arXiv subjects

Justin Forlano

Publications and source records attributed to Justin Forlano.

At least 19 recordsLinked to original sources

arXiv2026

Sub-critical well-posedness for the intermediate nonlinear Schr\"{o}dinger equation on the line

We continue our study of the well-posedness theory for the intermediate nonlinear Schr\"{o}dinger 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.

math.AP↗

arXiv2026

Local and global well-posedness for the extended Schr\"{o}dinger-Benjamin-Ono system

We study the well-posedness problem for the extended Schr\"{o}dinger-Benjamin-Ono system (eSBO) on the real line. This system couples a Schr\"{o}dinger field $u$ with a Benjamin-Ono type field $v$, including a term of the form $\partial_{x}(v^2)$. This latter term, just as in the case of the Benjamin-Ono equation, causes the system to become quasilinear and unsolvable via Picard iteration. We prove that eSBO is locally well-posed in $H^{s+\frac 12}(\mathbb{R})\times H^{s}(\mathbb{R})$ for any $s\geq 0$. In particular, this result covers the energy space at $s=\frac 12$, yielding global well-posedness in $H^{1}(\mathbb{R})\times H^{\frac 12}(\mathbb{R})$ with a small $L^2$-assumption on the Schr\"{o}dinger part of the initial data.

math.AP↗

arXiv2026

Well-posedness for the periodic Intermediate nonlinear Schr\"{o}dinger equation

We study the well-posedness for the intermediate nonlinear Schr\"{o}dinger 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.

math.AP↗

arXiv2026

Invariant Gibbs dynamics for the hyperbolic sinh-Gordon model

We study the hyperbolic defocusing sinh-Gordon model with parameter $\beta^2>0$ and its associated Gibbs dynamics on the two-dimensional torus. We establish global well-posedness of the model for a certain range of parameters $\beta^2>0$ with the corresponding Gibbs measure initial data and prove invariance of the Gibbs measure under the flow, thereby resolving a question posed by Oh, Robert, and Wang (2019). Our physical space approach hinges on developing a novel $L^\infty$-based well-posedness theory for wave equations with exponential-type nonlinearities, going beyond the classical $L^2$-based framework. This refinement allows us to fully leverage structural properties of Gaussian multiplicative chaos. As a by-product of our method, we also obtain an improved well-posedness theory for the hyperbolic Liouville model.

math.AP↗

arXiv2025

On the well-posedness of the intermediate nonlinear Schr\"{o}dinger equation on the line

We consider a family of intermediate nonlinear Schr\"{o}dinger 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.

math.AP↗

arXiv2025

Invariant Gibbs dynamics for the nonlinear Schr\"odinger equations on the disc

We consider the two-dimensional defocusing nonlinear Schr\"odinger equation (NLS) on the unit disc in the plane with the Gibbs initial data under radial symmetry. By using a type of random averaging operator ansatz, we build a strong local-in-time solution theory, and thus prove almost sure global well-posedness and invariance of the Gibbs measure via Bourgain's invariant measure argument. This work completes the program initiated by Tzvetkov (2006, 2008) on the construction of invariant Gibbs dynamics (of strong solutions) for NLS on the disc.

math.AP↗

arXiv2025

Improved quasi-invariance result for the periodic Benjamin-Ono-BBM equation

We extend recent results of Genovese-Luca-Tzvetkov (2022) regarding the quasi-invariance of Gaussian measures under the flow of the periodic Benjamin-Ono-BBM (BO-BBM) equation to the full range where BO-BBM is globally well-posed. The main difficulty is due to the critical nature of the dispersion which we overcome by combining the approach of Coe-Tolomeo (2024) with an iteration argument due to Forlano-Tolomeo (2024) to obtain long-time higher integrability bounds on the transported density.

math.AP↗

arXiv2024

Quasi-invariance of the Gaussian measure for the two-dimensional stochastic cubic nonlinear wave equation

We consider the stochastic damped nonlinear wave equation $\partial_t^{2}u+\partial_t u+u-\Delta u +u^{3} = \sqrt{2} {\langle{\nabla}\rangle^{-s}} \xi$ on the two-dimensional torus $\mathbb T^2$, where $\xi$ denotes a space-time white noise and $s>0$. We show that the measure $\vec{\mu}_s$ corresponding to the unique invariant measure for the flow of the associated linear equation is quasi-invariant under the nonlinear stochastic flow.

math.PR↗

arXiv2024

Unconditional deep-water limit of the intermediate long wave equation in low-regularity

In this paper, we establish the unconditional deep-water limit of the intermediate long wave equation (ILW) to the Benjamin-Ono equation (BO) in low-regularity Sobolev spaces on both the real line and the circle. Our main tool is new unconditional uniqueness results for ILW in $H^s$ when $s_0<s\leq \frac 14$ on the line and $s_0<s< \frac 12$ on the circle, where $s_0 = 3-\sqrt{33/4}\approx 0.1277$. Here, we adapt the strategy of Mo\c{s}incat-Pilod (2023) for BO to the setting of ILW by viewing ILW as a perturbation of BO and making use of the smoothing property of the perturbation term.

