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Eliseo Luongo

Publications and source records attributed to Eliseo Luongo.

At least 19 recordsLinked to original sources
arXiv2026

An Effective SPDE Model for a Schr\"odinger Equation with a Fluctuating Magnetic Potential

We consider the Schr\"odinger 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 Schr\"odinger equation with Gaussian fluctuations. We propose a stochastic Schr\"odinger 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

Global smooth solutions by high mode Lie-Transport noise for Logarithmically Hyperdissipative Navier-Stokes equations

We study a logarithmically hyperviscous Navier-Stokes model on the three-dimensional torus with Lie-transport noise, which includes both transport and stretching. We prove that, for noise of sufficiently large intensity and high frequency, the system admits a unique global smooth solution with probability arbitrarily close to one. Unlike previous works, this physically motivated noise does not preserve energy or enstrophy, but rather circulation. Global well-posedness is established through a probabilistic mechanism that produces effective dissipation via a scaling limit. Crucially, this approach bypasses the lack of conserved quantities and tames the singular nature of stochastic stretching.

math.PR↗
arXiv2026

Spectral instability and non-uniqueness for the Keller-Segel system

We show that the Cauchy problem associated with the parabolic-elliptic Keller-Segel model is locally ill-posed in $L^q(\mathbb{R}^n)$ for dimensions $n \in \{3,\dots,9\}$ and throughout the supercritical range $q\in [1,\frac{n}{2})$. An analog non-uniqueness result is given in the critical space of bounded functions taking values in $L^{n/2,\infty}(\R^n)$. The non-uniqueness is driven by an instability mechanism in self-similarity variables, in the spirit of the program proposed by Jia and \v{S}ver\'ak for the three-dimensional Navier-Stokes equations.

math.AP↗
arXiv2026

Averaging Dynamics and Wong-Zakai approximations for a Fast-Slow Navier-Stokes System Driven by fractional Brownian Motion

We study a slow-fast system of coupled two- and three-dimensional Navier-Stokes equations in which the fast component is perturbed by an additive fractional Brownian noise with Hurst parameter $H>\frac{1}{3}$. The system is analyzed using rough path theory, and the limiting behaviour strongly depends on the value of $H$. We prove convergence in law of the slow component to a Navier-Stokes system with an additional It\^o-Stokes drift when $H<\frac{1}{2}$. In contrast, for $H\in (\frac{1}{2},1)$, the limit equation features only a transport noise driven by a rough path.

math.PR↗
arXiv2025

Self-similar blowup from arbitrary data for supercritical wave maps with additive noise

We consider stochastically perturbed wave maps from $\mathbb{R}^{1+d}$ into $\mathbb{S}^d$, in all energy-supercritical dimensions $d \geq 3$. We show that corotational non-degenerate Gaussian additive noise leads to self-similar blowup with positive probability for any corotational initial data. The same result without noise is conjectured, but unknown, for large data.

math.AP↗
arXiv2025

On the effect of randomization on supercritical heat equations

Recently, in \cite{glogic2025non}, it has been shown that the focusing power nonlinearity heat equation \begin{equation}\label{Eq:Heat_abstract}\tag{NLH} \partial_t u -\Delta u = |u|^{p-1}u, \quad p>1, \end{equation} in dimensions $d \geq 3$ has non-unique local solutions in $L^q(\mathbb{R}^d)$ for $q < d(p-1)/2$ provided that $p < p_{JL}$, where $p_{JL}$ denotes the Joseph-Lundgren exponent. In this paper we investigate the effect of different randomizations on the well-posedness of the equation. First we show that adding a forcing term white in time and colored in space in \eqref{Eq:Heat_abstract} is not sufficient to improve the solution theory: namely, we prove non-uniqueness for local-in-time mild solutions of \eqref{Eq:Heat_abstract} with additive noise. Second, we discuss how randomizing the initial conditions of \eqref{Eq:Heat_abstract} affects its well-posedness.

math.PR↗
arXiv2025

Zero-noise selection and Large Deviations in $L^\infty_t L^p_x$ for the stochastic transport equation beyond DiPerna-Lions

We consider $L^\infty_t L^p_x$ solutions of the stochastic transport equation with drift in $L^\infty_t W^{1,q}_x$. We show strong existence and pathwise uniqueness of solutions in a regime of parameters $p,q$ for which non-unique weak solutions of the deterministic transport equation exist. When the intensity of the noise goes to zero, we prove that the solutions of the stochastic transport equation converge to the unique renormalized solution of the transport equation in the sense of DiPerna-Lions. Furthermore, we show that the convergence is governed by a Large Deviations Principle in the space $L^\infty_t L^p_x$. Since the space $L^\infty_t L^p_x$ is not separable, the weak convergence approach to Large Deviations by Budhiraja, Dupuis, and Maroulas is not directly applicable.

math.PR↗
arXiv2025

Non-uniqueness of mild solutions to supercritical heat equations

We consider the focusing power nonlinearity heat equation \begin{equation}\label{Eq:Heat_abstract}\tag{NLH} \partial_t u -\Delta u = |u|^{p-1}u, \quad p>1, \end{equation} in dimensions $d \geq 3$. It is well-known that if $p$ is large enough then \eqref{Eq:Heat_abstract} is unconditionally locally well-posed in $L^q(\mathbb{R}^d)$ for $q \geq d(p-1)/2$. We prove that this result is optimal in the sense that uniqueness of local solutions fails when $q < d(p-1)/2$ as long as $p < p_{JL}$, where $p_{JL}$ stands for the Joseph-Lundgren exponent. Our proof is based on the method that Jia-\v{S}ver\'ak proposed in \cite{JiaSve15} to show non-uniqueness of Leray solutions to incompressible 3d Navier-Stokes equations. In particular, we rigorously verify for \eqref{Eq:Heat_abstract} the (analogue of the) spectral assumption made in \cite{JiaSve15}. To our knowledge, this is the first rigorous implementation of the Jia-\v{S}ver\'ak method to a nonlinear parabolic equation without forcing.

