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Paolo Buttà

Publications and source records attributed to Paolo Buttà.

At least 19 recordsLinked to original sources
arXiv2026

Vorticity confinement for 2D incompressible flows in an infinite cylinder

We study the confinement of vorticity for two-dimensional incompressible flows in an infinite cylinder. For Navier-Stokes solutions with non-negative and compactly supported initial vorticity, we derive quantitative decay estimates showing that the vorticity mass outside regions whose distance from the initial support grows like $\sqrt{t\log^αt}$ (with $α>1$) or like $t^β$ (with $β>1/2$) becomes, respectively, super-polynomially or stretched-exponentially small. The analysis combines an iterative scheme with an antisymmetry property of the Biot-Savart kernel. In the Euler case, by coupling this approach with a first-moment estimate from [Commun. Math. Phys., 367, 1077-1093, 2019], we recover the confinement bound of [Commun. Math. Phys., 367, 1077-1093, 2019] and refine it slightly: the diameter of the vorticity support grows at most like $(t\log t)^{1/3}$, rather than $t^{1/3}\log^2 t$.

math.AP↗
arXiv2026

Cost Functions in Economic Complexity

Economic complexity algorithms aim to uncover the hidden capabilities that drive economic systems. Here, we present a fundamental reinterpretation of two of these algorithms, the Economic Complexity Index (ECI) and the Economic Fitness and Complexity (EFC), by reformulating them as optimization problems that minimize specific cost functions. We show that ECI computation is equivalent to finding eigenvectors of the network's transition matrix by minimizing the quadratic form associated with the network's Laplacian. For EFC, we derive a novel cost function that exploits the algorithm's intrinsic logarithmic structure and clarifies the role of the regularization parameter in its non-homogeneous version. Additionally, we establish the existence and uniqueness of its solution, providing theoretical foundations for its application. This optimization-based reformulation bridges economic complexity and established frameworks in spectral theory, network science, and optimization. The theoretical insights translate into practical computational advantages: we introduce a conservative, gradient-based update rule that substantially accelerates algorithmic convergence, with potential implications for a broader class of algorithms, including the Sinkhorn-Knopp method. Finally, we apply the energetic framework to a real-world trade network, demonstrating how link-wise energy provides a direct way to identify structurally relevant and vulnerable regions of the export matrix, thus complementing and enriching standard economic complexity analyses. Beyond advancing our theoretical understanding of economic complexity indicators, this work opens new pathways for algorithmic improvements and extends applicability to general network structures beyond traditional bipartite economic networks.

physics.soc-ph↗
arXiv2024

Leapfrogging vortex rings as scaling limit of Euler Equations

We consider an incompressible fluid with axial symmetry without swirl, assuming initial data such that the initial vorticity is very concentrated inside $N$ small disjoint rings of thickness $\varepsilon$, each one of vorticity mass and main radius of order $|\log\varepsilon|$. When $\varepsilon \to 0$, we show that, at least for small but positive times, the motion of the rings converges to a dynamical system firstly introduced in [NoDEA Nonlinear Diff. Eq. Appl. 6 (1999), 473-499]. In the special case of two vortex rings with large enough main radius, the result is improved reaching longer times, in such a way to cover the case of several overtakings between the rings, thus providing a mathematical rigorous derivation of the leapfrogging phenomenon.

math.AP↗
arXiv2024

Asymptotics of the $ϕ^4_1$ measure in the sharp interface limit

We consider the $ϕ^4_1$ measure in an interval of length $\ell$, defined by a symmetric double-well potential $W$ and inverse temperature $β$. Our results concern its asymptotic behavior in the joint limit $β, \ell \to \infty$, both in the subcritical regime $\ell \ll \rme^{βC_W}$ and in the supercritical regime $\ell \gg \rme^{βC_W}$, where $C_W$ denotes the surface tension. In the former case, in which the measure concentrates on the pure phases, we prove the corresponding large deviation principle. The associated rate function is the Modica-Mortola functional modified to take into account the entropy of the locations of the interfaces. Further, we provide the sharp asymptotics of the probability of having a given number of transitions between the two pure phases. In the supercritical regime, the measure does not longer concentrate and we show that the interfaces are asymptotically distributed according to a Poisson point process.

