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Tomoyuki Tanaka

Publications and source records attributed to Tomoyuki Tanaka.

15 recordsLinked to original sources
arXiv2025

Improved refined bilinear estimates and well-posedness for generalized KdV type equations on $\mathbb{R}$

We study the Cauchy problem for one-dimensional dispersive equations posed on $\mathbb{R} $, under the hypotheses that the dispersive operator behaves, for high frequencies, as a Fourier multiplier by $ i |ξ|^αξ$ with $ 1 \le α\le 2 $, and that the nonlinear term is of the form $ \partial_x f(u) $ where $f $ is a real analytic function satisfying certain conditions. We prove the unconditional local well-posedness of the Cauchy problem in $H^s(\mathbb{R}) $ for $ s\ge \frac{5-2α}{4} $ whenever $ 1\le α<\frac{3}{2} $, and for $ s>\frac{1}{2} $ whenever $α\in [\frac{3}{2},2] $. This result is optimal in the case $α\ge \frac{3}{2}$ in view of the restriction $ s>\frac{1}{2} $ required for the continuous embedding $ H^s(\mathbb{R}) \hookrightarrow L^\infty(\mathbb{R}) $. The main novelty of this work, compared to our previous studies, is an improvement of the refined linear and bilinear estimates on $\mathbb{R} $. Our local well-posedness results enable us to derive global existence of solutions for $ α\in [\frac{5}{4},2] $.

math.AP↗
arXiv2025

Refined bilinear Strichartz estimates with application to the well-posedness of periodic generalized KdV type equations

We improve our previous result [L. Molinet and T. Tanaka, Unconditional well-posedness for some nonlinear periodic one-dimensional dispersive equations, J. Funct. Anal. 283 (2022), 109490] on the Cauchy problem for one dimensional dispersive equations with a quite general nonlinearity in the periodic setting. Under the same hypotheses that the dispersive operator behaves for high frequencies as a Fourier multiplier by $ i |ξ|^αξ$ with $ 1 \le α\le 2 $, and that the nonlinear term is of the form $ \partial_x f(u) $ where $f $ is a real analytic function whose Taylor series around the origin has an infinite radius of convergence, we prove the unconditional LWP of the Cauchy problem in $H^s(\mathbb{T}) $ for $ s\ge 1-\fracα{4} $ with $ s>1/2 $. It is worth noting that this result is optimal in the case $α=2$ (generalized KdV equation) in view of the restriction $ s>1/2 $ for the continuous injection of $ H^s(\mathbb{T}) $ into $ L^\infty(\mathbb{T}) $. Our main new ingredient is the replacement of refined Strichartz estimates with refined bilinear estimates in the treatment of the worst resonant interactions. Such refined bilinear estimates already appeared in the work of Hani in the context of Schrödinger equations on a compact manifold. Finally, the main theorem yields global existence results for $ α\in [4/3,2] $.

math.AP↗
arXiv2025

Local well-posedness for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions

We consider the derivative nonlinear Schrödinger equation on the real line, with a background function $ψ(t,x)\in L^\infty(\mathbb{R}^2)$ that satisfies suitable conditions. Such a function may, for example, be a non-decaying solution of the equation, such as a dark soliton. By developing the energy method with correction terms, we prove that the Cauchy problem for perturbations around such an $L^\infty$ function is unconditionally locally well-posed in $ H^s(\mathbb{R}) $ for $ s>3/4 $. As a byproduct, we also establish local well-posedness in the Zhidkov space.

math.AP↗
arXiv2024

Remark on the local well-posedness for NLS with the modulated dispersion

We consider the Cauchy problem of the nonlinear Schrödinger equation with the modulated dispersion and power type nonlinearities in any spatial dimensions. We adapt the Young integral theory developed by Chouk-Gubinelli [K. Chouk and M, Gubinelli, Comm. Partial Differential Equations 40 (2015)] and multilinear estimates which are based on divisor counting, and show the local well-posedness. Our result generalizes the result by Chouk-Gubinelli.

math.AP↗
arXiv2023

Angular correlation of the two gamma rays produced in the thermal neutron capture on gadolinium-155 and gadolinium-157

The ANNRI-Gd collaboration studied in detail the single $γ$-ray spectrum produced from the thermal neutron capture on $^{155}$Gd and $^{157}$Gd in our previous publications. Gadolinium targets were exposed to a neutron beam provided by the Japan Spallation Neutron Source (JSNS) in J-PARC, Japan. In the present analysis, one new additional coaxial germanium crystal was used in the analysis in combination with the fourteen germanium crystals in the cluster detectors to study the angular correlation of the two $γ$ rays emitted in the same neutron capture. We present for the first time angular correlation functions for two $γ$ rays produced during the electromagnetic cascade transitions in the (n, $γ$) reactions on $^{\rm 155}$Gd and $^{\rm 157}$Gd. As expected, we observe the mild angular correlations for the strong, but rare transitions from the resonance state to the two energy levels of known spin-parities. Contrariwise, we observe negligibly small angular correlations for arbitrary pairs of two $γ$ rays produced in the majority of cascade transitions from the resonance state to the dense continuum states.

