arXiv subjects

Peter Hirvonen

Publications and source records attributed to Peter Hirvonen.

3 recordsLinked to original sources

arXiv2026

Superlinear nonlocal diffusion systems for multispecies populations: well-posedness and discrete chain rules

A nonlocal diffusion system describing the dynamics of multi-species populations and approximating the Shigesada-Kawasaki-Teramoto (SKT) model is analyzed in a bounded domain. The model consists of a system of integro-differential equations of Andreu-Mazón-Rossi-Toledo type, in which a nonlocal cross-diffusion operator acts on nonlinear diffusion potentials. We establish the global existence and uniqueness of strong solutions. The existence proof relies on suitable a priori estimates derived from an entropy inequality, whose proof requires the development of novel discrete chain-rule inequalities. As a by-product, these inequalities provide the basis for the construction of structure-preserving finite-volume schemes for the corresponding local SKT system. Uniqueness of solutions is established by means of a duality argument. Finally, one-dimensional numerical simulations illustrate the nonlocal-to-local limit and the qualitative behavior of solutions for superlinear and sublinear diffusion potentials.

math.AP↗

arXiv2026

A nonlocal Busenberg-Travis cross-diffusion system with nonlinear Brinkman law

A nonlocal Busenberg-Travis cross-diffusion system for segregating populations is analyzed in a bounded domain with no-flux boundary conditions. The velocities of the species solve a regularized Darcy law, which can be interpreted as a Brinkman equation. Compared to results in the literature, the density-pressure relation is assumed to be nonlinear. The global existence of weak solutions to this system is shown for a broad range of the exponents of the power-law nonlinearity, and the localization limit is proved. The proofs are based on uniform estimates coming from the Tsallis entropy inequality. Due to regularity issues, the original problem is approximated by various schemes, and the de-regularization limits are obtained through compactness arguments.

math.AP↗

arXiv2024

Analysis of a Poisson-Nernst-Planck cross-diffusion system with steric effects

A transient Poisson-Nernst-Planck system with steric effects is analyzed in a bounded domain with no-flux boundary conditions for the ion concentrations and mixed Dirichlet-Neumann boundary conditions for the electric potential. The steric repulsion of ions is modeled by a localized Lennard-Jones force, leading to cross-diffusion terms. The existence of global weak solutions, a weak--strong uniqueness property, and, in case of pure Neumann conditions, the exponential decay towards the thermal equilibrium state is proved. The main difficulties are the cross-diffusion terms and the different boundary conditions satisfied by the unknowns. These issues are overcome by exploiting the entropy structure of the equations and carefully taking into account the electric potential term. A numerical experiment illustrates the long-time behavior of the solutions when the potential satisfies mixed boundary conditions.

math.AP↗