math.PR↗
arXiv2024

Long time evolution of concentrated vortex rings with large radius

We study the time evolution of an incompressible fluid with axial symmetry without swirl when the vorticity is sharply concentrated on $N$ annuli of radii of the order of $r_0$ and thickness $\varepsilon$. We prove that when $r_0= |\log \varepsilon|^α$, $α>1$, the vorticity field of the fluid converges for $\varepsilon \to 0$ to the point vortex model, in an interval of time which diverges as $\log|\log\varepsilon|$. This generalizes previous result by Cavallaro and Marchioro in [J. Math. Phys. 62, 053102, (2021)], that assumed $α>2$ and in which the convergence was proved for short times only.

math.AP↗
arXiv2023

From particle systems to the BGK equation

In [Phys. Rev. 94 (1954), 511-525], P.L. Bhatnagar, E.P. Gross and M. Krook introduced a kinetic equation (the BGK equation), effective in physical situations where the Knudsen number is small compared to the scales where Boltzmann's equation can be applied, but not enough for using hydrodynamic equations. In this paper, we consider the stochastic particle system (inhomogeneous Kac model) underlying Bird's direct simulation Monte Carlo method (DSMC), with tuning of the scaled variables yielding kinetic and/or hydrodynamic descriptions. Although the BGK equation cannot be obtained from pure scaling, it does follow from a simple modification of the dynamics. This is proposed as a mathematical interpretation of some arguments in [Phys. Rev. 94 (1954), 511-525], complementing previous results in [Arch. Ration. Mech. Anal. 240 (2021), 785-808] and [Kinet. Relat. Models 16 (2023), 269-293].

math-ph↗
arXiv2022

Vanishing viscosity limit for concentrated vortex rings

We study the time evolution of a viscous incompressible fluid with axial symmetry without swirl, when the initial vorticity is very concentrated in $N$ disjoint rings. We show that in a suitable joint limit, in which both the thickness of the rings and the viscosity tend to zero, the vorticity remains concentrated in $N$ disjointed rings, each one of them performing a simple translation along the symmetry axis with constant speed.

math.AP↗
arXiv2022

Global time evolution of concentrated vortex rings

We study the time evolution of an incompressible fluid with axial symmetry without swirl, assuming initial data such that the initial vorticity is very concentrated inside $N$ small disjoint rings of thickness $\varepsilon$ and vorticity mass of the order of $|\log\varepsilon|^{ -1}$. When $\varepsilon \to 0$ we show that the motion of each vortex ring converges to a simple translation with constant speed (depending on the single ring) along the symmetry axis. We obtain a sharp localization of the vorticity support at time $t$ in the radial direction, whereas we state only a concentration property in the axial direction. This is obtained for arbitrary (but fixed) intervals of time. This study is the completion of a previous paper arXiv:1904.04785 [math-ph], where a sharp localization of the vorticity support was obtained both along the radial and axial directions, but the convergence for $\varepsilon\to 0$ worked only for short times.

math.AP↗
arXiv2022

A stochastic particle system approximating the BGK equation

We consider a stochastic $N$-particle system on a torus in which each particle moving freely can instantaneously thermalize according to the particle configuration at that instant. Following [2], we show that the propagation of chaos does hold and that the one-particle distribution converges to the solution of the BGK equation. The improvement with respect to [2] consists in the fact that here, as suggested by physical considerations, the thermalizing transition is driven only by the restriction of the particle configuration in a small neighborhood of the jumping particle. In other words, the Maxwellian distribution of the outgoing particle is computed via the empirical hydrodynamical fields associated to the fraction of particles sufficiently close to the test particle and not, as in [2], via the whole particle configuration. [2]: arXiv:2002.10535 (Journal reference: Arch. Ration. Mech. Anal. Vol. 240 (2021), pp. 785-808)

math-ph↗
arXiv2021

Time evolution of concentrated vortex rings

We study the time evolution of an incompressible fluid with axisymmetry without swirl when the vorticity is sharply concentrated. In particular, we consider $N$ disjoint vortex rings of size $\varepsilon$ and intensity of the order of $|\log\varepsilon|^{-1}$. We show that in the limit $\varepsilon\to 0$, when the density of vorticity becomes very large, the movement of each vortex ring converges to a simple translation, at least for a small but positive time.

