arXiv subjects

Debora Amadori

Publications and source records attributed to Debora Amadori.

15 recordsLinked to original sources

arXiv2026

Systems of nonlocal conservation laws: well-posedness and the singular limit for a nonlocal generalized Aw-Rascle-Zhang model

In this paper, we study a system of nonlocal conservation laws motivated by traffic flow: a nonlocal version of the generalized Aw-Rascle-Zhang (GARZ) model. The nonlocality arises from downstream spatial averaging of the velocity by a one-sided kernel. We prove the existence and uniqueness of weak solutions for initial data of bounded variation via a fixed-point argument in the nonlocal velocity. We also establish stability with respect to the initial datum and an approximation of weak solutions by strong solutions in $L^1$. Under additional, physically meaningful assumptions on the velocity and the initial datum, we obtain either a maximum principle for the density or invariant-region estimates. Finally, we study the singular limit as the nonlocal kernel converges to a Dirac distribution. Indeed, under additional assumptions, convergence to the unique local entropy solution can be proved. Some numerical simulations are provided, and the paper concludes with a discussion of open problems.

math.AP↗

arXiv2026

Decay of periodic entropy solutions to Euler-alignment systems with non-constant kernel

We consider a hydrodynamic model of flocking-type with pressure on the torus, with integrable interaction kernel and density bounded away from zero. We prove that, if an entropy weak solution exists, then its $L^2$ norm decays exponentially fast in time towards the mean values on the period. The proof relies on the study of a suitable energy functional that combines a strictly convex entropy for the system and a potential term, and this allows us to treat the nonlocal source term for a class of strictly positive convolution kernels in $L^1$.

math.AP↗

arXiv2026

A nonlocal Aw-Rascle-Zhang system with linear pressure term

In this paper, we study a nonlocal extension of the Aw-Rascle-Zhang traffic model, where the pressure-like term is modeled as a convolution between vehicle density and a kernel function. This formulation captures nonlocal driver interactions and aligns structurally with the Euler-alignment system studied in [23]. Using a sticky particle approximation, we construct entropy solutions to the equation for the cumulative density and prove convergence of approximate solutions to weak solutions of the nonlocal system. The analysis includes well-posedness, stability estimates, and an entropic selection principle.

math.AP↗

arXiv2026

Conservation Laws with Discontinuous Gradient-Dependent Flux: the Unstable Case

The paper is concerned with a scalar conservation law with discontinuous gradient-dependent flux. Namely, the flux is described by two different functions $f(u)$ or $g(u)$, when the gradient $u_x$ of the solution is positive or negative, respectively. We study here the unstable case where $f(u)>g(u)$ for all $u\in {\mathbb R}$. Assuming that both $f$ and $g$ are strictly convex, solutions to the Riemann problem are constructed. Even for a smooth initial data, examples show that the Cauchy problem can have infinitely many solutions. For an initial data which is piecewise monotone, i.e., increasing or decreasing on a finite number of intervals, a solution can be constructed globally in time. It is proved that such solution is unique under the additional requirement that the number of interfaces, where the flux switches between $f$ and $g$, remains as small as possible.

math.AP↗

arXiv2024

Conservation Laws with Discontinuous Gradient-Dependent Flux: the Stable Case

The paper is concerned with a scalar conservation law with discontinuous gradient-dependent flux. Namely, the flux is described by two different functions $f(u)$ or $g(u)$, when the gradient $u_x$ of the solution is positive or negative, respectively. We study here the stable case where $f(u)<g(u)$ for all $u\in {\mathbb R}$, with $f,g$ smooth but possibly not convex. A front tracking algorithm is introduced, proving that piecewise constant approximations converge to the trajectories of a contractive semigroup on $\mathbf{L}^1({\mathbb R})$. In the spatially periodic case, we prove that semigroup trajectories coincide with the unique limits of a suitable class of vanishing viscosity approximations.

math.AP↗

arXiv2023

Unconditional flocking for weak solutions to self-organized systems of Euler-type with all-to-all interaction kernel

We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in one-space dimension and establish that the global entropy weak solutions, constructed in [2] to the Cauchy problem for any $BV$ initial data that has finite total mass confined in a bounded interval and initial density uniformly positive therein, admit unconditional time-asymptotic flocking without any further assumptions on the initial data. In addition, we show that the convergence to a flocking profile occurs exponentially fast.

math.AP↗

arXiv2023

The mathematical theory of Hughes' model: a survey of results

We provide an overview of the results on Hughes' model for pedestrian movements available in the literature. After the first successful approaches to solving a regularised version of the model, researchers focused on the structure of the Riemann problem, which led to local-in-time existence results for Riemann-type data and paved the way for a WFT (Wave-Front Tracking) approach to the solution semigroup. In parallel, a DPA (Deterministic Particles Approximation) approach was developed in the spirit of follow-the-leader approximation results for scalar conservation laws. Beyond having proved to be powerful analytical tools, the WFT and the DPA approaches also led to interesting numerical results. However, only existence theorems on very specific classes of initial data (essentially ruling out non-classical shocks) have been available until very recently. A proper existence result using a DPA approach was proven not long ago in the case of a linear coupling with the density in the eikonal equation. Shortly after, a similar result was proven via a fixed point approach. We provide a detailed statement of the aforementioned results and sketch the main proofs. We also provide a brief overview of results that are related to Hughes' model, such as the derivation of a dynamic version of the model via a mean-field game strategy, an alternative optimal control approach, and a localized version of the model. We also present the main numerical results within the WFT and DPA frameworks.

