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Reika Fukuizumi

Publications and source records attributed to Reika Fukuizumi.

17 recordsLinked to original sources
arXiv2026

Ergodicity for a Constantin-Lax-Majda-DeGregorio model of turbulent flow

This paper presents a mathematical analysis of a one-dimensional model of turbulence based on a stochastic generalized Constantin-Lax-Majda-DeGregorio (gCLMG) equation. We focus on the specific case where the nonlinearity in the equation allows the existence of the anomalous enstrophy cascade, which is an inviscid conserved quantity, and some effective energy estimates for mathematical analysis. The existence of an invariant measure in the attractor is proved via the classical Krylov-Bogoliubov argument. The uniqueness of the measure and exponential mixing are proved under a sufficiently large viscosity condition, in which the nonlocal structure of the nonlinear term plays a prominent role. The construction of this invariant measure is the first step towards a theoretical understanding of turbulent phenomena that cause anomalous cascades in the zero viscous limit, viewed from the dynamical systems theory.

math.AP↗
arXiv2026

On the parabolic $Φ_3^4$ model for the harmonic oscillator II: global existence and invariant measures

We establish an a priori bound for the dynamical parabolic $Φ_3^4$ model with harmonic potential. This bound yields the global well-posedness of the equation and, via the Krylov-Bogoliubov method, the existence of an invariant measure, shown to be non-Gaussian. The argument builds on the strategy developed by Mourrat and Weber for the periodic $Φ_3^4$ model, with substantial modifications to handle the non-compact geometry of $\mathbb{R}^3$ and the spectral framework imposed by the harmonic oscillator. We further prove that this measure is unique in the small-coupling regime.

math.AP↗
arXiv2026

An Effective SPDE Model for a Schrödinger Equation with a Fluctuating Magnetic Potential

We consider the Schrödinger equation for a charged particle in a randomly fluctuating external magnetic field and establish a singular perturbation limit in which the solution converges to a deterministic Schrödinger equation with Gaussian fluctuations. We propose a stochastic Schrödinger equation incorporating the behavior of both the deterministic average and the fluctuations and show that it has the same limiting behavior as the original model, while providing a simpler way to study the distribution of relevant observables.

math.PR↗
arXiv2026

Besov spaces associated with the Harmonic oscillator

The Besov space associated with the harmonic oscillator is introduced and thoroughly explored in this paper. It provides a comprehensive summary of the fundamental concepts of the Besov spaces, their embedding properties, bilinear estimates, and related topics.

math.AP↗
arXiv2025

Renormalization of a 1d quadratic Schr{ö}dinger model with additive noise

The study is devoted to the interpretation and wellposedness of the stochastic NLS model \begin{equation*} (\imath \partial_t-Δ)u=|u|^2+\dot{B}, \quad u_0=0,\quad \quad t\in \mathbb{R}, \ x\in \mathbb{T}, \end{equation*} where $\dot{B}$ stands for a space-time fractional noise with index $H=(H_0,H_1)$ in a subset of $(0,1)^{2}$. We first establish that in the situation where $0<2H_0+H_1\leq 2$, the equation cannot be interpreted in a (classical) functional sense.\\ \indent Our investigations then focus on the rough regime corresponding to the condition $\frac74<2H_0+H_1\leq 2$. In this specific case, we exhibit an \textit{explicit} renormalization procedure allowing to restore the (local) convergence of the approximated solutions. We follow a pathwise-type approach emphasizing the distinction between the stochastic objects at the core of the dynamics and the general deterministic machinery.

math.AP↗
arXiv2025

On the parabolic $Φ_3^4$ model for the harmonic oscillator: diagrams and local existence

We prove the local wellposedness of the (renormalized) parabolic $Φ^4_3$ model associated with the harmonic oscillator on $\mathbb{R}^3$, that is, the equation formally written as \begin{equation*} \partial_t X + HX= -X^3+\infty\cdot X + ξ, \quad t>0, \quad x \in \mathbb{R}^3, \end{equation*} where $H:=-Δ_{\mathbb{R}^3} +|x|^2$ and $ξ$ denotes a space-time white noise. This model is closely related to the Gross-Pitaevskii equation which is used in the description of Bose-Einstein condensation. Our overall formulation of the problem, based on the so-called paracontrolled calculus, follows the strategy outlined by Mourrat and Weber for the $Φ^4_3$ model on the three-dimensional torus. Significant effort is then required to adapt, within the framework imposed by the harmonic oscillator, the key tools that contribute to the success of this method-particularly the construction of stochastic diagrams at the core of the dynamics.

math.PR↗
arXiv2023

The stochastic fast logarithmic equation in $\mathbb{R}^{d}$ with multiplicative Stratonovich noise

This paper is concerned with the existence and uniqueness of the solution for the stochastic fast logarithmic equation with Stratonovich multiplicative noise in $\mathbb{R}^{d}$ for $d\geqslant 3$. It provides an answer to a critical case (morally speaking, corresponding to the porous media operator $ΔX^m$ for $m=0$) left as an open problem in the paper Barbu-Röckner-Russo (Journal de Mathématiques Pures et Appliquées,103(4):1024--1052, 2015). We face several technical difficulties related both to the degeneracy properties of the logarithm and to the fact that the problem is treated in an unbounded domain. Firstly, the order in which the approximations are considered is very important and different from previous methods. Secondly, the energy estimates needed in the last step can only be achieved with a well-chosen Stratonovich-type rectification of the noise.

