arXiv subjects

Lara Trussardi

Publications and source records attributed to Lara Trussardi.

12 recordsLinked to original sources

arXiv2026

A framework for continuum modeling of opinion dynamics on a network based on probability of connections

We propose a modeling framework to develop a continuum description of opinion dynamics on networks as an alternative to other models, like the ones based on graphons. In a nutshell, the continuum model that we propose aims at approximating the distribution of opinions as well as the probability that two given opinions are connected. To illustrate our framework, we focus on a simple model of consensus dynamics on a network and derive a continuum description using techniques inspired by mean-field limits. We also discuss the limitations of this approach and suggest extensions to account for dynamic networks with evolving connections, stochastic effects, and directional interactions.

math.AP↗

arXiv2025

Local asymptotics for the nonlocal Swift-Hohenberg equation

The nonlocal-to-local asymptotics investigation for evolutionary problems is a central topic both in the theory of PDEs and in functional analysis. More recently, it became the main core of the mathematical analysis of phase-separation models. In this paper we focus on the Swift-Hohenberg equations which are key benchmark models in pattern formation problems and amplitude equations. We prove well-posedness of the nonlocal Swift-Hohenberg equation, and study the nonlocal-to-local asymptotics with one and two nonlocal contributions under homogeneous Neumann boundary conditions using suitable energy estimates on the nonlocal problems.

math.AP↗

arXiv2024

On the rate of convergence of Yosida approximation for the nonlocal Cahn-Hilliard equation

It is well-known that one can construct solutions to the nonlocal Cahn-Hilliard equation with singular potentials via Yosida approximation with parameter $λ\to 0$. The usual method is based on compactness arguments and does not provide any rate of convergence. Here, we fill the gap and we obtain an explicit convergence rate $\sqrtλ$. The proof is based on the theory of maximal monotone operators and an observation that the nonlocal operator is of Hilbert-Schmidt type. Our estimate can provide convergence result for the Galerkin methods where the parameter $λ$ could be linked to the discretization parameters, yielding appropriate error estimates.

math.NA↗

arXiv2023

Local asymptotics and optimal control for a viscous Cahn-Hilliard-Reaction-Diffusion model for tumor growth

In this paper we study nonlocal-to-local asymptotics for a tumor-growth model coupling a viscous Cahn-Hilliard equation describing the tumor proportion with a reaction-diffusion equation for the nutrient phase parameter. First, we prove that solutions to the nonlocal Cahn-Hilliard system converge, as the nonlocality parameter tends to zero, to solutions to its local counterpart. Second, we provide first-order optimality conditions for an optimal control problem on the local model, accounting also for chemotaxis, and both for regular or singular potentials, without any additional regularity assumptions on the solution operator. The proof is based on an approximation of the local control problem by means of suitable nonlocal ones, and on proving nonlocal-to-local convergence both for the corresponding dual systems and for the associated first-order optimality conditions.

math.AP↗

arXiv2020

A minimizing-movements approach to GENERIC systems

We present a new time discretization scheme adapted to the structure of GENERIC systems. The scheme is variational in nature and is based on a conditional incremental minimization. The GENERIC structure of the scheme provides stability and convergence of the scheme. We prove that the scheme can be rigorously implemented in the case of the damped harmonic oscillator. Numerical evidence is collected, illustrating the performance of the method.

math.NA↗

arXiv2019

Local asymptotics for nonlocal convective Cahn-Hilliard equations with W^{1,1} kernel and singular potential

We prove existence of solutions and study the nonlocal-to-local asymptotics for nonlocal, convective, Cahn-Hilliard equations in the case of a W^{1,1} convolution kernel and under homogeneous Neumann conditions. Any type of potential, possibly also of double-obstacle or logarithmic type, is included. Additionally, we highlight variants and extensions to the setting of periodic boundary conditions and viscosity contributions, as well as connections with the general theory of evolutionary convergence of gradient flows.

math.AP↗

arXiv2019

Two structure-preserving time discretizations for gradient flows

The equality between dissipation and energy drop is a structural property of gradient-flow dynamics. The classical implicit Euler scheme fails to reproduce this equality at the discrete level. We discuss two modifications of the Euler scheme satisfying an exact energy equality at the discrete level. Existence of discrete solutions and their convergence as the fineness of the partition goes to zero are discussed. Eventually, we address extensions to generalized gradient flows, GENERIC flows, and curves of maximal slope in metric spaces.

