Solutions to aggregation–diffusion equations with nonlinear mobility constructed via a deterministic particle approximation

Abstract

We investigate the existence of weak type solutions for a class of aggregation-diffusion PDEs with nonlinear mobility obtained as large particle limit of a suitable nonlocal version of the follow-the-leader scheme, which is interpreted as the discrete Lagrangian approximation of the target continuity equation. We restrict the analysis to bounded, nonnegative initial data with bounded variation and away from vacuum, supported in a closed interval with zero-velocity boundary conditions. The main novelties of this work concern the presence of a nonlinear mobility term and the non strict monotonicity of the diffusion function. As a consequence, our result applies also to strongly degenerate diffusion equations. The results are complemented with some numerical simulations.

Multiple patterns formation for an aggregation/diffusion predator-prey system

“…Numerics and perspectives. In this section, we study numerically solutions to system (3) using two different methods, the finite volume method introduced in [12,13] and the particles method studied in [23,28]. We validate the results about the existence of the mixed steady state and the multiple bumps steady states.…”

“…Note that since the values cm i l satisfy (28) we have that B i l = 0. After the manipulations above, equation F i l [p; 0](z) = 0 reads as…”

Opinion formation systems via deterministic particles approximation

“…we deduce that all the terms of I 3 except the last one can be estimated from above by Cσ N u , where C is some positive constant depending on ζ, T and on the proper bounds on D u , φ u , P uh provided by the assumptions (D), (Dif) and (P). Concerning the last term of I 3 , standard computations, the above estimate on |Ẇ i | and (24) and (25)…”

“…where λ is a fixed diffusion coefficient. Let us observe, that, accordingly to the results in [25,28,35], it is possible to generalize the diffusion law (5) to more general non linear expressions. Indeed, the quantity…”

“…Once here, if U(t, w) were the inverse of W(t, z) and u(t, w) were ∂ w U(t, w) then, inspired by the results in [19,25], see also [34,35], u would be a weak solutions of the following aggregation-diffusion PDE…”