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arXiv2022

Unconditional well-posedness for some nonlinear periodic one-dimensional dispersive equations

We consider the Cauchy problem for one-dimensional dispersive equations with a general nonlinearity in the periodic setting. Our main hypotheses are both that the dispersive operator behaves for high frequencies as a Fourier multiplier by $ i |ξ|^αξ$, with $ 1\le α\le 2 $, and that the nonlinear term is of the form $ \partial_x f(u) $ where $ f $ is the sum of an entire series with infinite radius of convergence. Under these conditions, we prove the unconditional local well-posedness of the Cauchy problem in $H^{s}(\mathbb{T})$ for $ s\ge 1-\fracα{2(α+1)}$. This leads to some global existence results above the energy space $ H^{α/2}(\mathbb{T}) $, for $ α\in [\sqrt{2},2]$.

math.AP↗
arXiv2020

On the critical decay for the wave equation with a cubic convolution in 3D

We consider the wave equation with a cubic convolution $\partial_t^2 u-Δu=(|x|^{-γ}*u^2)u$ in three space dimensions. Here, $0<γ<3$ and $*$ stands for the convolution in the space variables. It is well known that if initial data are smooth, small and compactly supported, then $γ\ge2$ assures unique global existence of solutions. On the other hand, it is also well known that solutions blow up in finite time for initial data whose decay rate is not rapid enough even when $2\le γ<3$. In this paper, we consider the Cauchy problem for $2\le γ<3$ in the space-time weighted $L^\infty$ space in which functions have critical decay rate. When $γ=2$, we give an optimal estimate of the lifespan. This gives an affirmative answer to the Kubo conjecture (see Remark right after Theorem 2.1 in Kubo(2004)). When $2<γ<3$, we also prove unique global existence of solutions for small data.

math.AP↗
arXiv2020

Well-posedness for the fourth-order Schrödinger equation with third order derivative nonlinearities

We study the Cauchy problem to the semilinear fourth-order Schrödinger equations: \begin{equation}\label{0-1}\tag{4NLS} \begin{cases} i\partial_t u+\partial_x^4u=G\left(\left\{\partial_x^{k}u\right\}_{k\le γ},\left\{\partial_x^{k}\bar{u}\right\}_{k\le γ}\right), & t>0,\ x\in \mathbb{R}, \\ \ \ \ u|_{t=0}=u_0\in H^s(\mathbb{R}), \end{cases} \end{equation} where $γ\in \{1,2,3\}$ and the unknown function $u=u(t,x)$ is complex valued. In this paper, we consider the nonlinearity $G$ of the polynomial \[ G(z)=G(z_1,\cdots,z_{2(γ+1)}) :=\sum_{m\le |α|\le l}C_αz^α, \] for $z\in \mathbb{C}^{2(γ+1)}$, where $m,l\in\mathbb{N}$ with $3\le m\le l$ and $C_α\in \mathbb{C}$ with $α\in (\mathbb{N}\cup \{0\})^{2(γ+1)}$ is a constant. The purpose of the present paper is to prove well-posedness of the problem (\ref{0-1}) in the lower order Sobolev space $H^s(\mathbb{R})$ or with more general nonlinearities than previous results. Our proof of the main results is based on the contraction mapping principle on a suitable function space employed by D. Pornnopparath (2018). To obtain the key linear and bilinear estimates, we construct a suitable decomposition of the Duhamel term introduced by I. Bejenaru, A. D. Ionescu, C. E. Kenig, and D. Tataru (2011). Moreover we discuss scattering of global solutions and the optimality for the regularity of our well-posedness results, namely we prove that the flow map is not smooth in several cases.

math.AP↗
arXiv2020

Critical exponent for the wave equation with a time-dependent scale invariant damping and a cubic convolution

In the present paper, we study the Cauchy problem for the wave equation with a time-dependent scale invariant damping $\frac{2}{1+t}\partial_t v$ and a cubic convolution $(|x|^{-γ}*v^2)v$ with $γ\in \left(-\frac{1}{2},3\right)$ in three spatial dimension for initial data $\left(v(x,0),\partial_tv(x,0)\right)\in C^2(\mathbb{R}^3)\times C^1(\mathbb{R}^3)$ with a compact support, where $v=v(x,t)$ is an unknown function to the problem on $\mathbb{R}^3\times[0,T)$. Here $T$ denotes a maximal existence time of $v$. The first aim of the present paper is to prove unique global existence of the solution to the problem and asymptotic behavior of the solution in the supercritical case $γ\in (0,3)$, and show a lower estimate of the lifespan in the critical or subcritical case $γ\in \left(-\frac{1}{2},0\right]$. The essential part for their proofs is to derive a weaker estimate under the weaker condition than the case without damping and to recover the weakness by the effect of the dissipative term. The second aim of the present paper is to prove a small data blow-up and the almost sharp upper estimate of the lifespan for positive data with a compact support in the subcritical case $γ\in \left(-\frac{1}{2},0\right)$. The essential part for the proof is to refine the argument for the proof of Theorem 6.1 in \cite{H20} to obtain the upper estimate of the lifespan. Our two results determine that a critical exponent $γ_c$ which divides global existence and blow-up for small solutions is $0$, namely $γ_c=0$. As the result, we can see that the critical exponent shift from $2$ to $0$ due to the effect of the scale invariant damping term.

