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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 ε . We prove that when r 0 = | log ε | α , α > 1 , the vorticity field of the fluid converges for ε → 0 to the point vortex model, in an interval of time which diverges as log | log ε | . 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. [ABSTRACT FROM AUTHOR]

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 ε and vorticity mass of the order of | log ε | - 1 . When ε → 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 [5], where a sharp localization of the vorticity support was obtained both along the radial and axial directions, but the convergence for ε → 0 worked only for short times. [ABSTRACT FROM AUTHOR]

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 converges to the solution of the BGK equation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 and intensity of the order of | log ε | - 1 . We show that in the limit ε → 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. [ABSTRACT FROM AUTHOR]

We discuss the sharp interface limit of the action functional associated with 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 deviation asymptotics with respect to the mean curvature flow. [ABSTRACT FROM AUTHOR]

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 ≤ 3 and that the potential has quartic growth, we prove existence and uniqueness of the solution as a process u in L 2 with continuous paths, satisfying almost surely the regularity properties u ∈ C ([ 0 , T ] ; H 1) and u ∈ L 2 ([ 0 , T ] ; H 2) . [ABSTRACT FROM AUTHOR]

Consider the Allen-Cahn equation on the d-dimensional torus, d = 2, 3, in the sharp interface limit. As 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 asymptotics in a joint sharp interface and small noise limit. Relying on previous results on the variational convergence of the action functional, we prove the large deviations upper bound. The corresponding rate function is finite only when there exists a time evolving interface of codimension one between the two stable phases. The zero level set of this rate function is given by the evolution by mean curvature in the sense of Brakke. Finally, the rate function can be written in terms of the sum of two non-negative quantities: the first measures how much the velocity of the interface deviates from its mean curvature, while the second is due to the possible occurrence of nucleation events. [ABSTRACT FROM AUTHOR]

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 this system, with a control of the growth in time of the local energy, and we give a nontrivial bound on the velocity of propagation of a perturbation. This is an extension to the case $$\sigma _1 < 2\sigma _2-1$$ of some already known results obtained for $$\sigma _1 \ge 2\sigma _2-1$$ . [ABSTRACT FROM AUTHOR]

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 hydrodynamic approximation, we prove the presence of such a drift, providing an analytical expression for it. This expression is then shown to be an antisymmetric function of the three densities. The antisymmetry of the drift can also be inferred from a symmetry property of the underlying microscopic dynamics. [ABSTRACT FROM AUTHOR]

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, ) for inverse power-like singular interactions. [ABSTRACT FROM AUTHOR]

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 any bound ( runaway effect). Moreover, the body asymptotically reaches a uniformly accelerated motion with acceleration E/ M. We then discuss at a heuristic level the case in which Ψ( r) diverges at short distances like gr, g, α>0, by showing that the runaway effect still occurs if α<2. [ABSTRACT FROM AUTHOR]

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 fluctuations by analyzing the $${\phi^4_1}$$ -measure with Dobrushin boundary conditions. In particular, we obtain a non-trivial limit in a suitable scaling in which the length of the interval diverges and the temperature vanishes. The limiting state is not translation invariant and describes a localized interface. This result can be seen as the probabilistic counterpart of the variational convergence of the associated excess free energy. [ABSTRACT FROM AUTHOR]

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 prove that, in a suitable scaling limit, the front evolves according to a one-dimensional diffusion process with a non-linear drift accounting for a “soft” repulsion from the boundary. We finally show how a “hard” repulsion can be obtained by an extra diffusive scaling. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 of an invariant, one dimensional manifold connecting the stable and metastable phases. This is the unstable manifold of a distinguished, spatially nonhomogeneous, stationary solution, called the critical droplet.(4, 10) We show that the points on the manifold are droplets longer or shorter than the critical one, and that their motion is very slow in agreement with the theory of metastable patterns. We also obtain a new proof of the existence of the critical droplet, which is supplied with a local uniqueness result. [ABSTRACT FROM AUTHOR]

