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Mamoru Okamoto

Publications and source records attributed to Mamoru Okamoto.

At least 19 recordsLinked to original sources
arXiv2026

Norm inflation for quadratic derivative fractional nonlinear Schr\"odinger equations

We consider the Cauchy problem for quadratic derivative fractional nonlinear Schr\"odinger equations on $\mathbb{R}$ or $\mathbb{T}$. We determine the sharp exponents of the fractional derivatives for which the Cauchy problem is well-posed in the Sobolev space. Thanks to the global well-posedness result established by Nakanishi and Wang (2025), we can expand the solution as a sum of iterated terms. By deriving estimates for each iterated term, we establish norm inflation with infinite loss of regularity, which in particular implies ill-posedness.

math.AP↗
arXiv2025

Well- and Ill-posedness of the Cauchy problem for derivative fractional nonlinear Schrödinger equations on the torus

We consider the Cauchy problem for derivative fractional Schrödinger equations (fNLS) on the torus $\mathbb T$. Recently, the second and third authors established a necessary and sufficient condition on the nonlinearity for well-posedness of semi-linear Schrödinger equations on $\mathbb T$. In this paper, we extend this result to derivative fNLS. More precisely, we prove that the necessary and sufficient condition on the nonlinearity is the same as that for semi-linear Schrödinger equations. However, since we can not employ a gauge transformation for derivative fNLS, we use the modified energy method to prove well-posedness. We need to inductively construct correction terms for the modified energy when the fractional Laplacian is of order between $1$ and $\frac 32$. For the ill-posedness, we prove the non-existence of solutions to the Cauchy problem by exploiting a Cauchy-Riemann-type operator that appears in nonlinear interactions.

math.AP↗
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.

math.AP↗
arXiv2025

Global well-posedness of the energy-critical stochastic nonlinear Schrödinger equation on the three-dimensional torus

We study the Cauchy problem of the defocusing energy-critical stochastic nonlinear Schrödinger equation (SNLS) on the three dimensional torus, forced by an additive noise. We adapt the atomic spaces framework in the context of the energy-critical nonlinear Schrödinger equation, and employ probabilistic perturbation arguments in the context of stochastic PDEs, establishing the global well-posedness of the defocusing energy-critical quintic SNLS in the energy space. It is the first global well-posedness result for the periodic SNLS in a critical space.

math.AP↗
arXiv2025

Well- and ill-posedness of the Cauchy problem for semi-linear Schrödinger equations on the torus

We consider the Cauchy problem for semi-linear Schrödinger equations on the torus $\mathbb T$. We establish a necessary and sufficient condition on the polynomial nonlinearity for the Cauchy problem to be well-posed in the Sobolev space $H^s(\mathbb T)$ for $s>\frac 52$. For the well-posedness, we use the energy estimates and the gauge transformation. For the ill-posedness, we prove the non-existence of solutions to the Cauchy problem.

math.AP↗
arXiv2024

Stochastic quantization of the $Φ^3_3$-model

(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) We study the construction of the $Φ^3_3$-measure and complete the program on the (non-)construction of the focusing Gibbs measures, initiated by Lebowitz, Rose, and Speer (1988). This problem turns out to be critical, exhibiting the following phase transition. In the weakly nonlinear regime, we prove normalizability of the $Φ^3_3$-measure and show that it is singular with respect to the massive Gaussian free field. Moreover, we show that there exists a shifted measure with respect to which the $Φ^3_3$-measure is absolutely continuous. In the strongly nonlinear regime, by further developing the machinery introduced by the authors (2020), we establish non-normalizability of the $Φ^3_3$-measure. Due to the singularity of the $Φ^3_3$-measure with respect to the massive Gaussian free field, this non-normalizability part poses a particular challenge as compared to our previous works. In order to overcome this issue, we first construct a $σ$-finite version of the $Φ^3_3$-measure and show that this measure is not normalizable. Furthermore, we prove that the truncated $Φ^3_3$-measures have no weak limit in a natural space, even up to a subsequence. We also study the dynamical problem. By adapting the paracontrolled approach, in particular from the works by Gubinelli, Koch, and the first author (2018) and by the authors (2020), we prove almost sure global well-posedness of the hyperbolic $Φ^3_3$-model and invariance of the Gibbs measure in the weakly nonlinear regime. In the globalization part, we introduce a new, conceptually simple and straightforward approach, where we directly work with the (truncated) Gibbs measure, using the variational formula and ideas from theory of optimal transport.

