Paolo Buttà

• Ph.D. in Mathematical Physics
• Professor (Full) at Sapienza University of Rome

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \se...

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...

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 th...

We consider interacting particle dynamics with Vicsek-type interactions, and their macroscopic Partial Differential Equation (PDE) limit, in the non-mean-field regime; that is, we consider the case in which each particle/agent in the system interacts only with a prescribed subset of the particles in the system (for example, those within a certain d...

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...

We consider interacting particle dynamics with Vicsek type interactions, and their macroscopic PDE limit, in the non-mean-field regime; that is, we consider the case in which each particle/agent in the system interacts only with a prescribed subset of the particles in the system (for example, those within a certain distance). In this non-mean-field...

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 $$\varepsilon $$ ε and vorticity mass of the order of $$|\log \varepsilon |^{ -1}$$ | log ε | - 1 . When $$\varepsilon \rightarrow 0$$ ε → 0 ,...

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...

We consider a stochastic \begin{document}$ N $\end{document}-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 BG...

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 the one-particle distribution function conve...

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...

We study the existence and uniqueness of the time evolution of a system of infinitely many individuals, moving in a tunnel and subjected to a Cucker–Smale type alignment dynamics with compactly supported communication kernels and to short-range repulsive interactions to avoid collisions.

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 desc...

We study the solutions of a generalized Allen–Cahn equation deduced from a Landau energy functional, endowed with a non–constant higher order stiffness. We assume the stiffness to be a positive function of the field and we discuss the stability of the stationary solutions proving both linear and local non–linear stability.

We study the existence and uniqueness of the time evolution of a system of infinitely many individuals, moving in a tunnel and subjected to a Cucker-Smale type alignment dynamics with compactly supported communication kernels and to short-range repulsive interactions to avoid collisions.

We study the solutions of a generalized Allen-Cahn equation deduced from a Landau energy functional, endowed with a non-constant higher order stiffness. We assume the stiffness to be a positive function of the field and we discuss the stability of the stationary solutions proving both linear and local non-linear stability.

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 ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \us...

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...

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 descr...

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...

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 homogen...

We present a new method of 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 homo...

We present a non-linear non-homogeneous fitness-complexity algorithm where the presence of non homogeneous terms guarantees 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 method is parameter free. This new algorithm reproduces the findings...

We introduce and analyse a continuum model for an interacting particle system of Vicsek type. Such a model is given by a non-linear kinetic partial differential equation (PDE) describing the time-evolution of the density $f_t$ of a collection of interacting particles confined to move on the one-dimensional torus. The density profile $f_t$ is a func...

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. T...

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 respe...

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 respe...

We introduce a class of stochastic Allen-Cahn equations with a mobility coefficient and colored noise. For initial data with finite free energy, we analyze the corresponding Cauchy problem on the $d$-dimensional torus in the time interval $[0,T]$. Assuming that $d\le 3$ and that the potential has quartic growth, we prove existence and uniqueness of...

Consider the Allen-Cahn equation on the $d$-dimensional torus, $d=2,3$, in the sharp interface limit. As it is well known, the limiting dynamics is described by the motion by mean curvature of the interface between the two stable phases. Here, we analyze a stochastic perturbation of the Allen-Cahn equation and describe its large deviation asymptoti...

We study the time evolution of an incompressible Euler fluid with planar symmetry when the vorticity is initially concentrated in small disks. We discuss about 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...

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 origi...

We consider an unbounded lattice and at each point of this lattice an anharmonic oscillator, that interacts with its first neighborhoods via a pair potential $V$ and is subjected to a restoring force of potential $U$. We assume that $U$ and $V$ are even nonnegative polynomials of degree $2\sigma_1$ and $2\sigma_2$. We study the time evolution of th...

Consider the Allen-Cahn equation on the $d$-dimensional torus, $d=2,3$, in the sharp interface limit. As it is well known, the limiting dynamics is described by the motion by mean curvature of the interface between the two stable phases. Here, we analyze a stochastic perturbation of the Allen-Cahn equation and describe its large deviation asymptoti...

We consider an unbounded lattice and at each point of this lattice an anharmonic oscillator, that interacts with its first neighborhoods via a pair potential $V$ and is subjected to a restoring force of potential $U$. We assume that $U$ and $V$ are even nonnegative polynomials of degree $2\sigma_1$ and $2\sigma_2$. We study the time evolution of th...

In this chapter the problem of viscous friction is considered when the medium is described by a gas of free particles in the mean field approximation. We give necessary conditions on the body/medium interaction to have a microscopic model of viscous friction and, conversely, we show that the runaway particle effect takes place in the case of bounde...

