arXiv subjects

Andrea Corli

Publications and source records attributed to Andrea Corli.

At least 19 recordsLinked to original sources

arXiv2026

Shock wavefronts for parabolic equations with sign-changing diffusivity

We consider a reaction-diffusion equation in a one-dimensional space, where the diffusion coefficient changes sign from positive to negative and back to positive. The reaction term is bistable, with its interior zero located in the region where the diffusivity is negative. The model does not admit continuous wavefronts, i.e., continuous traveling waves that connect the steady states $0$ and $1$. We prove the existence of a family of shock wavefronts, that is, wavefronts with profiles exhibiting a jump discontinuity. We investigate the properties of these profiles and their propagation speeds. Finally, we apply the results to a recently proposed model describing the movement of a population composed of both isolated and grouped individuals.

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arXiv2023

The role of convection in the existence of wavefronts for biased movements

We investigate a model, inspired by (Johnston et al., Sci. Rep., 7:42134, 2017), to describe the movement of a biological population which consists of isolated and grouped organisms. We introduce biases in the movements and then obtain a scalar reaction-diffusion equation which includes a convective term as a consequence of the biases. We focus on the case the diffusivity makes the parabolic equation of forward-backward-forward type and the reaction term models a strong Allee effect, with the Allee parameter lying between the two internal zeros of the diffusion. In such a case, the unbiased equation (i.e., without convection) possesses no smooth traveling-wave solutions; on the contrary, in the presence of convection, we show that traveling-wave solutions do exist for some significant choices of the parameters. We also study the sign of their speeds, which provides information on the long term behavior of the population, namely, its survival or extinction.

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arXiv2021

Coherence and flow-maximization of a one-way valve

We consider a mathematical model for the gas flow through a one-way valve and focus on two issues. First, we propose a way to eliminate the chattering (the fast switch on and off of the valve) by slightly modifying the design of the valve. This mathematically amounts to the construction of a coupling Riemann solver with a suitable stability property, namely, coherence. We provide a numerical comparison of the behavior of the two valves. Second, we analyze, both analytically and numerically, for several significative situations, the maximization of the flow through the modified valve according to a control parameter of the valve and time.

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arXiv2021

Wavefronts for degenerate diffusion-convection reaction equations with sign-changing diffusivity

We consider in this paper a diffusion-convection reaction equation in one space dimension. The main assumptions are about the reaction term, which is monostable, and the diffusivity, which changes sign once or twice; then, we deal with a forward-backward parabolic equation. Our main results concern the existence of globally defined traveling waves, which connect two equilibria and cross both regions where the diffusivity is positive and regions where it is negative. We also investigate the monotony of the profiles and show the appearance of sharp behaviours at the points where the diffusivity degenerates. In particular, if such points are interior points, then the sharp behaviours are new and unusual.

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arXiv2021

Diffusion-convection reaction equations with sign-changing diffusivity and bistable reaction term

We consider a reaction-diffusion equation with a convection term in one space variable, where the diffusion changes sign from the positive to the negative and the reaction term is bistable. We study the existence of wavefront solutions, their uniqueness and regularity. The presence of convection reveals several new features of wavefronts: according to the mutual positions of the diffusivity and reaction, profiles can occur either for a single value of the speed or for a bounded interval of such values; uniqueness (up to shifts) is lost; moreover, plateaus of arbitrary length can appear; profiles can be singular where the diffusion vanishes.

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arXiv2020

Uniqueness and nonuniqueness of fronts for degenerate diffusion-convection reaction equations

We consider a scalar parabolic equation in one spatial dimension. The equation is constituted by a convective term, a reaction term with one or two equilibria, and a positive diffusivity which can however vanish. We prove the existence and several properties of traveling-wave solutions to such an equation. In particular, we provide a sharp estimate for the minimal speed of the profiles and improve previous results about the regularity of wavefronts. Moreover, we show the existence of an infinite number of semi-wavefronts with the same speed.

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arXiv2018

A model for traffic flow encoding hysteresis

A simple macroscopic model for the vehicular traffic flow with hysteresis is proposed. The model includes drivers' hysteresis behavior into the classical Lighthill-Whitham-Richard (LWR) model. One novelty of the model is how the hysteresis is modeled based on hysteresis behavior reported in literature. By comparing wave patterns producible by the classical LWR model versus this model, the effect of hysteresis can be revealed. For this purpose, wave pattern analysis is carried out by construction of viscous waves and Riemann solvers of the model. Examples of solutions for initial data corresponding to some real world setups or experimental observations are constructed. Many of them qualitatively agree with experimental observations and intuition. These result shows that the model not only keeps the successful part of the LWR model, but also demonstrates many wave patterns observed real world traffic that LWR fails to exhibit, such as stop-and-go waves and many other traffic patterns. A mechanism for formation and decay of stop-and-go patterns and traffic jams is revealed. Furthermore, acceleration shocks in congested zone are predicted, which can serve as fronts of stop-and-go patterns.

