Bruna M, Burger M, Esposito A, Schulz S (2022)

Publication Type: Journal article

Publication year: 2022


Book Volume: 54

Pages Range: 5662-5697

Journal Issue: 6

DOI: 10.1137/21M1462039


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.

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Bruna, M., Burger, M., Esposito, A., & Schulz, S. (2022). WELL-POSEDNESS OF AN INTEGRO-DIFFERENTIAL MODEL FOR ACTIVE BROWNIAN PARTICLES. SIAM Journal on Mathematical Analysis, 54(6), 5662-5697. https://dx.doi.org/10.1137/21M1462039


Bruna, Maria, et al. "WELL-POSEDNESS OF AN INTEGRO-DIFFERENTIAL MODEL FOR ACTIVE BROWNIAN PARTICLES." SIAM Journal on Mathematical Analysis 54.6 (2022): 5662-5697.

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