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Well-posedness of an integro-differential model for active Brownian particles
- Publication Year :
- 2022
- Publisher :
- Society for Industrial and Applied Mathematics, 2022.
-
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 anyalysis 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.<br />Comment: 35 pages
Details
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....2a8580e52c3aab809f37d47652ac7d96
- Full Text :
- https://doi.org/10.17863/cam.90770