Ono equation by normal form approach

We show unconditional uniqueness of solutions to the Cauchy problem associated with the Benjamin-Ono equation under the periodic boundary condition with initial data given in $H^s$ for $s>1/6$. This improves the previous unconditional uniqueness result in $H^{1/2}$ by Molinet and Pilod (2012). Our proof is based on a gauge transform and integration by parts in the time variable.

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arXiv2022

Invariance of the Gibbs measures for periodic generalized Korteweg-de Vries equations

In this paper, we study the Gibbs measures for periodic generalized Korteweg-de Vries equations (gKdV) with quartic or higher nonlinearities. In order to bypass the analytical ill-posedness of the equation in the Sobolev support of the Gibbs measures, we establish deterministic well-posedness of the gauged gKdV equations within the framework of the Fourier-Lebesgue spaces. Our argument relies on bilinear and trilinear Strichartz estimates adapted to the Fourier-Lebesgue setting. Then, following Bourgain's invariant measure argument, we construct almost sure global-in-time dynamics and show invariance of the Gibbs measures for the gauged equations. These results can be brought back to the ungauged side by inverting the gauge transformation and exploiting the invariance of the Gibbs measures under spatial translations. We thus complete the program initiated by Bourgain (1994) on the invariance of the Gibbs measures for periodic gKdV equations.

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arXiv2021

Well-posedness of the Cauchy Problem for the Kinetic DNLS on $\mathbf{T}$

We consider the Cauchy problem for the kinetic derivative nonlinear Schrödinger equation on the torus: \[ \partial_t u - i \partial_x^2 u = α\partial_x \big( |u|^2 u \big) + β\partial_x \big[ H \big( |u|^2 \big) u \big] , \quad (t, x) \in [0,T] \times \mathbf{T}, \] where the constants $α,β$ are such that $α\in \mathbf{R}$ and $β<0$, and $H$ denotes the Hilbert transform. This equation has dissipative nature, and the energy method is applicable to prove local well-posedness of the Cauchy problem in Sobolev spaces $H^s$ for $s>3/2$. However, the gauge transform technique, which is useful for dealing with the derivative loss in the nonlinearity when $β=0$, cannot be directly adapted due to the presence of the Hilbert transform. In particular, there has been no result on local well-posedness in low regularity spaces or global solvability of the Cauchy problem. In this article, we shall prove local and global well-posedness of the Cauchy problem for small initial data in $H^s(\mathbf{T})$, $s>1/2$. To this end, we make use of the parabolic-type smoothing effect arising from the resonant part of the nonlocal nonlinear term $β\partial_x [H(|u|^2)u]$, in addition to the usual dispersive-type smoothing effect for nonlinear Schrödinger equations with cubic nonlinearities. As by-products of the proof, we also obtain smoothing effect and backward-in-time ill-posedness results.

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arXiv2021

Characterization of three-dimensional Euler flows supported on finitely many Fourier modes

Recently, the Nash-style convex integration has been becoming the main scheme for the mathematical study of turbulence, and the main building block of it has been either Beltrami flow (finite mode) or Mikado flow (compactly supported in the physical side). On the other hand, in physics, it is observed that turbulence is composed of a hierarchy of scale-by-scale vortex stretching. Thus our mathematical motivation in this study is to find another type of building blocks accompanied by vortex stretching and scale locality (possibly finitely many Fourier modes). In this paper, we give a complete list of solutions to the 3D Euler equations with finitely many Fourier modes, which is an extension of the corresponding 2D result by Elgindi-Hu-Šverák (2017). In particular, we show that there is no 3D Euler flows with finitely many Fourier modes, except for stationary 2D-like flows and Beltrami flows. We also discuss the case when viscosity and Coriolis effect are present.

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arXiv2021

Unconditional uniqueness of solutions for nonlinear dispersive equations

When a solution to the Cauchy problem for nonlinear dispersive equations is obtained by a fixed point argument using auxiliary function spaces, it is non-trivial to ensure uniqueness of solutions in a natural space such as the class of continuous curves in the data space. This property is called unconditional uniqueness, and proving it often requires some additional work. In the last decade, unconditional uniqueness has been shown for some canonical nonlinear dispersive equations by an integration-by-parts technique, which can be regarded as a variant of the (Poincaré-Dulac) normal form reduction. In this article, we aim to provide an abstract framework for establishing unconditional uniqueness as well as existence of certain weak solutions via infinite iteration of the normal form reduction. In particular, in an abstract setting we find two sets of fundamental estimates, each of which can be used repeatedly to generate all multilinear estimates of arbitrarily high degrees required in this scheme. Then, we confirm versatility of the framework by applying it to various equations, including the cubic nonlinear Schrödinger equation (NLS) in higher dimension, the cubic NLS with fractional Laplacians, the cubic derivative NLS, and the Zakharov system, for which new results on unconditional uniqueness are obtained under the periodic boundary condition.

