1,720,968 research outputs found
Stability of the 1D IBVP for a non autonomous scalar conservation law
No full textWe prove the stability with respect to the flux of solutions to initial-boundary value problems for scalar non autonomous conservation laws in one space dimension. Key estimates are obtained through a careful construction of the solution
Erratum to Hyperbolic predators vs. parabolic prey [Commun. Math. Sci., 13, 2, (2015) 369-400]
No full textWe correct an error in the proof of the main result in [R.M. Colombo and E. Rossi, Commun. Math. Sci., 13(2):369-400, 2015]. The theorem, with all the provided estimates, remains true. The online version is corrected
Rigorous estimates on balance laws in bounded domains
No full textThe initial-boundary value problem for a general balance law in a bounded domain is proved to be well posed. Indeed, we show the existence of an entropy solution, its uniqueness and its Lipschitz continuity as a function of time, of the initial datum and of the boundary datum. The proof follows the general lines in [4], striving to provide a rigorous treatment and detailed references
Modelling crowd movements in domains with boundaries
No full textThis paper contributes to the macroscopic modeling of crowd movements. The presented model is non local, i.e., it takes into account interactions among pedestrians at different distances. Particular care is given to how non local interactions are influenced by walls, obstacles and exits. The resulting dynamics captures various well known patterns of crowd movements, such as the clogging of exits and the spontaneous formation of queues. The careful choice of obstacles near an exit is shown to be able to reduce evacuation times. An ad hoc numerical algorithm is detailed, some of its properties discussed and the convergence of the corresponding approximate solutions is investigated
Hyperbolic predators vs. parabolic prey
No full textWe present a nonlinear predator-prey system consisting of a nonlocal conservation law for predators coupled with a parabolic equation for prey. The drift term in the predators' equation is a nonlocal function of the prey density, so that the movement of the predators can be directed towards regions with high prey density. Moreover, Lotka-Volterra type right hand sides describe the feeding. A theorem ensuring existence, uniqueness, continuous dependence of weak solutions, and various stability estimates is proved, in any space dimension. Numerical integrations show a few qualitative features of the solutions
Biological and industrial models motivating nonlocal conservation laws: A review of analytic and numerical results
No full textThis paper is devoted to the overview of recent results concerning nonlocal systems of conservation laws. First, we present a predator - prey model and, second, a model for the laser cutting of metals. In both cases, these equations lead to interesting pattern formation
A modeling framework for biological pest control
Full text linkWe present an analytic framework where biological pest control can be simulated. Control is enforced through the choice of a time and space dependent function representing the deployment of a species of predators that feed on pests. A sample of different strategies aimed at reducing the presence of pests is considered, evaluated and compared. The strategies explicitly taken into account range, for instance, from the uniform deployment of predators on all the available area over a short/long time interval, to the alternated insertion of predators in different specific regions, to the release of predators in suitably selected regions. The effect of each strategy is measured through a suitably defined cost, essentially representing the total amount of prey present over a given time interval over all the considered region, but the variation in time of the total amount of pests is also evaluated. The analytic framework is provided by an integro-differential hyperbolic-parabolic system of partial differential equations. While prey diffuse according to the usual Laplace operator, predators hunt for prey, moving at finite speed towards regions of higher prey density
IBVPs for scalar conservation laws with time discontinuous fluxes
No full textThe initial boundary value problem for a class of scalar nonautonomous conservation laws in 1 space dimension is proved to be well posed and stable with respect to variations in the flux. Targeting applications to traffic, the regularity assumptions on the flow are extended to a merely L ∞ dependence on time. These results ensure, for instance, the well-posedness of a class of vehicular traffic models with time-dependent speed limits. A traffic management problem is then shown to admit an optimal solution
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