Global classical solutions of Keller-Segel-(Navier)-Stokes system with nonlinear motility functions

This paper establishes the global existence of solutions to the following Keller-Segel-(Navier)-Stokes system with nonlinear motility functions {n(t) + u . del n = del . (gamma(c)del n - n phi(c)del c), x is an element of Omega, t > 0, c(t) + u . del c = d Delta c + n - c, x is an element of Omega, t > 0, (*) u(t) + kappa(u .del)u = Delta u + del P + n del phi, x is an element of Omega, t > 0, del . u = 0, x is an element of Omega, t > 0 in a bounded domain Omega subset of R-N (2 <= N <= 3) with smooth boundary, where the density-dependent motility functions gamma(c) and phi(c) denote the diffusive and chemotactic coefficients, respectively. The major technical difficulty in the analysis is the possible degeneracy of diffusion. Assume that N = 2, kappa not equal 0 or N = 3, kappa = 0. Then for all reasonably regular initial data, an associated no-flux type initial-boundary value problem (*) admits a global classical solution when gamma(c) > 0 and phi(c) > 0 are smooth on [0, infinity) and satisfy inf(c>0) d gamma(c)/c phi(c)(c phi(c) + d - gamma(c))(+) > N/2. (C) 2022 Elsevier Inc. All rights reserved.

Automated summary

This paper establishes the global existence of solutions for the Keller-Segel-(Navier)-Stokes system with nonlinear motility functions. The system describes a biological diffusion and chemotaxis process, taking into account possible diffusion degeneracy. Under certain conditions, it is proven that the system has a global classical solution for reasonable initial data.