math-ph↗
arXiv2020

Particle approximation of the BGK equation

In this paper we prove the convergence of a suitable particle system towards the BGK model. More precisely, we consider an interacting stochastic particle system in which each particle can instantaneously thermalize locally. We show that, under a suitable scaling limit, propagation of chaos does hold and that the one-particle distribution function converges to the solution of the BGK equation.

math-ph↗
arXiv2019

Some aspects of the inertial spin model for flocks and related kinetic equations

In this paper we study the macroscopic behavior of the inertial spin (IS) model. This model has been recently proposed to describe the collective dynamics of flocks of birds, and its main feature is the presence of an auxiliary dynamical variable, a sort of internal spin, which conveys the interaction among the birds with the effect of better describing the turning of flocks. After discussing the geometrical and mechanical properties of the IS model, we show that, in the case of constant interaction among the birds, its mean-field limit is described by a non-linear Fokker-Planck equation, whose equilibria are fully characterized. Finally, in the case of non-constant interactions, we derive the kinetic equation for the mean-field limit of the model in the absence of thermal noise, and explore its macroscopic behavior by analyzing the mono-kinetic solutions.

math-ph↗
arXiv2019

A non-linear kinetic model of self-propelled particles with multiple equilibria

We introduce and analyse a continuum model for an interacting particle system of Vicsek type. The model is given by a non-linear kinetic partial differential equation (PDE) describing the time-evolution of the density $f_t$, in the single particle phase-space, of a collection of interacting particles confined to move on the one-dimensional torus. The corresponding stochastic differential equation for the position and velocity of the particles is a conditional McKean-Vlasov type of evolution (conditional in the sense that the process depends on its own law through its own conditional expectation). In this paper, we study existence and uniqueness of the solution of the PDE in consideration. Challenges arise from the fact that the PDE is neither elliptic (the linear part is only {\em hypoelliptic}) nor in gradient form. Moreover, for some specific choices of the interaction function and for the simplified case in which the density profile does not depend on the spatial variable, we show that the model exhibits multiple stationary states (corresponding to the particles forming a coordinated clockwise/anticlockwise rotational motion) and we study convergence to such states as well. Finally, we prove mean-field convergence of an appropriate $N$-particles system to the solution of our PDE: more precisely, we show that the empirical measures of such a particle system converge weakly, as $N \rightarrow \infty$, to the solution of the PDE.

math-ph↗
arXiv2018

A new and stable estimation method of country economic fitness and product complexity

We present a new metric estimating fitness of countries and complexity of products by exploiting a non-linear non-homogeneous map applied to the publicly available information on the goods exported by a country. The non homogeneous terms guarantee both convergence and stability. After a suitable rescaling of the relevant quantities, the non homogeneous terms are eventually set to zero so that this new metric is parameter free. This new map almost reproduces the results of the original homogeneous metrics already defined in literature and allows for an approximate analytic solution in case of actual binarized matrices based on the Revealed Comparative Advantage (RCA) indicator. This solution is connected with a new quantity describing the neighborhood of nodes in bipartite graphs, representing in this work the relations between countries and exported products. Moreover, we define the new indicator of country net-efficiency quantifying how a country efficiently invests in capabilities able to generate innovative complex high quality products. Eventually, we demonstrate analytically the local convergence of the algorithm involved.

econ.GN↗
arXiv2018

On large deviations of interface motions for statistical mechanics models

We discuss the sharp interface limit of the action functional associated to either the Glauber dynamics for Ising systems with Kac potentials or the Glauber+Kawasaki process. The corresponding limiting functionals, for which we provide explicit formulae of the mobility and transport coefficients, describe the large deviations asymptotics with respect to the mean curvature flow.

math-ph↗
arXiv2018

Long time evolution of concentrated Euler flows with planar symmetry

We study the time evolution of an incompressible Euler fluid with planar symmetry when the vorticity is initially concentrated in small disks. We discuss how long this concentration persists, showing that in some cases this happens for quite long times. Moreover, we analyze a toy model that shows a similar behavior and gives some hints on the original problem.

math-ph↗