arXiv subjects

Andreia Chapouto

Publications and source records attributed to Andreia Chapouto.

18 recordsLinked to original sources

arXiv2026

Gauge transform for the Korteweg-de Vries equation and well-posedness below the $H^{-1}$-scale

We propose a new formulation of the Korteweg-de Vries equation (KdV) on the real line, via a gauge transform. While KdV and the gauged equation are equivalent for smooth solutions, the latter is better behaved at low regularity in Fourier-Lebesgue spaces. In particular, the admissible regularities go beyond the $H^{-1}$-scale, which is a well-known threshold for KdV. As a byproduct, by reversing the gauge transform, we are able to improve on the known theory for KdV and derive sharp local well-posedness in Fourier-Lebesgue spaces with large integrability exponent. Our strategy is based on an infinite normal form reduction and Fourier restriction estimates, together with a thorough exploitation of algebraic cancellations. Additionally, our method is totally independent of the KdV completely integrable structure, and extends to other non-integrable models with quadratic nonlinearities.

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arXiv2026

Sub-critical well-posedness for the intermediate nonlinear Schrödinger equation on the line

We continue our study of the well-posedness theory for the intermediate nonlinear Schrödinger equation (INLS). Firstly, we prove that INLS is locally well-posed in $H^s (\mathbb{R})$ for any $s>0$. This improves on our previous result of local well-posedness for any $s>\frac 14$, and covers the full scaling-subcritical range for INLS. In particular, we also obtain the local well-posedness for the continuum Calogero-Moser equation without chirality assumption in the full scaling-subcritical range. Our method relies on a gauge transformation, the derivation of a closed system for four auxiliary variables, and nonlinear smoothing estimates, but not on the completely integrable nature of these equations. Secondly, for the integrable models, we prove global well-posedness for $0<s<\frac12$ for initial data with small $L^2$-norm. Moreover, we show that our global well-posedness result applies whenever $L^2$-equicontinuous sets are preserved by the flow, and so the small-data restriction would be removed by an a-priori equicontinuity result. Our argument relies on a novel family of conserved quantities based upon the Lax pair we discovered in our prior work.

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arXiv2026

Nonlinear PDEs with modulated dispersion III: multiplicative noises

We investigate pathwise well-posedness of the stochastic modulated Korteweg-de Vries equation (KdV) on the circle with a multiplicative noise, where a time non-homogeneous modulation acts on the linear dispersion term. (i) In the Young case (= fractional-in-time case with Hurst parameter greater than $\frac 12$), we establish a new regularization-by-noise phenomenon on the stochastic convolution in a pathwise manner, where a gain of spatial regularity becomes (arbitrarily) larger for more irregular modulations. We then prove that, given any $s \in \mathbb R$ and any multiplicative Young noise, however rough it is in space, the stochastic modulated KdV is pathwise locally well-posed in $H^s(\mathbb T)$, provided that the modulation is sufficiently irregular. (ii) In the rough case (= white-in-time case), irregularity of the modulation does not induce any smoothing on the stochastic convolution, and in fact, there is a slight loss in the spatial regularity. In this case, by slightly regularizing the multiplicative noise term, we prove pathwise local well-posedness in $H^s(\mathbb T)$ for any given $s \in \mathbb R$, provided that the noise is sufficiently smooth in space. We achieve these goals by combining (i) the sewing lemma approach to the nonlinear Young integration theory, introduced by Chouk and the second author (2014), and (ii) the pathwise construction of stochastic convolutions as Young or rough integrals via the random tensor estimate and the sewing lemma, introduced by the first, fourth, and fifth authors (2026). In the appendix, we also present an example of regularization by noise for a stochastic modulated Schrödinger equation with a multiplicative Young noise.