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arXiv2020

Unconditional local well-posedness for periodic NLS

The nonlinear Schrödinger equations with nonlinearities $|u|^{2k}u$ on the $d$-dimensional torus are considered for arbitrary positive integers $k$ and $d$. The solution of the Cauchy problem is shown to be unique in the class $C_tH^s_x$ for a certain range of scale-subcritical regularities $s$, which is almost optimal in the case $d\geq 4$ or $k\geq 2$. The proof is based on various multilinear estimates and the infinite normal form reduction argument.

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arXiv2019

Unconditional uniqueness for the periodic modified Benjamin-Ono equation by normal form approach

We show that the solution (in the sense of distribution) to the Cauchy problem with the periodic boundary condition associated with the modified Benjamin-Ono equation is unique in $L^\infty_t(H^s(\mathbb{T}))$ for $s>1/2$. The proof is based on the analysis of a normal form equation obtained by infinitely many reduction steps using integration by parts in time after a suitable gauge transform.

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arXiv2018

Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition

We consider a mass-critical system of nonlinear Schödinger equations \begin{align*} \begin{cases} i\partial_t u +Δu =\bar{u}v,\\ i\partial_t v +κΔv =u^2, \end{cases} (t,x)\in \mathbb{R}\times \mathbb{R}^4, \end{align*} where $(u,v)$ is a $\mathbb{C}^2$-valued unknown function and $κ>0$ is a constant. If $κ=1/2$, we say the equation satisfies mass-resonance condition. We are interested in the scattering problem of this equation under the condition $M(u,v)<M(ϕ,ψ)$, where $M(u,v)$ denotes the mass and $(ϕ,ψ)$ is a ground state. In the mass-resonance case, we prove scattering by the argument of Dodson \cite{MR3406535}. Scattering is also obtained without mass-resonance condition under the restriction that $(u,v)$ is radially symmetric.

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arXiv2018

Blow-up of the radially symmetric solutions for the quadratic nonlinear Schrödinger system without mass-resonance

We consider the quadratic nonlinear Schrödinger system \begin{align*} \begin{cases} i\partial_t u +Δu =v \overline{u},\\ i\partial_t v +κΔv =u^2, \end{cases} \text{ on } I \times \mathbb{R}^d, \end{align*} where $1\leq d \leq 6$ and $κ>0$. In the lower dimensional case $d=1,2,3$, it is known that the $H^1$-solution is global in time. On the other hand, there are finite time blow-up solutions when $d=4,5,6$ and $κ=1/2$. The condition of $κ=1/2$ is called mass-resonance. In this paper, we prove finite time blow-up under radially symmetric assumption when $d=5,6$ and $κ\neq 1/2$ and we show blow-up or grow-up when $d=4$.

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arXiv2018

A remark on norm inflation for nonlinear Schrödinger equations

We consider semilinear Schrödinger equations with nonlinearity that is a polynomial in the unknown function and its complex conjugate, on $\mathbb{R}^d$ or on the torus. Norm inflation (ill-posedness) of the associated initial value problem is proved in Sobolev spaces of negative indices. To this end, we apply the argument of Iwabuchi and Ogawa (2012), who treated quadratic nonlinearities. This method can be applied whether the spatial domain is non-periodic or periodic and whether the nonlinearity is gauge/scale-invariant or not.

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arXiv2018

Ill-Posedness of the Third Order NLS Equation with Raman Scattering Term

We consider the ill-posedness and well-posedness of the Cauchy problem for the third order NLS equation with Raman scattering term on the one dimensional torus. It is regarded as a mathematical model for the photonic crystal fiber oscillator. Regarding the ill-posedness, we show the nonexistence of solutions in the Sobolev space and the norm inflation of the data-solution map under slightly different conditions, respectively. We also prove the local unique existence of solutions in the analytic function space.

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arXiv2017

Global solvability of the rotating Navier-Stokes equations with fractional Laplacian in a periodic domain

We consider existence of global solutions to equations for three-dimensional rotating fluids in a periodic frame provided by a sufficiently large Coriolis force. The Coriolis force appears in almost all of the models of meteorology and geophysics dealing with large-scale phenomena. In the spatially decaying case, Koh, Lee and Takada (2014) showed existence for the large times of solutions of the rotating Euler equations provided by the large Coriolis force. In this case the resonant equation does not appear anymore. In the periodic case, however, the resonant equation appears, and thus the main subject in this case is to show existence of global solutions to the resonant equation. Research in this direction was initiated by Babin, Mahalov and Nicolaenko (1999) who treated the rotating Navier-Stokes equations on general periodic domains. On the other hand, Golse, Mahalov and Nicolaenko (2008) considered bursting dynamics of the resonant equation in the case of a cylinder with no viscosity. Thus we may not expect to show global existence of solutions to the resonant equation without viscosity in the periodic case. In this paper we show existence of global solutions for fractional Laplacian case (with its power strictly less than the usual Laplacian) in the periodic domain with the same period in each direction. The main ingredient is an improved estimate on resonant three-wave interactions, which is based on a combinatorial argument.

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