math.AP↗
arXiv2024

Background Vlasov equations and Young measures for passive scalar and vector advection equations under special stochastic scaling limits

In the last few years it was proved that scalar passive quantities subject to suitable stochastic transport noise, and more recently that also vector passive quantities subject to suitable stochastic transport and stretching noise, weakly converge to the solutions of deterministic equations with a diffusion term. In the background of these stochastic models, we introduce stochastic Vlasov equations which gives additional information on the fluctuations and oscillations of solutions: we prove convergence to non-trivial Young measures satisfying limit PDEs with suitable diffusion terms. In the case of a passive vector field the background Vlasov equation adds completely new statistical information to the stochastic advection equation.

math.PR↗
arXiv2024

Global well-posedness of 2D Navier-Stokes with Dirichlet boundary fractional noise

In this paper, we prove the global well-posedness and interior regularity for the 2D Navier-Stokes equations driven by a fractional noise acting as an inhomogeneous Dirichlet-type boundary condition. The model describes a vertical slice of the ocean with a relative motion between the two surfaces and can be thought of as a stochastic variant of the Couette flow. The relative motion of the surfaces is modeled by a Gaussian noise which is coloured in space and fractional in time with Hurst parameter greater than 3/4.

math.AP↗
arXiv2024

Mean-Field Magnetohydrodynamics Models as Scaling Limits of Stochastic Induction Equations

We study the asymptotic properties of a stochastic model for the induction equations of the magnetic field in a three dimensional periodic domain. The turbulent velocity field driving the electromotive force on the magnetic field is modeled by a noise white in time. For this model we rigorously take a scaling limit leading to a deterministic model. While in case of isotropic turbulence this produces an additional dissipation in the limit model which influences also the decay rate of the Magnetic field in the stochastic model, the case of turbulence devoloped in a preferential direction allows us to find a dynamo effect.

math.PR↗
arXiv2024

On the It\^o-Stratonovich Diffusion Limit for the Magnetic Field in a 3D Thin Domain

We introduce a stochastic model for a passive magnetic field in a three dimensional thin domain. The velocity field, white in time and modelling phenomenologically a turbulent fluid, acts on the magnetic field as a transport-stretching noise. We prove, in a quantitative way, that, in the simultaneous scaling limit of the thickness of the thin layer and the separation of scales, the mean on the thin direction of the magnetic field is close to the solution of the equation of the magnetic field with additional dissipation. In certain choice of noises with correlation between their components, without mirror symmetry and with a non zero mean helicity, we identify an alpha-term, in addition to the extra dissipation term. However, it does not produce dynamo; consequently, we extend a no-dynamo theorem to thin layers.

math.PR↗
arXiv2023

Existence of Invariant Measures for Stochastic Inviscid Multi-Layer Quasi-Geostrophic Equations

We consider an inviscid 3-layer quasi-geostrophic model with stochastic forcing in a 2D bounded domain. After establishing well-posedness of such system under natural regularity assumptions on the initial condition and the (additive) noise, we prove the existence of an invariant measure supported on bounded functions by means of the Krylov-Bogoliubov approach developed by Ferrario and Bessaih (Comm. Math. Phys. 377, 2020).

math.PR↗
arXiv2023

Gibbs Equilibrium Fluctuations of Point Vortex Dynamics

We consider a system of N point vortices in a bounded domain with null total circulation, whose statistics are given by the Canonical Gibbs Ensemble at inverse temperature $\beta\geq 0$. We prove that the space-time fluctuation field around the (constant) Mean Field limit satisfies when $N\to\infty$ a generalized version of 2-dimensional Euler dynamics preserving the Gaussian Energy-Enstrophy ensemble.

math.PR↗
arXiv2023

Global well-Posedness and Interior Regularity of 2D Navier-Stokes Equations with Stochastic Boundary Conditions

The paper is devoted to the analysis of the global well-posedness and the interior regularity of the 2D Navier-Stokes equations with inhomogeneous stochastic boundary conditions. The noise, white in time and coloured in space, can be interpreted as the physical law describing the driving mechanism on the atmosphere-ocean interface, i.e. as a balance of the shear stress of the ocean and the horizontal wind force.

math.AP↗
arXiv2023

Large Deviations Principle for the Inviscid Limit of Fluid Dynamic Systems in 2D Bounded Domains

Using a weak convergence approach, we establish a Large Deviation Principle (LDP) for the solutions of fluid dynamic systems in two-dimensional bounded domains subjected to no-slip boundary conditions and perturbed by additive noise. Our analysis considers the convergence of both viscosity and noise intensity to zero. Specifically, we focus on three important scenarios: Navier-Stokes equations in a Kato-type regime, Navier-Stokes equations for fluids with circularly symmetric flows and Second-Grade Fluid equations. In all three cases, we demonstrate the validity of the LDP, taking into account the critical topology $C([0,T];L^2).$

math.PR↗
arXiv2022

Uniform Approximation of 2D Navier-Stokes Equations with Vorticity Creation by Stochastic Interacting Particle Systems

We consider a stochastic interacting particle system in a bounded domain with reflecting boundary, including creation of new particles on the boundary prescribed by a given source term. We show that such particle system approximates 2d Navier-Stokes equations in vorticity form and impermeable boundary, the creation of particles modeling vorticity creation at the boundary. Kernel smoothing, more specifically smoothing by means of the Neumann heat semigroup on the space domain, allows to establish uniform convergence of regularized empirical measures to (weak solutions of) Navier-Stokes equations.

math.AP↗