math.AP↗
arXiv2020

Gamma Ray Spectra from Thermal Neutron Capture on Gadolinium-155 and Natural Gadolinium

Natural gadolinium is widely used for its excellent thermal neutron capture cross section, because of its two major isotopes: $^{\rm 155}$Gd and $^{\rm 157}$Gd. We measured the $γ$-ray spectra produced from the thermal neutron capture on targets comprising a natural gadolinium film and enriched $^{\rm 155}$Gd (in Gd$_{2}$O$_{3}$ powder) in the energy range from 0.11 MeV to 8.0 MeV, using the ANNRI germanium spectrometer at MLF, J-PARC. The freshly analysed data of the $^{\rm 155}$Gd(n, $γ$) reaction are used to improve our previously developed model (ANNRI-Gd model) for the $^{\rm 157}$Gd(n, $γ$) reaction, and its performance confirmed with the independent data from the $^{\rm nat}$Gd(n, $γ$) reaction. This article completes the development of an efficient Monte Carlo model required to simulate and analyse particle interactions involving the thermal neutron captures on gadolinium in any relevant future experiments.

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arXiv2020

Small data blow-up for the wave equation with a time-dependent scale invariant damping and a cubic convolution for slowly decaying initial data

In the present paper, we study the Cauchy problem for the wave equation with a time-dependent scale invariant damping, i.e.$\frac{2}{1+t}\partial_t v$ and a cubic convolution $(|x|^{-γ}*v^2)v$ with $γ\in (0,n)$, where $v=v(x,t)$ is an unknown function on $\mathbb{R}^n\times[0,T)$. Our aim of the present paper is to prove a small data blow-up result and show an upper estimate of lifespan of the problem for slowly decaying positive initial data $(v(x,0),\partial_t v(x,0))$ such as $\partial_t v(x,0)=O(|x|^{-(1+ν)})$ as $|x|\rightarrow\infty$. Here $ν$ belongs to the scaling supercritical case $ν<\frac{n-γ}{2}$. Our main new contribution is to estimate the convolution term in high spatial dimensions, i.e. $n\ge 4$. This paper is the first blow-up result to treat wave equations with the cubic convolution in high spatial dimensions ($n\ge 4$).

math.AP↗
arXiv2019

Local well-posedness for fourth order Benjamin-Ono type equations

We continue to study the local well-posedness for higher order Benjamin-Ono type equations, especially fourth order equations. The proof is based on the energy methods with correction terms. Although one of correction terms can eliminate the highest order derivative loss in the energy inequality, it may yield a lower order derivative loss than the worst term. In order to cancel this derivative loss, we define correction terms inductively.

math.AP↗
arXiv2018

Local well-posedness for third order Benjamin-Ono type equations on the torus

We consider the Cauchy problem of third order Benjamin-Ono type equations on the torus. Nonlinear terms may yield derivative losses, which prevents us from using the classical energy method. In order to overcome that difficulty, we add a correction term into the energy. We also use the Bona-Smith type argument to show the continuous dependence.

math.AP↗
arXiv2018

Gamma Ray Spectrum from Thermal Neutron Capture on Gadolinium-157

We have measured the $γ$-ray energy spectrum from the thermal neutron capture, ${}^{157}$Gd$(n,γ){}^{158}$Gd, on an enriched $^{157}$Gd target (Gd$_{2}$O$_{3}$) in the energy range from 0.11 MeV up to about 8 MeV. The target was placed inside the germanium spectrometer of the ANNRI detector at J-PARC and exposed to a neutron beam from the Japan Spallation Neutron Source (JSNS). Radioactive sources ($^{60}$Co, $^{137}$Cs, and $^{152}$Eu) and the reaction $^{35}$Cl($n$,$γ$) were used to determine the spectrometer's detection efficiency for $γ$ rays at energies from 0.3 to 8.5 MeV. Using a Geant4-based Monte Carlo simulation of the detector and based on our data, we have developed a model to describe the $γ$-ray spectrum from the thermal ${}^{157}$Gd($n$,$γ$) reaction. While we include the strength information of 15 prominent peaks above 5 MeV and associated peaks below 1.6 MeV from our data directly into the model, we rely on the theoretical inputs of nuclear level density and the photon strength function of ${}^{158}$Gd to describe the continuum $γ$-ray spectrum from the ${}^{157}$Gd($n$,$γ$) reaction. Our model combines these two components. The results of the comparison between the observed $γ$-ray spectra from the reaction and the model are reported in detail.

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