In this chapter we study the motion of a body immersed in a Vlasov system. Such a choice for the medium allows to overcome problems connected to the irregular motion of the body occurring when it interacts with a gas of point particles. On the contrary, in case of a Vlasov system, the motion is expected to be regular. The interaction body/medium is...

In this chapter we study the unsteady motion of a sphere immersed in a Stokes fluid, that is a linear approximation of a fluid governed by the Navier–Stokes equation. The equation of motion for the sphere leads to an integro-differential equation, and we are interested in the asymptotic behavior in time of the solution. We show that the velocity of...

In this chapter we study the problem of viscous friction in the framework of microscopic models of classical point particles. The system body/medium is modeled by the dynamics of a heavy particle (the body), subjected to a constant force and interacting with infinitely many identical particles (the medium). We discuss conditions on the body/medium...

We discuss the effect of structure-preserving perturbations on complex or real Hamiltonian eigenproblems and characterize the structured worst-case effect perturbations. We derive explicit expressions for the maximal Hamiltonian perturbations.

Let $A$ be either a complex or real matrix with all distinct eigenvalues. We propose a new method for the computation of both the unstructured and the real-structured (if the matrix is real) distance $w_{\mathbb K}(A)$ (where ${\mathbb K}=\mathbb C$ if general complex matrices are considered and ${\mathbb K} ={\mathbb R}$ if only real matrices are...

We consider the Cahn-Hilliard equation in one space dimension, perturbed by the derivative of a space and time white noise of intensity $\epsilon^{\frac 12}$, and we investigate the effect of the noise, as $\epsilon \to 0$, on the solutions when the initial condition is a front that separates the two stable phases. We prove that, given $\gamma< \fr...

In this paper we discuss the time evolution of a classical Hamiltonian system composed of infinitely many particles mutually interacting via a pair potential with a hard core and confined in an unbounded domain D of ℝ3.

In a recent work, Bodineau and Derrida analyzed the phase fluctuations in the ABC model. In particular, they computed the asymptotic variance and, on the basis of numerical simulations, they conjectured the presence of a drift, which they guessed to be an antisymmetric function of the three densities. By assuming the validity of the fluctuating hyd...

We investigate the ground and low energy states of a one dimensional non local free energy functional describing at a mean field level a spin system with both ferromagnetic and antiferromagnetic interactions. In particular, the antiferromagnetic interaction is assumed to have a range much larger than the ferromagnetic one. The competition between t...

We discuss the effect of structure-preserving perturbations on complex or real Hamiltonian eigenproblems and characterize the structured worst-case effect perturbations. We derive significant expressions for both the structured condition numbers and the worst-case effect Hamiltonian perturbations. It is shown that, for purely imaginary eigenvalues,...

We study the existence and uniqueness of the time evolution via the Newton law of a two dimensional system of infinitely many particles with very singular mutual interactions. It is an improvement of the result by Fritz and Dobrushin given in (Comm. Math. Phys. 57:67–81, 1977) for inverse power-like singular interactions.

The computation of the structured pseudospectral abscissa and radius (with respect to the Frobenius norm) of a Toeplitz matrix is discussed and two algorithms based on a low rank property to construct extremal perturbations are presented. The algorithms are inspired by those considered in [SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1166-1192] for t...

We consider the van der Waals' free energy functional, with a scaling small parameter epsilon, in the plane domain given by the first quadrant, and inhomogeneous Dirichlet boundary conditions. The boundary data are chosen in such a way that the interface between the pure phases tends to be horizontal and is pinned at some point on the y-axis which...

We study the linear stability problem of the stationary solution ψ* = −cos y for the Euler equation on a 2-dimensional flat torus of sides 2πL and 2π. We show that ψ* is stable if L ∈ (0, 1) and that exponentially unstable modes occur in a right neighborhood of L = n for any integer n. As a corollary, we gain exponentially instability for any L lar...

We study the motion of a classical point body of mass M, moving under the action of a constant force of intensity E and immersed in a Vlasov fluid of free particles, interacting with the body via a bounded short range potential Ψ. We prove that if its initial velocity is large enough then the body escapes to infinity increasing its speed without an...

We study the two-dimensional Navier-Stokes system on a flat cylinder with the usual Dirichlet boundary conditions for the velocity field u. We formulate the problem as an infinite system of ODE's for the natural Fourier components of the vorticity, and the boundary conditions are taken into account by adding a vorticity production at the boundary....

