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Gigliola Staffilani

Publications and source records attributed to Gigliola Staffilani.

At least 37 records · Page 2Linked to original sources
arXiv2025

Formation of condensations for non-radial solutions to 3-wave kinetic equations

We consider in this work a $2$-dimensional $3$-wave kinetic equation describing the dynamics of the thermal cloud outside a Bose-Einstein Condensate. We construct global non-radial mild solutions for the equation. Those mild solutions are the summation of Dirac masses on circles. We prove that in each spatial direction, either Dirac masses at the origin, which are the so-called Bose-Einstein condensates, can be formed in finite time or the solutions converge to Bose-Einstein condensates as time evolves to infinity. We also describe a dynamics of the formation of the Bose-Einstein condensates latter case. In this case, on each direction, the solutions accumulate around circles close to the origin at growth rates at least linearly in time.

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arXiv2025

Well-posedness and invariant measures for complex valued modified KdV equation

We consider the one dimensional periodic complex valued mKdV, which corresponds to the first equation above cubic NLS in the associated integrable hierarchy. Our main result is the construction of a sequence of invariant measures supported on Sobolev spaces with increasing regularity. The fact that we work with complex valued functions makes the analysis of the invariance much harder compared to the real valued case, that can be handled instead following the ideas used by Zhidkov [73].

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arXiv2024

Well-Posedness and regularity properties of 2d $β$-plane stochastic Navier-Stokes equations in a periodic channel

We consider the 2d $β$-plane stochastic Navier-Stokes equations in a periodic channel. We prove the well-posedness and existence of the stationary measure, as well as certain regularity estimates concerning the support of the stationary measure. The mentioned estimates are crucial for the rigorous study of the cascade phenomena in this equation [8]. To the best of our knowledge, this is the first mathematically rigorous treatment of these equations involving both the stochastic noise and the Coriolis force.

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arXiv2024

On the energy transfer towards large values of wavenumbers for solutions of 4-wave kinetic equations

Inspired by the fundamental work of Escobedo and Velazquez \cite{EscobedoVelazquez:2015:FTB,EscobedoVelazquez:2015:OTT}, we prove that solutions of 4-wave kinetic equations, under very general forms of the dispersion relations, exhibit the transfer of energy towards large values of wavenumbers as time evolves. We also provide a global well-posedness theory for mild solutions of the equation under general forms of the dispersion relations.

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arXiv2024

Condensation and non-condensation times for 4-wave kinetic equations

Inspired by the pioneering work of Escobedo and Velazquez \cite{EscobedoVelazquez:2015:FTB,EscobedoVelazquez:2015:OTT}, we prove that solutions of 4-wave kinetic equations, under very general forms of dispersion relations, develop condensation at the origin in finite time, under weaker conditions on the initial data than the ones considered in \cite{EscobedoVelazquez:2015:FTB,EscobedoVelazquez:2015:OTT}. We also provide some estimates on the non-condensation times of the solutions.

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arXiv2024

Energy transfer for solutions to the nonlinear Schrödinger equation on irrational tori

We analyze the energy transfer for solutions to the defocusing cubic nonlinear Schrödinger (NLS) initial value problem on 2D irrational tori. Moreover we complement the analytic study with numerical experimentation. As a biproduct of our investigation we also prove that the quasi-resonant part of the NLS initial value problem we consider, in both the focusing and defocusing case, is globally well-posed for initial data of finite mass.

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arXiv2024

On the wave turbulence theory for a stochastic KdV type equation

Starting from the stochastic Zakharov-Kuznetsov equation, a multidimensional KdV type equation on a hypercubic lattice, we provide a derivation of the 3-wave kinetic equation. We show that the two point correlation function can be asymptotically expressed as the solution of the 3-wave kinetic equation at the kinetic limit under very general assumptions: the initial condition is out of equilibrium, the dimension is $d\ge 2$, the smallness of the nonlinearity $λ$ is allowed to be independent of the size of the lattice, the weak noise is chosen not to compete with the weak nonlinearity and not to inject energy into the equation. Unlike the cubic nonlinear Schrödinger equation, for which such a general result is commonly expected without the noise, the kinetic description of the deterministic lattice ZK equation is unlikely to happen. One of the key reasons is that the dispersion relation of the lattice ZK equation leads to a singular manifold, on which not only $3$-wave interactions but also all $m$-wave interactions ($m\ge3$) are allowed to happen. This phenomenon has been first observed by Lukkarinen \cite{lukkarinen2007asymptotics} as a counterexample for which one of the main tools to derive kinetic equations from wave equations (the suppression of crossings) fails to hold true. To the best of our knowledge, the work provides the first rigorous derivation of nonlinear 3-wave kinetic equations. Also this is the first derivation for wave kinetic equations in the lattice setting and out-of-equilibrium.