math.PR↗
arXiv2022

Pattern formation in 2d stochastic anisotropic Swift-Hohenberg equation

In this paper, we study a phenomenological model for pattern formation in electroconvection, and the effect of noise on the pattern. As such model we consider an anisotropic Swift-Hohenberg equation adding an additive noise. We prove the existence of a global solution of that equation on the two dimensional torus. In addition, inserting a scaling parameter, we consider the equation on a large domain near its change of stability. We observe numerically that, under the appropriate scaling, its solutions can be approximated by a periodic wave, which is modulated by the solutions to a stochastic Ginzburg-Landau equation.

math.AP↗
arXiv2022

Non relativistic and ultra relativistic limits in 2d stochastic nonlinear damped Klein-Gordon equation

We study the non relativistic and ultra relativistic limits in the two-dimensional nonlinear damped Klein-Gordon equation driven by a space-time white noise on the torus. In order to take the limits, it is crucial to clarify the parameter dependence in the estimates of solution. In this paper we present two methods to confirm this parameter dependence. One is the classical, simple energy method. Another is the method via Strichartz estimates.

math.AP↗
arXiv2021

Stochastic Schrödinger-Lohe model

The Schrödinger-Lohe model consists of wave functions interacting with each other, according to a system of Schrödinger equations with a specific coupling such that all wave functions evolve on the $L^2$ unit ball. This model has been extensively studied over the last decade and it was shown that under suitable assumptions on the initial state, if one waits long enough all the wave functions become arbitrarily close to each other, which we call a synchronization. In this paper, we consider a stochastic perturbation of the Schrödinger-Lohe model and show a weak version of synchronization for this perturbed model in the case of two oscillators.

math.PR↗
arXiv2021

Two dimensional Gross-Pitaevskii equation with space-time white noise

In this paper we consider the two-dimensional stochastic Gross-Pitaevskii equation, which is a model to describe Bose-Einstein condensation at positive temperature. The equation is a complex Ginzburg-Landau equation with a harmonic potential and an additive space-time white noise. We study the well-posedness of the model using an inhomogeneous Wick renormalization due to the potential, and prove the existence of an invariant measure and of stationary martingale solutions.

math.PR↗
arXiv2019

Scattering for the $L^2$ supercritical point NLS

We consider the 1D nonlinear Schrödinger equation with focusing point nonlinearity. "Point" means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This equation is used to model a Kerr-type medium with a narrow strip in the optic fibre. There are several mathematical studies on this equation and the local/global existence of solution, blow-up occurrence and blow-up profile have been investigated. In this paper we focus on the asymptotic behavior of the global solution, i.e, we show that the global solution scatters as t tends to minus/plus infinity in the $L^2$ supercritical case. The main argument we use is due to Kenig-Merle, but it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation.

math.AP↗
arXiv2017

Long time behavior of Gross-Pitaevskii equation at positive temperature

The stochastic Gross-Pitaevskii equation is used as a model to describe Bose-Einstein condensation at positive temperature. The equation is a complex Ginzburg Landau equation with a trapping potential and an additive space-time white noise. Two important questions for this system are the global existence of solutions in the support of the Gibbs measure, and the convergence of those solutions to the equilibrium for large time. In this paper, we give a proof of these two results in one space dimension. In order to prove the convergence to equilibrium, we use the associated purely dissipative equation as an auxiliary equation, for which the convergence may be obtained using standard techniques. Global existence is obtained for all initial data, and not almost surely with respect to the invariant measure.

math.AP↗
arXiv2012

Derivation of the Bose-Hubbard model in the multiscale limit

In this paper we consider a one-dimensional non-linear Schroedinger equation (NLSE) with a periodic potential. In the semiclassical limit we prove that the stationary solutions of the Bose-Hubbard equation approximate the stationary solutions of the (NLSE). In particular, in the limit of large nonlinearity strength the stationary solutions turn out to be localized on a single lattice site of the periodic potential; as a result the phase transition from superfluid to Mott-insulator phase for Bose-Einstein condensates in a one-dimensional periodic lattice is rigorously proved.

math-ph↗
arXiv2011

Bifurcation and stability for Nonlinear Schroedinger equations with double well potential in the semiclassical limit

We consider the stationary solutions for a class of Schroedinger equations with a symmetric double-well potential and a nonlinear perturbation. Here, in the semiclassical limit we prove that the reduction to a finite-mode approximation give the stationary solutions, up to an exponentially small term, and that symmetry-breaking bifurcation occurs at a given value for the strength of the nonlinear term. The kind of bifurcation picture only depends on the non-linearity power. We then discuss the stability/instability properties of each branch of the stationary solutions. Finally, we consider an explicit one-dimensional toy model where the double well potential is given by means of a couple of attractive Dirac's delta pointwise interactions.

math.AP↗
arXiv2007

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

We study analytically and numerically the stability of the standing waves for a nonlinear Schrödinger equation with a point defect and a power type nonlinearity. A main difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves, and it is overcome by a perturbation method and continuation arguments. Among others, in the case of a repulsive defect, we show that the standing wave solution is stable in $\hurad$ and unstable in $\hu$ under subcritical nonlinearity. Further we investigate the nature of instability: under critical or supercritical nonlinear interaction, we prove the instability by blowup in the repulsive case by showing a virial theorem and using a minimization method involving two constraints. In the subcritical radial case, unstable bound states cannot collapse, but rather narrow down until they reach the stable regime (a {\em finite-width instability}). In the non-radial repulsive case, all bound states are unstable, and the instability is manifested by a lateral drift away from the defect, sometimes in combination with a finite-width instability or a blowup instability.

nlin.PS↗