math.NA↗

arXiv2019

Nonlocal-to-local convergence of Cahn-Hilliard equations: Neumann boundary conditions and viscosity terms

We consider a class of nonlocal viscous Cahn-Hilliard equations with Neumann boundary conditions for the chemical potential. The double-well potential is allowed to be singular (e.g. of logarithmic type), while the singularity of the convolution kernel does not fall in any available existence theory under Neumann boundary conditions. We prove well-posedness for the nonlocal equation in a suitable variational sense. Secondly, we show that the solutions to the nonlocal equation converge to the corresponding solutions to the local equation, as the convolution kernels approximate a Dirac delta. The asymptotic behaviour is analyzed by means of monotone analysis and Gamma convergence results, both when the limiting local Cahn-Hilliard equation is of viscous type and of pure type.

math.AP↗

arXiv2019

Degenerate nonlocal Cahn-Hilliard equations: well-posedness, regularity and local asymptotics

Existence and uniqueness of solutions for nonlocal Cahn-Hilliard equations with degenerate potential is shown. The nonlocality is described by means of a symmetric singular kernel not falling within the framework of any previous existence theory. A convection term is also taken into account. Building upon this novel existence result, we prove convergence of solutions for this class of nonlocal Cahn-Hilliard equations to their local counterparts, as the nonlocal convolution kernels approximate a Dirac delta. Eventually, we show that, under suitable assumptions on the data, the solutions to the nonlocal Cahn-Hilliard equations exhibit further regularity, and the nonlocal-to-local convergence is verified in a stronger topology.

math.AP↗

arXiv2019

From nonlocal to local Cahn-Hilliard equation

In this paper we prove the convergence of a nonlocal version of the Cahn-Hilliard equation to its local counterpart as the nonlocal convolution kernel is scaled using suitable approximations of a Dirac delta in a periodic boundary conditions setting. This convergence result strongly relies on the dynamics of the problem. More precisely, the $H^{-1}$-gradient flow structure of the equation allows to deduce uniform $H^1$ estimates for solutions of the nonlocal Cahn-Hilliard equation and, together with a Poincaré type inequality by Ponce, provides the compactness argument that allows to prove the convergence result.

math.AP↗

arXiv2016

A meeting point of entropy and bifurcations in cross-diffusion herding

A cross-diffusion system modeling the information herding of individuals is analyzed in a bounded domain with no-flux boundary conditions. The variables are the species' density and an influence function which modifies the information state of the individuals. The cross-diffusion term may stabilize or destabilize the system. Furthermore, it allows for a formal gradient-flow or entropy structure. Exploiting this structure, the global-in-time existence of weak solutions and the exponential decay to the constant steady state is proved in certain parameter regimes. This approach does not extend to all parameters. We investigate local bifurcations from homogeneous steady states analytically to determine whether this defines the validity boundary. This analysis shows that generically there is a gap in the parameter regime between the entropy approach validity and the first local bifurcation. Next, we use numerical continuation methods to track the bifurcating non-homogeneous steady states globally and to determine non-trivial stationary solutions related to herding behaviour. In summary, we find that the main boundaries in the parameter regime are given by the first local bifurcation point, the degeneracy of the diffusion matrix and a certain entropy decay validity condition. We study several parameter limits analytically as well as numerically, with a focus on the role of changing a linear damping parameter as well as a parameter controlling the cross-diffusion. We suggest that our paradigm of comparing bifurcation-generated obstructions to the parameter validity of global-functional methods could also be of relevance for many other models beyond the one studied here.

math.AP↗

arXiv2016

A kinetic equation for economic value estimation with irrationality and herding

A kinetic inhomogeneous Boltzmann-type equation is proposed to model the dynamics of the number of agents in a large market depending on the estimated value of an asset and the rationality of the agents. The interaction rules take into account the interplay of the agents with sources of public information, herding phenomena, and irrationality of the individuals. In the formal grazing collision limit, a nonlinear nonlocal Fokker-Planck equation with anisotropic (or incomplete) diffusion is derived. The existence of global-in-time weak solutions to the Fokker-Planck initial-boundary-value problem is proved. Numerical experiments for the Boltzmann equation highlight the importance of the reliability of public information in the formation of bubbles and crashes. The use of Bollinger bands in the simulations shows how herding may lead to strong trends with low volatility of the asset prices, but eventually also to abrupt corrections.

math.AP↗