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arXiv2026

Fourier restriction norm method adapted to controlled paths: stochastic wave equations

We investigate the pathwise well-posedness issue of the stochastic nonlinear wave equation (SNLW) with a multiplicative noise. While the Ito solution theory (= random field solution theory) was established in the '80s, its pathwise well-posedness has remained a challenging open problem for over forty years. By building a unified framework for the Fourier restriction norm method adapted to the $U^p$- and $V^p$-spaces, due to Koch and Tataru (2007), and the Young/rough integration theory via the sewing lemma and controlled paths due to Gubinelli (2004) along with the random tensor estimate for multiple stochastic integrals with respect to (fractional) Brownian motions, we establish pathwise local well-posedness of SNLW in optimal regularity ranges. In particular, in the one-dimensional case with a white-in-time noise, our result covers the case of an almost space-time white noise, which is optimal within the framework of one-parameter rough paths.

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arXiv2026

Well-posedness for the periodic Intermediate nonlinear Schrödinger equation

We study the well-posedness for the intermediate nonlinear Schrödinger equation (INLS) with periodic boundary conditions. Using a gauge transform, we obtain large data local well-posedness in $H^{s}(\mathbb{T})$ for any $s\geq \frac 12$. We extend this result to global well-posedness under a small $L^2$-norm constraint by exploiting the complete integrability of the continuum Calogero-Moser equation (CCM). We also establish additional results such as the unconditional well-posedness in the energy space and the convergence of solutions to INLS to those of CCM in the infinite-depth limit.

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arXiv2026

On the singular nature of shallow-water convergence of the intermediate long wave equation on the real line

We investigate regularity properties of the solution map for the intermediate long wave equation (ILW) on the real line. More precisely, we study the scaled ILW which was shown to converge to the Korteweg-de Vries equation (KdV) in $L^2(\mathbb R)$ in the shallow-water limit in a recent work by the first, third, and fourth authors with T. Zhao (2025). By decomposing the dynamics into the low frequency part and the residual part, we show that, when the depth parameter is sufficiently small, the solution map for the low frequency part is analytic in $L^2(\mathbb R)$, while the solution map for the residual part fails to be $C^2$. Moreover, we establish shallow-water convergence in $L^2(\mathbb R)$ of the low frequency dynamics to KdV. This explains the mechanism of the regularity gain of the solution map in the shallow-water limit.

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arXiv2025

Shallow-water convergence of the intermediate long wave equation in $L^2$

We continue our study on the convergence issue of the intermediate long wave equation (ILW) on both the real line and the circle. In particular, we establish convergence of the scaled ILW dynamics to that of the Korteweg-de Vries equation (KdV) in the shallow-water limit at the $L^2$-level. Together with the recent work by the first three authors and D. Pilod (2024) on the deep-water convergence in $L^2$, this work completes the well-posedness and convergence study of ILW on both geometries within the $L^2$-framework. Our proof equally applies to both geometries and is based on the following two ingredients: the complete integrability of ILW and the normal form method. More precisely, by making use of the Lax pair structure and the perturbation determinant for ILW, recently introduced by Harrop-Griffths, Killip, and Vişan (2025), we first establish weakly uniform (in small depth parameters) equicontinuity in $L^2$ of solutions to the scaled ILW, providing a control on the high frequency part of solutions. Then, we treat the low frequency part by implementing a perturbative argument based on an infinite iteration of normal form reductions for KdV.

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arXiv2025

On the well-posedness of the intermediate nonlinear Schrödinger equation on the line

We consider a family of intermediate nonlinear Schrödinger equations (INLS) on the real line, which includes the continuum Calogero-Moser models (CCM). We prove that INLS is locally well-posed in $H^{s}(\mathbb{R})$ for any $s>\frac 14$, which improves upon the previous best result of $s>\frac 12$ by de Moura-Pilod (2008). This result is also new in the special case of CCM, as the initial condition is not required to lie in any Hardy space. Our approach is based on a gauge transformation, exploiting the remarkable structure of the nonlinearity together with bilinear Strichartz estimates, which allows to recover some of the derivative loss. This turns out to be sufficient to establish our main results for CCM in the Hardy space. For INLS and CCM outside of the Hardy space, the main difficulty comes from the lack of the Hardy space assumption, which we overcome by implementing a refined decomposition of the solutions, which observes a nonlinear smoothing effect in part of the solution. We also discover a new Lax pair for INLS and use it to establish global well-posedness in $H^{s}(\mathbb{R})$ for any $s>\frac 14$ under the additional assumption of small $L^2$-norm.