We study the motion of a classical point body of mass M, moving under the action of a constant force of intensity E and immersed in a Vlasov fluid of free particles, interacting with the body via a bounded short range potential Psi. We prove that if its initial velocity is large enough then the body escapes to infinity increasing its speed without...

We consider a stochastic perturbation of the Allen–Cahn equation in a bounded interval [−a,b] with boundary conditions fixing the different phases at a and b. We investigate the asymptotic behavior of the front separating the two stable phases in the limit ε→0, when the intensity of the noise is \(\sqrt{\varepsilon}\) and a,b→∞ with ε. In particula...

We consider the van der Waals free energy functional in a bounded interval with inhomogeneous Dirichlet boundary conditions imposing the two stable phases at the endpoints. We compute the asymptotic free energy cost, as the length of the interval diverges, of shifting the interface from the midpoint. We then discuss the effect of thermal fluctuatio...

We rigorously prove the existence of directed transport for a certain class of ac-driven nonlinear one-dimensional systems, namely the generation of transport with a preferred direction in the absence of a net driving force.

We consider a stochastically perturbed reaction diffusion equation in a bounded interval, with boundary conditions imposing the two stable phases at the endpoints. We investigate the asymptotic behavior of the front separating the two stable phases, as the intensity of the noise vanishes and the size of the interval diverges. In particular, we prov...

We investigate the problem of the existence of trajectories asymptotic to elliptic equilibria of Hamiltonian systems in the presence of resonances.

We investigate the problem of the existence of trajectories asymptotic to elliptic equilibria of Hamiltonian systems in the presence of resonances.

Endothelial cells are responsible for the formation of the capillary blood vessel network. We describe a system of endothelial cells by means of two-dimensional molecular dynamics simulations of point-like particles. Cells' motion is governed by the gradient of the concentration of a chemical substance that they produce (chemotaxis). The typical ti...

We give a not trivial upper bound on the velocity of disturbances in an infinitely extended anharmonic system at thermal equilibrium. The proof is achieved by combining a control on the non equilibrium dynamics with an explicit use of the state invariance with respect to the time evolution.

We consider a stochastically perturbed reaction diffusion equation in a bounded interval, with boundary conditions imposing the two stable phases at the endpoints. We investigate the asymptotic behavior of the front separating the two stable phases, as the intensity of the noise vanishes and the size of the interval diverges. In particular, we prov...

We give a not trivial upper bound on the velocity of disturbances in an infinitely extended anharmonic system at thermal equilibrium. The proof is achieved by combining a control on the non equilibrium dynamics with an explicit use of the state invariance with respect to the time evolution.

We study the time evolution of a system of infinitely many charged particles confined by an external magnetic field in an unbounded cylindrical conductor and mutually interacting via the Coulomb force. We prove the existence, uniqueness and quasi-locality of the motion. Moreover, we give some nontrivial bounds on its long time behavior.

We show how a polymer in two dimensions with a self-repelling interaction of Kac type exhibits a diffusive–ballistic transition if considered on the appropriate scale.

We investigate the existence of nontranslation invariant (periodic) density profiles, for systems interacting via translation invariant long-range potentials, as minimizers of local mean field free energy functionals. The existence of a second-order transition from a uniform to a nonuniform density at a specified temperature is proven for a class o...

We study a recent model of random networks based on the presence of an intrinsic character of the vertices called fitness. The vertex fitnesses are drawn from a given probability distribution density. The edges between pairs of vertices are drawn according to a linking probability function depending on the fitnesses of the two vertices involved. We...

We consider an infinite Hamiltonian system in one space dimension, given by a charged particle subjected to a constant electric field and interacting with an infinitely extended system of particles. We discuss conditions on the particle/medium interaction which are necessary for the charged particle to reach a finite limiting velocity. We assume th...

We study a recent model of random networks based on the presence of an intrinsic character of the vertices called fitness. The vertices fitnesses are drawn from a given probability distribution density. The edges between pair of vertices are drawn according to a linking probability function depending on the fitnesses of the two vertices involved. W...

In this paper we consider a nonlocal evolution equation in one dimension, which describes the dynamics of a ferromagnetic system in the mean field approximation. In the presence of a small magnetic field, it admits two stationary and homogeneous solutions, representing the stable and metastable phases of the physical system. We prove the existence...

We study the time evolution of a charged particle moving in a medium under the action of a constant electric field E. In the framework of fully Hamiltonian models, we discuss conditions on the particle/medium interaction which are necessary for the particle to reach a finite limit velocity. We first consider the case when the charged particle is co...