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arXiv2018

Viscous profiles in models of collective movements with negative diffusivities

In this paper we consider an advection-diffusion equation, in one space dimension, whose diffusivity can be negative. Such equations arise in particular in the modeling of vehicular traffic flows or crowds dynamics, where a negative diffusivity simulates aggregation phenomena. We focus on traveling-wave solutions that connect two states whose diffusivity has different signs; under some geometric conditions we prove the existence, uniqueness (in a suitable class of solutions avoiding plateaus) and sharpness of the corresponding profiles. Such results are then extended to the case of end states where the diffusivity is positive but it becomes negative in some interval between them. Also the vanishing-viscosity limit is considered. At last, we provide and discuss several examples of diffusivities that change sign and show that our conditions are satisfied for a large class of them in correspondence of real data.

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arXiv2018

Vanishing Viscosity Limits of Scalar Equations with Degenerate Diffusivity

We consider a scalar, possibly degenerate parabolic equation with a source term, in several space dimensions. For initial data with bounded variation we prove the existence of solutions to the initial-value problem. Then we show that these solutions converge, in the vanishing-viscosity limit, to the Kruzhkov entropy solution of the corresponding hyperbolic equation. The proof exploits the H-measure compactness in several space dimensions.

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arXiv2017

The reflection of an ionized shock wave

In a previous paper we studied the thermodynamic and kinetic theory for an ionized gas, in one space dimension; in this paper we provide an application of those results to the reflection of a shock wave in an electromagnetic shock tube. Under some reasonable limitations, which fully agree with experimental data, we prove that both the incident and the reflected shock waves satisfy the Lax entropy conditions; this result holds even outside genuinely nonlinear regions, which are present in the model. We show that the temperature increases in a significant way behind the incident shock front but the degree of ionization does not undergo a similar growth. On the contrary, the degree of ionization increases substantially behind the reflected shock front. We explain these phenomena by means of the concavity of the Hugoniot loci. Therefore, our results not only fit perfectly but explain what was remarked in experiments.

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arXiv2017

Coupling conditions for isothermal gas flow and applications to valves

We consider an isothermal gas flowing through a straight pipe and study the effects of a two-way electronic valve on the flow. The valve is either open or closed according to the pressure gradient and is assumed to act without any time or reaction delay. We first give a notion of coupling solution for the corresponding Riemann problem; then, we highlight and investigate several important properties for the solver, such as coherence, consistence, continuity on initial data and invariant domains. In particular, the notion of coherence introduced here is new and related to commuting behaviors of valves. We provide explicit conditions on the initial data in order that each of these properties is satisfied. The modeling we propose can be easily extended to a very wide class of valves.

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arXiv2017

The system of ionized gas dynamics

The aim of this paper is to study a system of three equations for ionized gas dynamics at high temperature, in one spatial dimension. In addition to the mass density, pressure and particle velocity, a further quantity is needed, namely, the degree of ionization. The system is supplemented by the first and second law of thermodynamics and by an equation of state; all of them involve the degree of ionization. At last, under the assumption of thermal equilibrium, the system is closed by requiring Saha's ionization equation. The geometric properties of the system are rather complicated: in particular, we prove the loss of convexity (genuine nonlinearity) for both forward and backward characteristic fields, and hence the loss of concavity of the physical entropy. This takes place in a small bounded region, which we are able to characterize by numerical estimates on the state functions. The structure of shock waves is also studied by a detailed analysis of the Hugoniot locus, which will be used in a forthcoming paper to study the shock tube problem.

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arXiv2017

Sharp profiles in models of collective movements

We consider a parabolic partial differential equation that can be understood as a simple model for crowds flows. Our main assumption is that the diffusivity and the source/sink term vanish at the same point; the nonhomogeneous term is different from zero at any other point and so the equation is not monostable. We investigate the existence, regularity and monotone properties of semi-wavefront solutions as well as their convergence to wavefront solutions.

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arXiv2017

Semi-wavefront solutions in models of collective movements with density-dependent diffusivity

This paper deals with a nonhomogeneous scalar parabolic equation with possibly degenerate diffusion term; the process has only one stationary state. The equation can be interpreted as modeling collective movements (crowd dynamics, for instance). We first prove the existence of semi-wavefront solutions for every wave speed; their properties are investigated. Then, a family of travelling wave solutions is constructed by a suitable combination of the previous semi-wavefront solutions. Proofs exploit comparison-type techniques and are carried out in the case of one spatial variable; the extension to the general case is straightforward.

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arXiv2017

Traveling waves for degenerate diffusive equations on networks

In this paper we consider a scalar parabolic equation on a star graph; the model is quite general but what we have in mind is the description of traffic flows at a crossroad. In particular, we do not necessarily require the continuity of the unknown function at the node of the graph and, moreover, the diffusivity can be degenerate. Our main result concerns a necessary and sufficient algebraic condition for the existence of traveling waves in the graph. We also study in great detail some examples corresponding to quadratic and logarithmic flux functions, for different diffusivities, to which our results apply.

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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.

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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.

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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.

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