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arXiv2023

The wave maps equation and Brownian paths

We discuss the $(1+1)$-dimensional wave maps equation with values in a compact Riemannian manifold $\mathcal{M}$. Motivated by the Gibbs measure problem, we consider Brownian paths on the manifold $\mathcal{M}$ as initial data. Our main theorem is the probabilistic local well-posedness of the associated initial value problem. The analysis in this setting combines analytic, geometric, and probabilistic methods.

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arXiv2023

Trend to equilibrium for flows with random diffusion

Motivated by the possibility of noise to cure equations of finite-time blowup, recent work arXiv:2109.09892 by the second and third named authors showed that with quantifiable high probability, random diffusion restores global existence for a large class of active scalar equations in arbitrary dimension with possibly singular velocity fields. This class includes Hamiltonian flows, such as the SQG equation and its generalizations, and gradient flows, such as the Patlak-Keller-Segel equation. A question left open is the asymptotic behavior of the solutions, in particular, whether they converge to a steady state. We answer this question by showing that the solutions from arXiv:2109.09892 in the periodic setting converge in Gevrey norm exponentially fast to the uniform distribution as time $t\rightarrow\infty$.

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arXiv2023

On the wave turbulence theory for the beam wave equation

Starting from the beam wave equation, which has a Schrödinger structure, on a hypercubic lattice of size $L$, with weak nonlinearity of strength $λ$, we show that the two point correlation function can be asymptotically expressed as the solution of the 3-wave kinetic equation at the kinetic limit under very general assumptions. The limit is taken in the physical order: first taking $L\to\infty$, then letting $λ\to0$. The initial condition is assumed out of equilibrium and the physical dimension is $d= 3$. The combination of previous works on the derivations of the wave kinetic equations from the lattice ZK equation [42,74] (with stochasticity, due to the singularity of the lattice ZK dispersion relation) and the present work (without stochasticity, thanks to the Schrödinger structure) give a detailed and general picture of how sensitive the kinetic descriptions are with respect to the regularities of the dispersion relations in the physical order of taking the limits. In terms of technical novelties, the current paper introduces new tools that produce the first obtained control of the Duhamel expansions when the number of collisions is of order $\mathcal{O}(λ^{-γ})$ (with $γ>0$) in the nonlinear context. To the best of our knowledge, such a strong control is a major step in understanding the long time dynamics of weakly nonlinear interactions of waves, and until now it could only be achieved in the linear context (see the work of Erdos-Salmhofer-Yau [34]).

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arXiv2022

On the wave turbulence theory for a stochastic KdV type equation -- Generalization for the inhomogeneous kinetic limit

Starting from a stochastic Zakharov-Kuznetsov (ZK) equation on a lattice, the previous work [ST21] by the last two authors gave a derivation of the homogeneous 3-wave kinetic equation at the kinetic limit under very general assumptions: the initial condition is out of equilibrium, the dimension $d\ge 2$, the smallness of the nonlinearity $λ$ is allowed to be independent of the size of the lattice, the weak noise is chosen not to compete with the weak nonlinearity and not to inject energy into the equation. In the present work, we build on the framework of [ST21], following the formal derivation of Spohn [Spo06] and inspired by previous work [HO21] of the first author and Olla, so that the inhomogeneous 3-wave kinetic equation can also be obtained at the kinetic limit under analogous assumptions. Similar to the homogeneous case -- and unlike the cubic nonlinear Schrödinger equation -- the inhomogeneous kinetic description of the deterministic lattice ZK equation is unlikely to happen due to the vanishing of the dispersion relation on a certain singular manifold on which not only $3$-wave interactions but also all $n$-wave interactions ($n\ge3$) are allowed to happen, a phenomenon first observed by Lukkarinen [Luk07]. To the best of our knowledge, our work provides the first rigorous derivation of a nonlinear inhomogeneous wave kinetic equation in the kinetic limit.

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arXiv2022

On decaying properties of nonlinear Schrödinger equations

In this paper we discuss quantitative (pointwise) decay estimates for solutions to the 3D cubic defocusing Nonlinear Schrödinger equation with various initial data, deterministic and random. We show that nonlinear solutions enjoy the same decay rate as the linear ones. The regularity assumption on the initial data is much lower than in previous results (see \cite{fan2021decay} and the references therein) and moreover we quantify the decay, which is another novelty of this work. Furthermore, we show that the (physical) randomization of the initial data can be used to replace the $L^1$-data assumption (see \cite{fan2022note} for the necessity of the $L^1$-data assumption). At last, we note that this method can be also applied to derive decay estimates for other nonlinear dispersive equations.