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arXiv2025

A continuum of invariant measures for the periodic KdV and mKdV equations

We consider the real-valued defocusing modified Korteweg-de Vries equation (mKdV) on the circle. Based on the complete integrability of mKdV, Killip-Vişan-Zhang (2018) discovered a conserved quantity which they used to prove low regularity a priori bounds for solutions. It has been an open question if this conserved quantity can be used to define invariant measures supported at fractional Sobolev regularities. Motivated by this question, we construct probability measures supported on $H^s(\mathbb{T})$ for $0<s<1/2$ invariant under the mKdV flow. We then use the Miura transform to obtain invariant measures for the Korteweg-de Vries equation, whose supports are rougher than the white noise measure. We also obtain analogous results for the defocusing cubic nonlinear Schrödinger equation. These invariant measures cover the lowest possible regularities for which the flows of these equations are well-posed.

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arXiv2024

Deep-water and shallow-water limits of statistical equilibria for the intermediate long wave equation

We study the construction of invariant measures associated with higher order conservation laws of the intermediate long wave equation (ILW) and their convergence properties in the deep-water and shallow-water limits. By exploiting its complete integrability, we first carry out detailed analysis on the construction of appropriate conservation laws of ILW at the $H^\frac k2$-level for each $k \in \mathbb{N}$, and establish their convergence to those of the Benjamin-Ono equation (BO) in the deep-water limit and to those of the Korteweg-de Vries equation (KdV) in the shallow-water limit. In particular, in the shallow-water limit, we prove rather striking 2-to-1 collapse of the conservation laws of ILW to those of KdV. Such 2-to-1 collapse is novel in the literature and, to our knowledge, this is the first construction of a complete family of shallow-water conservation laws with non-trivial shallow-water limits. We then construct an infinite sequence of generalized Gibbs measures for ILW associated with these conservation laws and prove their convergence to the corresponding (invariant) generalized Gibbs measures for BO and KdV in the respective limits. Finally, for $k \ge 3$, we establish invariance of these measures under ILW dynamics, and also convergence in the respective limits of the ILW dynamics at each equilibrium state to the corresponding invariant dynamics for BO and KdV constructed by Deng, Tzvetkov, and Visciglia (2010-2015) and Zhidkov (1996), respectively. In particular, in the shallow-water limit, we establish 2-to-1 collapse at the level of the generalized Gibbs measures as well as the invariant ILW dynamics. As a byproduct of our analysis, we also prove invariance of the generalized Gibbs measure associated with the $H^2$-conservation law of KdV, which seems to be missing in the literature.

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arXiv2024

Deep-water limit of the intermediate long wave equation in $L^2$

We study the well-posedness issue of the intermediate long wave equation (ILW) on both the real line and the circle. By applying the gauge transform for the Benjamin-Ono equation (BO) and adapting the $L^2$ well-posedness argument for BO by Molinet and the fourth author (2012), we prove global well-posedness of ILW in $L^2$ on both the real line and the circle. In the periodic setting, this provides the first low regularity well-posedness of ILW. We then establish convergence of the ILW dynamics to the BO dynamics in the deep-water limit at the $L^2$-level.

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arXiv2024

Intermediate long wave equation in negative Sobolev spaces

We study the intermediate long wave equation (ILW) in negative Sobolev spaces. In particular, despite the lack of scaling invariance, we identify the regularity $s = -\frac 12$ as the critical regularity for ILW with any depth parameter, by establishing the following two results. (i) By viewing ILW as a perturbation of the Benjamin-Ono equation (BO) and exploiting the complete integrability of BO, we establish a global-in-time a priori bound on the $H^s$-norm of a solution to ILW for $ - \frac 12 < s < 0$. (ii) By making use of explicit solutions, we prove that ILW is ill-posed in $H^s$ for $s < - \frac 12$. Our results apply to both the real line case and the periodic case.