math.PR↗
arXiv2024

Focusing $Φ^4_3$-model with a Hartree-type nonlinearity

(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) Lebowitz, Rose, and Speer (1988) initiated the study of focusing Gibbs measures, which was continued by Brydges and Slade (1996), Bourgain (1997, 1999), and Carlen, Fröhlich, and Lebowitz (2016) among others. In this paper, we complete the program on the (non-)construction of the focusing Hartree Gibbs measures in the three-dimensional setting. More precisely, we study a focusing $Φ^4_3$-model with a Hartree-type nonlinearity, where the potential for the Hartree nonlinearity is given by the Bessel potential of order $β$. We first construct the focusing Hartree $Φ^4_3$-measure for $β> 2$, while we prove its non-normalizability for $β< 2$. Furthermore, we establish the following phase transition at the critical value $β= 2$: normalizability in the weakly nonlinear regime and non-normalizability in the strongly nonlinear regime. We then study the canonical stochastic quantization of the focusing Hartree $Φ^4_3$-measure, namely, the three-dimensional stochastic damped nonlinear wave equation (SdNLW) with a cubic nonlinearity of Hartree-type, forced by an additive space-time white noise, and prove almost sure global well-posedness and invariance of the focusing Hartree $Φ^4_3$-measure for $β> 2$ (and $β= 2$ in the weakly nonlinear regime). In view of the non-normalizability result, our almost sure global well-posedness result is sharp. In Appendix, we also discuss the (parabolic) stochastic quantization for the focusing Hartree $Φ^4_3$-measure. We also construct the defocusing Hartree $Φ^4_3$-measure for $β> 0$.

math.PR↗
arXiv2024

A remark on randomization of a general function of negative regularity

In the study of partial differential equations (PDEs) with random initial data and singular stochastic PDEs with random forcing, we typically decompose a classically ill-defined solution map into two steps, where, in the first step, we use stochastic analysis to construct various stochastic objects. The simplest kind of such stochastic objects is the Wick powers of a basic stochastic term (namely a random linear solution, a stochastic convolution, or their sum). In the case of randomized initial data of a general function of negative regularity for studying nonlinear wave equations (NLW), we show necessity of imposing additional Fourier-Lebesgue regularity for constructing Wick powers by exhibiting examples of functions slightly outside $L^2(\mathbb T^d)$ such that the associated Wick powers do not exist. This shows that probabilistic well-posedness theory for NLW with general randomized initial data fails in negative Sobolev spaces (even with renormalization). Similar examples also apply to stochastic NLW and stochastic nonlinear heat equations with general white-in-time stochastic forcing, showing necessity of appropriate Fourier-Lebesgue $γ$-radonifying regularity in the construction of the Wick powers of the associated stochastic convolution.

math.AP↗
arXiv2023

Norm inflation for the viscous nonlinear wave equation

In this article, we study the ill-posedness of the viscous nonlinear wave equation for any polynomial nonlinearity in negative Sobolev spaces. In particular, we prove a norm inflation result above the scaling critical regularity in some cases. We also show failure of $C^k$-continuity, for $k$ the power of the nonlinearity, up to some regularity threshold.

math.AP↗
arXiv2023

Uniqueness and non-uniqueness of the Gaussian free field evolution under the two-dimensional Wick ordered cubic wave equation

We study the nonlinear wave equation (NLW) on the two-dimensional torus $\mathbb T^2$ with Gaussian random initial data on $H^s(\mathbb T^2) \times H^{s-1}(\mathbb T^2)$, $s < 0$, distributed according to the base Gaussian free field $μ$ associated with the invariant Gibbs measure studied by Thomann and the first author (2020). In particular, we investigate the approximation property of the corresponding solution by smooth (random) solutions. Our main results in this paper are two-fold. (i) We show that the solution map for the renormalized cubic NLW defined on the Gaussian free field $μ$ is the unique extension of the solution map defined for smoothed Gaussian initial data obtained by mollification, independent of mollification kernels. (ii) We also show that there is a regularization of the Gaussian initial data so that the corresponding smooth solutions almost surely have no limit in the natural topology. This second result in particular states that one can not use arbitrary smooth approximation for the renormalized cubic NLW dynamics. As a preliminary step for proving (ii), we establish a (deterministic) norm inflation result at general initial data for the (unrenormalized) cubic NLW on $\mathbb T^d$ and $\mathbb R^d$ in negative Sobolev spaces, extending the norm inflation result by Christ, Colliander, and Tao (2003).

