Abstract
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.
Highlights
We are interested in the advection–diffusion equation ρt + f (ρ)x = (D(ρ)ρx)x, t ≥ 0, x ∈ R
Malaguti modeling can be made in the framework of crowds dynamics; in this case, as we propose in this paper, it may simulates panic behaviors in overcrowded environments [9]
We prove the vanishing viscosity limit
Summary
Experimental data show that this is not the case [18,23]: such pairs usually cover a two-dimensional region To reproduce this effect, either one considers second-order models [1,34,45] or, as in this paper, introduces a diffusive term. Either one considers second-order models [1,34,45] or, as in this paper, introduces a diffusive term In the latter case, the physical flow is q = f (ρ) − D(ρ)ρx, see [4,5,7,32], and the density-flow pairs correctly cover a full two-dimensional region in the (ρ, q)-plane.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