In this paper we consider a non local evolution equation in one dimension, which describes the dynamics of a ferromagnetic system in the mean eld approximation. In the presence of a small external magnetic eld, this equation admits two stationary homogeneous solutions, which represent the stable and metastable phases of the physical system. We prov...

We analyze the long time behavior of an infinitely extended system of particles in one dimension, evolving according to the Newton laws and interacting via a non-negative superstable Kac potential (x)=(x), (0,1]. We first prove that the velocity of a particle grows at most linearly in time, with rate of order . We next study the motion of a fast...

We prove a convexity property of the surface tension corresponding to a non-local, anisotropic free-energy functional of van der Waals type which implies that the Wulff shape is strictly convex and smooth. We also prove that the transport coefficients of the limiting anisotropic motion by mean curvature obtained in [33] are strictly positive and eq...

We prove a convexity property of the surface tension of non local, anisotropic free energy functionals which implies that the Wulff shape is strictly convex and smooth. We also prove that the transport coefficients of the limit anisotropic motion by mean curvature obtained in [33] are strictly positive and equal to the stiffness parameters determin...

We study a non local evolution and define the interface in terms of a local equilibrium condition. We prove that in a diffusive scaling limit the local equilibrium condition propagates in time thus defining an interface evolution which is given by a motion by mean curvature. The analysis extend through all times before the appearance of singulariti...

We introduce a microscopic and stochastic model of phase field type and discuss its macroscopic limits.

We consider a stochastic spin system coupled to a linear diffusion process. The coupling is such that there is a locally conserved quantity. The equilibrium states are the corresponding canonical Gibbs measures. We prove that, under a diffusive scaling limit, the macroscopic density of the conserved quantity solves a non–linear diffusion equation....

We consider a Ginzburg–Landau equation in the interval [–Ö{e}\sqrt {\varepsilon } and reaction term being the derivative of a function which has two equal–depth wells at 1, but is not symmetric. When =0, the equation has equilibrium solutions that are increasing, and connect –1 with +1. We call them instantons, and we study the evolution of the so...

We investigate the time evolution of a model system of interacting particles, moving in a $d$-dimensional torus. The microscopic dynamics are first order in time with velocities set equal to the negative gradient of a potential energy term $\Psi$ plus independent Brownian motions: $\Psi$ is the sum of pair potentials, $V(r)+\gamma^d J(\gamma r)$, t...

We consider the one-dimensional planar rotator and classical Heisenberg models with a ferromagnetic Kac potential J (r)=J(yr), J with compact support. Below the Lebowitz-Penrose critical temperature the limit (mean-field) theory exhibits a phase transition with a continuum of equilibrium states, indexed by the magnetization vectors m s, s any unit...

. We consider a Ginzburg-Landau equation in the interval [Gamma" Gamma1 ; " Gamma1 ], " ? 0, with Neumann boundary conditions, perturbed by an additive white noise of strength p ", and reaction term being the derivative of a function which has two equal depth wells at Sigma1, but is not symmetric. When " = 0, the equation has equilibrium solutions...

We consider a Ginzburg–Landau equation in the interval [––, –], >0, 1, with Neumann boundary conditions, perturbed by an additive white noise of strength We prove that if the initial datum is close to an "instanton" then, in the limit 0+, the solution stays close to some instanton for times that may grow as fast as any inverse power of , as long as...

We consider an Ising system in d≥2 dimensions with ferromagnetic spin-spin interactions -J γ (x,y)σ(x)σ(y), x,y∈ℤ d , where J γ (x,y) scales like a Kac potential. We prove that when the temperature is below the mean field critical value, for any γ small enough (i.e. when the range of the interaction is long but finite), there are only two pure homo...

The convergence to a motion by mean curvature by diffusively scaling a nonlocal evolution equation, describing the macroscopic behavior of a ferromagnetic spin system with Kac interaction and Glauber dynamics has recently been proved. The convergence is proven up to the times when the motion by curvature is regular. Here we show the convergence at...

We consider models of interface dynamics derived from Ising systems with Kac interactions and we prove the validity of the Einstein relation=, where is the proportionality coefficient in the motion by curvature, is the interface mobility, and is the surface tension.

We study the linear stability problem of the stationary solution = cosy for the Euler equation on a 2-dimensional at torus of sides 2 L and 2 . We show that is stable if L2 (0; 1) and that exponentially unstable modes occur in a right neighborhood of L = n for any integer n. As a corollary, we gain exponentially instability for any L large enough a...