math.AP↗
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.

math.AP↗
arXiv2020

Comparing the stochastic nonlinear wave and heat equations: a case study

We study the two-dimensional stochastic nonlinear wave equation (SNLW) and stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional derivative (of order $α> 0$) of a space-time white noise. In particular, we show that the well-posedness theory breaks at $α= \frac 12$ for SNLW and at $α= 1$ for SNLH. This provides a first example showing that SNLW behaves less favorably than SNLH. (i) As for SNLW, Deya (2020) essentially proved its local well-posedness for $0 < α< \frac 12$. We first revisit this argument and establish multilinear smoothing of order $\frac 14$ on the second order stochastic term in the spirit of a recent work by Gubinelli, Koch, and Oh (2018). This allows us to simplify the local well-posedness argument for some range of $α$. On the other hand, when $α\geq \frac 12$, we show that SNLW is ill-posed in the sense that the second order stochastic term is not a continuous function of time with values in spatial distributions. This shows that a standard method such as the Da Prato-Debussche trick or its variant, based on a higher order expansion, breaks down for $α\ge \frac 12$. (ii) As for SNLH, we establish analogous results with a threshold given by $α= 1$. These examples show that in the case of rough noises, the existing well-posedness theory for singular stochastic PDEs breaks down before reaching the critical values ($α= \frac 34$ in the wave case and $α= 2$ in the heat case) predicted by the scaling analysis (due to Deng, Nahmod, and Yue (2019) in the wave case and due to Hairer (2014) in the heat case).

math.AP↗
arXiv2020

A remark on norm inflation for nonlinear wave equations

In this note, we study the ill-posedness of nonlinear wave equations (NLW). Namely, we show that NLW experiences norm inflation at every initial data in negative Sobolev spaces. This result covers a gap left open in a paper of Christ, Colliander, and Tao (2003) and extends the result by Oh, Tzvetkov, and the second author (2019) to non-cubic integer nonlinearities. In particular, for some low dimensional cases, we obtain norm inflation above the scaling critical regularity. We also prove ill-posedness for NLW, via norm inflation at general initial data, in negative regularity Fourier-Lebesgue and Fourier-amalgam spaces.

math.AP↗
arXiv2020

A remark on triviality for the two-dimensional stochastic nonlinear wave equation

We consider the two-dimensional stochastic damped nonlinear wave equation (SdNLW) with the cubic nonlinearity, forced by a space-time white noise. In particular, we investigate the limiting behavior of solutions to SdNLW with regularized noises and establish triviality results in the spirit of the work by Hairer, Ryser, and Weber (2012). More precisely, without renormalization of the nonlinearity, we establish the following two limiting behaviors; (i) in the strong noise regime, we show that solutions to SdNLW with regularized noises tend to 0 as the regularization is removed and (ii) in the weak noise regime, we show that solutions to SdNLW with regularized noises converge to a solution to a deterministic damped nonlinear wave equation with an additional mass term.

math.AP↗
arXiv2020

Two floating camphor particles interacting through lateral capillary force

We consider a mathematical model for a two-particle system driven by the spatial gradient of a concentration field of chemicals with conservative attractive interactions in one dimension. This setup corresponds to an experimental system with floating camphor particles at a water surface. Repulsive interaction is introduced, as well as self-propelling force, through the concentration field of camphor molecules at the water surface. Here we newly adopt the attractive lateral capillary force due to the deformation of the water surface. The particles experience competing dissipative repulsion and conservative attraction. We numerically investigated the mathematical model, and found six different modes of motion. The theoretical approach revealed that some of such mode transitions can be understood in terms of bifurcation.

nlin.PS↗
arXiv2020

On the stochastic nonlinear Schrödinger equations at critical regularities

We consider the Cauchy problem for the defocusing stochastic nonlinear Schrödinger equations (SNLS) with an additive noise in the mass-critical and energy-critical settings. By adapting the probabilistic perturbation argument employed in the context of the random data Cauchy theory by the first author with Bényi and Pocovnicu (2015) to the current stochastic PDE setting, we present a concise argument to establish global well-posedness of the mass-critical and energy-critical SNLS.

math.AP↗