math.AP↗

arXiv2024

Invariant measures for mKdV and KdV in infinite volume

We construct dynamics for the defocusing real-valued (Miura) mKdV equation on the real line with initial data distributed according to Gibbs measure. We also prove that Gibbs measure is invariant under these dynamics. On the way, we provide a new proof of the invariance of the Gibbs measure under mKdV on the torus. Building on these results, we construct new measure-preserving dynamics for the KdV equation on the whole real line. Samples from this family of measures exhibit the same local regularity as white-noise, but completely different statistics!

math.AP↗

arXiv2023

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.

math.AP↗

arXiv2023

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-Vi\c{s}an-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 Schr\"{o}dinger equation. These invariant measures cover the lowest possible regularities for which the flows of these equations are well-posed.

math.AP↗

arXiv2022

A remark on the well-posedness of the modified KdV equation in $L^2$

We study the real-valued modified KdV equation on the real line and the circle, in both the focusing and the defocusing case. By employing the method of commuting flows introduced by Killip and Vi\c{s}an (2019), we prove global well-posedness in $H^{s}$ for $0\leq s<\tfrac{1}{2}$. On the line, we show how the arguments in the recent paper by Harrop-Griffiths, Killip, and Vi\c{s}an (2020) may be simplified in the higher regularity regime $s\geq 0$. On the circle, we provide an alternative proof of the sharp global well-posedness in $L^2$ due to Kappeler and Topalov (2005), and also extend this to the large-data focusing case.

math.AP↗

arXiv2022

Quasi-invariance of Gaussian measures of negative regularity for fractional nonlinear Schr\"odinger equations

We consider the Cauchy problem for the fractional nonlinear Schr\"{o}dinger equation (FNLS) on the one-dimensional torus with cubic nonlinearity and high dispersion parameter $\alpha > 1$, subject to a Gaussian random initial data of negative Sobolev regularity $\sigma<s-\frac{1}{2}$, for $s \le \frac 12$. We show that for all $s_{*}(\alpha) <s\leq \frac{1}{2}$, the equation is almost surely globally well-posed. Moreover, the associated Gaussian measure supported on $H^{s}(\mathbb T)$ is quasi-invariant under the flow of the equation. For $\alpha < \frac{1}{20}(17 + 3\sqrt{21}) \approx 1.537$, the regularity of the initial data is lower than the one provided by the deterministic well-posedness theory. We obtain this result by following the approach of DiPerna-Lions (1989); first showing global-in-time bounds for the solution of the infinite-dimensional Liouville equation for the transport of the Gaussian measure, and then transferring these bounds to the solution of the equation by adapting Bourgain's invariant measure argument to the quasi-invariance setting. This allows us to bootstrap almost sure global bounds for the solution of (FNLS) from its probabilistic local well-posedness theory.

math.AP↗

arXiv2021

Transport of Gaussian measures under the flow of one-dimensional fractional nonlinear Schr\"{o}dinger equations

We study the transport property of Gaussian measures on Sobolev spaces of periodic functions under the dynamics of the one-dimensional cubic fractional nonlinear Schr\"{o}dinger equation. For the case of second-order dispersion or greater, we establish an optimal regularity result for the quasi-invariance of these Gaussian measures, following the approach by Debussche and Tsutsumi [15]. Moreover, we obtain an explicit formula for the Radon-Nikodym derivative and, as a corollary, a formula for the two-point function arising in wave turbulence theory. We also obtain improved regularity results in the weakly dispersive case, extending those in [20]. Our proof combines the approach introduced by Planchon, Tzvetkov and Visciglia [47] and that of Debussche and Tsutsumi [15].

math.AP↗

arXiv2021

On the unique ergodicity for a class of 2 dimensional stochastic wave equations

We study the global-in-time dynamics for a stochastic semilinear wave equation with cubic defocusing nonlinearity and additive noise, posed on the $2$-dimensional torus. The noise is taken to be slightly more regular than space-time white noise. In this setting, we show existence and uniqueness of an invariant measure for the Markov semigroup generated by the flow over an appropriately chosen Banach space. This extends a result of the second author to a situation where the invariant measure is not explicitly known.

math.AP↗

arXiv2019

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↗

arXiv2019

Almost sure global well posedness for the BBM equation with infinite $L^{2}$ initial data

We consider the probabilistic Cauchy problem for the Benjamin-Bona-Mahony equation (BBM) on the one-dimensional torus $\mathbb{T}$ with initial data below $L^{2}(\mathbb{T})$. With respect to random initial data of strictly negative Sobolev regularity, we prove that BBM is almost surely globally well-posed. The argument employs the $I$-method to obtain an a priori bound on the growth of the `residual' part of the solution. We then discuss the stability properties of the solution map in the deterministically ill-posed regime.

math.AP↗