math.AP↗

arXiv2022

BV solutions for a hydrodynamic model of flocking--type with all-to-all interaction kernel

We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in one-space dimension and establish global existence of entropy weak solutions with concentration to the Cauchy problem for any BV initial data that has finite total mass confined in a bounded interval and initial density uniformly positive therein. In addition, under a suitable condition on the initial data, we show that entropy weak solutions with concentration admit time-asymptotic flocking.

math.AP↗

arXiv2021

Exact Response Theory and Kuramoto dynamics

The dynamics of Kuramoto oscillators is investigated in terms of the exact response theory based on the Dissipation Function, which has been introduced in the field of nonequilibrium molecular dynamics. While linear response theory is a cornerstone of nonequilibrium statistical mechanics, it does not apply, in general, to systems undergoing phase transitions. Indeed, even a small perturbation may in that case result in a large modification of the state. An exact theory is instead expected to handle such situations. The Kuramoto dynamics, which undergoes synchronization transitions, is thus investigated as a testbed for the exact theory mentioned above. A comparison between the two approaches shows how the linear theory fails, while the exact theory yields the correct response.

cond-mat.stat-mech↗

arXiv2021

On the decay in $W^{1,\infty}$ for the 1D semilinear damped wave equation on a bounded domain

In this paper we study a semilinear wave equation with nonlinear, time-dependent damping in one space dimension. For this problem, we prove a well-posedness result in $W^{1,\infty}$ in the space-time domain $(0,1)\times [0,+\infty)$. Then we address the problem of the time-asymptotic stability of the zero solution and show that, under appropriate conditions, the solution decays to zero at an exponential rate in the space $W^{1,\infty}$. The proofs are based on the analysis of the corresponding semilinear system for the first order derivatives, for which we show a contractive property of the invariant domain.

math.AP↗

arXiv2018

Decay of approximate solutions for the damped semilinear wave equation on a bounded 1d domain

In this paper we study the long time behavior for a semilinear wave equation with space-dependent and nonlinear damping term. After rewriting the equation as a first order system, we define a class of approximate solutions that employ tipical tools of hyperbolic systems of conservation laws, such as the Riemann problem. By recasting the problem as a discrete-time nonhomogeneous system, which is related to a probabilistic interpretation of the solution, we provide a strategy to study its long-time behavior uniformly with respect to the mesh size parameter $Δx=1/N\to 0$. The proof makes use of the Birkhoff decomposition of doubly stochastic matrices and of accurate estimates on the iteration system as $N\to\infty$. Under appropriate assumptions on the nonlinearity, we prove the exponential convergence in $L^\infty$ of the solution to the first order system towards a stationary solution, as $t\to+\infty$, as well as uniform error estimates for the approximate solutions.

math.AP↗

arXiv2015

Global existence of solutions for a multi-phase flow: a bubble in a liquid tube and related cases

In this paper we study the problem of the global existence (in time) of weak, entropic solutions to a system of three hyperbolic conservation laws, in one space dimension, for large initial data. The system models the dynamics of phase transitions in an isothermal fluid; in Lagrangian coordinates, the phase interfaces are represented as stationary contact discontinuities. We focus on the persistence of solutions consisting in three bulk phases separated by two interfaces. Under some stability conditions on the phase configuration and by a suitable front tracking algorithm we show that, if the BV-norm of the initial data is less than an explicit (large) threshold, then the Cauchy problem has global solutions.

math.AP↗

arXiv2015

Global existence of solutions for a multi-phase flow: a drop in a gas-tube

In this paper we study the flow of an inviscid fluid composed by three different phases. The model is a simple hyperbolic system of three conservation laws, in Lagrangian coordinates, where the phase interfaces are stationary. Our main result concerns the global existence of weak entropic solutions to the initial-value problem for large initial data.

math.AP↗

arXiv2014

Global weak solutions for a model of two-phase flow with a single interface

We consider a simple nonlinear hyperbolic system modeling the flow of an inviscid fluid. The model includes as state variable the mass density fraction of the vapor in the fluid and then phase transitions can be taken into consideration; moreover, phase interfaces are contact discontinuities for the system. We focus on the special case of initial data consisting of two different phases separated by an interface. We find explicit bounds on the (possibly large) initial data in order that weak entropic solutions exist for all times. The proof exploits a carefully tailored version of the front tracking scheme.

math.AP↗

arXiv2007

On a model of multiphase flow

We consider a hyperbolic system of three conservation laws in one space variable. The system is a model for fluid flow allowing phase transitions; in this case the state variables are the specific volume, the velocity and the mass density fraction of the vapor in the fluid. For a class of initial data having large total variation we prove the global existence of solutions to the Cauchy problem.

math.AP↗