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arXiv2022

A rigorous derivation of the Hamiltonian structure for the Vlasov equation

We consider the Vlasov equation in any spatial dimension, which has long been known to be an infinite-dimensional Hamiltonian system whose bracket structure is of Lie-Poisson type. In parallel, it is classical that the Vlasov equation is a mean-field limit for a pairwise interacting Newtonian system. Motivated by this knowledge, we provide a rigorous derivation of the Hamiltonian structure of the Vlasov equation, both the Hamiltonian functional and Poisson bracket, directly from the many-body problem. One may view this work as a classical counterpart to arXiv:1908.03847, which provided a rigorous derivation of the Hamiltonian structure of the cubic nonlinear Schrödinger equation from the many-body problem for interacting bosons in a certain infinite particle number limit, the first result of its kind. In particular, our work settles a question of Marsden, Morrison, and Weinstein on providing a "statistical basis" for the bracket structure of the Vlasov equation.

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arXiv2021

Large deviations principle for the cubic NLS equation

In this paper, we present a probabilistic study of rare phenomena of the cubic nonlinear Schrödinger equation on the torus in a weakly nonlinear setting. This equation has been used as a model to numerically study the formation of rogue waves in deep sea. Our results are twofold: first, we introduce a notion of criticality and prove a Large Deviations Principle (LDP) for the subcritical and critical cases. Second, we study the most likely initial conditions that lead to the formation of a rogue wave, from a theoretical and numerical point of view. Finally, we propose several open questions for future research.

math.AP↗
arXiv2021

Global solutions of aggregation equations and other flows with random diffusion

Aggregation equations, such as the parabolic-elliptic Patlak-Keller-Segel model, are known to have an optimal threshold for global existence vs. finite-time blow-up. In particular, if the diffusion is absent, then all smooth solutions with finite second moment can exist only locally in time. Nevertheless, one can ask whether global existence can be restored by adding a suitable noise to the equation, so that the dynamics are now stochastic. Inspired by the work of Buckmaster et al. arXiv:1806.03734 showing that, with high probability, the inviscid SQG equation with random diffusion has global classical solutions, we investigate whether suitable random diffusion can restore global existence for a large class of active scalar equations in arbitrary dimension with possibly singular velocity fields. This class includes Hamiltonian flows, such as the SQG equation and its generalizations, and gradient flows, such as those arising in aggregation models. For this class, we show global existence of solutions in Gevrey-type Fourier-Lebesgue spaces with quantifiable high probability.

math.AP↗
arXiv2021

Sharp Strichartz estimates for some variable coefficient Schrödinger operators on $\mathbb{R}\times\mathbb{T}^2$

In the first part of the paper we continue the study of solutions to Schrödinger equations with a time singularity in the dispersive relation and in the periodic setting. In the second we show that if the Schrödinger operator involves a Laplace operator with variable coefficients with a particular dependence on the space variables, then one can prove Strichartz estimates at the same regularity as that needed for constant coefficients. Our work presents a two dimensional analysis, but we expect that with the obvious adjustments similar results are available in higher dimensions.

math.AP↗
arXiv2021

Uniqueness of Solutions to the Spectral Hierarchy in Kinetic Wave Turbulence Theory

In arXiv:1201.4067 and arXiv:1611.08030, Eyink and Shi and Chibbaro et al., respectively, formally derived an infinite, coupled hierarchy of equations for the spectral correlation functions of a system of weakly interacting nonlinear dispersive waves with random phases in the standard kinetic limit. Analogously to the relationship between the Boltzmann hierarchy and Boltzmann equation, this spectral hierarchy admits a special class of factorized solutions, where each factor is a solution to the wave kinetic equation (WKE). A question left open by these works and highly relevant for the mathematical derivation of the WKE is whether solutions of the spectral hierarchy are unique, in particular whether factorized initial data necessarily lead to factorized solutions. In this article, we affirmatively answer this question in the case of 4-wave interactions by showing, for the first time, that this spectral hierarchy is well-posed in an appropriate function space. Our proof draws on work of Chen and Pavlović for the Gross-Pitaevskii hierarchy in quantum many-body theory and of Germain et al. for the well-posedness of the WKE.

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