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@article{faucris.286398910,
abstract = {We propose a general strategy for solving nonlinear integro-differential evolution problems with periodic boundary conditions, where no direct maximum/minimum principle is available. This is motivated by the study of recent macroscopic models for active Brownian particles with repulsive interactions, consisting of advection-diffusion processes in the space of particle position and orientation. We focus on one of such models, namely a semilinear parabolic equation with a nonlinear active drift term, whereby the velocity depends on the particle orientation and angle-independent overall particle density (leading to a nonlocal term by integrating out the angular variable). The main idea of the existence analysis is to exploit a priori estimates from (approximate) entropy dissipation. The global existence and uniqueness of weak solutions is shown using a two-step Galerkin approximation with appropriate cutoff in order to obtain nonnegativity, an upper bound on the overall density, and preserve a priori estimates. Our analysis naturally includes the case of finite systems, corresponding to the case of a finite number of directions. The Duhamel principle is then used to obtain additional regularity of the solution, namely continuity in time-space. Motivated by the class of initial data relevant for the application, which includes perfectly aligned particles (same orientation), we extend the well-posedness result to very weak solutions allowing distributional initial data with low regularity.},
author = {Bruna, Maria and Burger, Martin and Esposito, Antonio and Schulz, Simon},
doi = {10.1137/21M1462039},
faupublication = {yes},
journal = {SIAM Journal on Mathematical Analysis},
keywords = {active particles; Galerkin approximation; parabolic equations; periodic heat kernel; space-periodic problems},
note = {CRIS-Team Scopus Importer:2022-12-09},
pages = {5662-5697},
peerreviewed = {Yes},
title = {{WELL}-{POSEDNESS} {OF} {AN} {INTEGRO}-{DIFFERENTIAL} {MODEL} {FOR} {ACTIVE} {BROWNIAN} {PARTICLES}},
volume = {54},
year = {2022}
}