: 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 confined in an unbounded tube of ℝ3. The electric field E is directed along the symmetry axis of the tube and the particle also interacts with an infinitely many particle system. The background system initial conditions are chosen in a set which is typical for any reasonable thermodynamic (equilibrium or non-equilibrium) state. We prove that, for large E and bounded interactions between the charged particle and the background, the velocity v(t) of the charged particle does not reach a finite limit velocity, but it increases to infinite as: |v(t)-Et|≤C0(1+t), where C0 is a constant independent of E. As a corollary we obtain that, if the initial conditions of the background system are distributed according to any Gibbs state, then the average velocity of the charged particle diverges as time goes to infinite. This result is obtained for E large enough in comparison with the mean energy of the Gibbs state. We next study the one-dimensional case, in which the estimates can be improved. We finally discuss, at an heuristic level, the existence of a finite limit velocity for unbounded interactions, and give some suggestions about the case of small electric fields. [ABSTRACT FROM AUTHOR]

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 φ γ( x)= γφ( γx), γ∈(0, 1]. We first prove that the velocity of a particle grows at most linearly in time, with rate of order γ. We next study the motion of a fast particle interacting with a background of slow particles, and we prove that its velocity remains almost unchanged for a very long time (at least proportional to γ−1 times the velocity itself). Finally we shortly discuss the so called “Vlasov limit,” when time and space are scaled by a factor γ. [ABSTRACT FROM AUTHOR]

We consider a conservative system of stochastic PDE's, namely a one dimensional phase field model perturbed by an additive space-time white noise. We prove a global existence and uniqueness result in a space of continuous functions on. This result is obtained by extending previous results of Doering [3] on the stochastic Allen-Cahn equation. [ABSTRACT FROM AUTHOR]

We consider a conservative system of stochastic PDE's, namely a weakly coupled, one dimensional phase field model with additive noise. We study the fluctuations of the front proving that, in a suitable scaling limit, the front evolves according to a non-Markov process, solution of a linear stochastic equation with long memory drift. [ABSTRACT FROM AUTHOR]

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 equal to the stiffness parameters determined by the surface tension. [ABSTRACT FROM AUTHOR]

We investigate the time evolution of a model system of interacting particles moving in a d-dimensional torus. The microscopic dynamics is first order in time with velocities set equal to the negative gradient of a potential energy term Ψ plus independent Brownian motions: Ψ is the sum of pair potentials, V( r)+ γd J( γr); the second term has the form of a Kac potential with inverse range γ. Using diffusive hydrodynamic scaling (spatial scale γ−1, temporal scale γ−2) we obtain, in the limit γ↓0, a diffusive-type integrodifferential equation describing the time evolution of the macroscopic density profile. [ABSTRACT FROM AUTHOR]

We consider a Ginzburg–Landau equation in the interval [−ε−1, ε−1], ε>0, with Neumann boundary conditions, perturbed by an additive white noise of strength $$\sqrt {\varepsilon } $$ and reaction term being the derivative of a function which has two equal–depth wells at ±1, but is not symmetric. When ε=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 solutions of the perturbed equation in the limit ε→0+, when the initial datum is close to an instanton. We prove that, for times that may be of the order of ε−1, the solution stays close to some instanton whose center, suitably normalized, converges to a Brownian motion plus a drift. This drift is known to be zero in the symmetric case, and, using a perturbative analysis, we show that if the nonsymmetric part of the reaction term is sufficiently small, it determines the sign of the drift. [ABSTRACT FROM AUTHOR]

We consider the one-dimensional planar rotator and classical Heisenberg models with a ferromagnetic Kac potential Jγ(r)=γ 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 mβ s, s any unit vector and mβ the Curie–Weiss spontaneous magnetization. We prove a large-deviation principle for the associated Gibbs measures. Then we study the system in the limit γ ↓ 0 below the above critical temperature. We prove that the norm of the empirical spin average in blocks of order γ−1 converges to mβ, uniformly in intervals of order γ− p, for any p ≥ 1. We also give a lower bound to the scale on which the change of phase occurs, by showing that the empirical spin average is approximately constant on intervals having length of order γ-1-λwith λ∈(0,1) small enough. [ABSTRACT FROM AUTHOR]

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 all times in the two-dimensional case. Since, in this case, the only singularity is the shrinking to a point of a closed curve, we verify that the curve actually disappears past the singularity. [ABSTRACT FROM AUTHOR]