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arXiv2023

Global dynamics for the stochastic nonlinear beam equations on the four-dimensional torus

We study global-in-time dynamics of the stochastic nonlinear beam equations (SNLB) with an additive space-time white noise, posed on the four-dimensional torus. The roughness of the noise leads us to introducing a time-dependent renormalization, after which we show that SNLB is pathwise locally well-posed in all subcritical and most of the critical regimes. For the (renormalized) defocusing cubic SNLB, we establish pathwise global well-posedness below the energy space, by adapting a hybrid argument of Gubinelli- Koch-Oh-Tolomeo (2022) that combines the $I$-method with a Gronwall-type argument. Lastly, we show almost sure global well-posedness and invariance of the Gibbs measure for the stochastic damped nonlinear beam equations in the defocusing case.

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arXiv2022

Bounded solutions of KdV: uniqueness and the loss of almost periodicity

We address two pressing questions in the theory of the Korteweg--de Vries (KdV) equation. First, we show the uniqueness of solutions to KdV that are merely bounded, without any further decay, regularity, periodicity, or almost periodicity assumptions. The second question, emphasized by Deift, regards whether almost periodic initial data leads to almost periodic solutions to KdV. Building on the new observation that this is false for the Airy equation, we construct an example of almost periodic initial data whose KdV evolution remains bounded, but fails to be almost periodic at a later time. Our uniqueness result ensures that the solution constructed is the unique development of this initial data.

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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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arXiv2020

A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces

We study the complex-valued modified Korteweg-de Vries equation (mKdV) on the circle. We first consider the real-valued setting and show global well-posedness of the (usual) renormalized mKdV equation in the Fourier-Lebesgue spaces. In the complex-valued setting, we observe that the momentum plays an important role in the well-posedness theory. In particular, we prove that the complex-valued mKdV equation is ill-posed in the sense of non-existence of solutions when the momentum is infinite, in the spirit of the work on the nonlinear Schrödinger equation by Guo-Oh (2018). This non-existence result motivates the introduction of the second renormalized mKdV equation, which we propose as the correct model in the complex-valued setting outside of $H^\frac12(\mathbb{T})$. Furthermore, imposing a new notion of finite momentum for the initial data, at low regularity, we show existence of solutions to the complex-valued mKdV equation. In particular, we require an energy estimate, from which conservation of momentum follows.

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arXiv2020

A refined well-posedness result for the modified KdV equation in the Fourier-Lebesgue spaces

We study the well-posedness of the complex-valued modified Korteweg-de Vries equation (mKdV) on the circle at low regularity. In our previous work (2019), we introduced the second renormalized mKdV equation, based on the conservation of momentum, which we proposed as the correct model to study the complex-valued mKdV outside of $H^\frac12(\mathbb{T})$. Here, we employ the method introduced by Deng-Nahmod-Yue (2019) to prove local well-posedness of the second renormalized mKdV equation in the Fourier-Lebesgue spaces $\mathcal{F}L^{s,p}(\mathbb{T})$ for $s\geq \frac12$ and $1\leq p <\infty$. As a byproduct of this well-posedness result, we show ill-posedness of the complex-valued mKdV without the second renormalization for initial data in these Fourier-Lebesgue spaces with infinite momentum.

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arXiv2015

Existence of positive solutions for an approximation of stationary mean-field games

Here, we consider a regularized mean-field game model that features a low-order regularization. We prove the existence of solutions with positive density. To do so, we combine a priori estimates with the continuation method. In contrast with high-order regularizations, the low-order regularizations are easier to implement numerically. Moreover, our methods give a theoretical foundation for this approach.

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