Global Existence and Temporal Decay in Keller-Segel Models Coupled to Fluid Equations

Abstract

We consider a Keller-Segel model coupled to the incompressible Navier-Stokes equations in spatial dimensions two and three. We establish the local existence of regular solutions and present some blow-up criteria for both cases that equations of oxygen concentration is of parabolic or hyperbolic type. We also prove global existence and decay estimate in time under the some smallness conditions of initial data.

Stabilization in a two-dimensional chemotaxis-Navier–Stokes system

“…with respect to the norm of n in L ∞ (Ω), in either finite or infinite time (cf. also [4] for some refined extensibility criteria for local-in-time smooth solutions).…”

“…[31] and [40]). If the range of m is further restricited by assuming m > 4 3 , then global existence of, possibly unbounded, solutions can be derived even in presence of the full nonlinear Navier-Stokes equations ( [34]). As alternative blow-up preventing mechanisms, the authors in [2] and [34] identify certain saturation effects at large cell densities in the cross-diffusive term, as well as the inclusion of logistic-type cell kinetics with quadratic death terms in (1.2), in both cases leading to corresponding results on global existence of weak solutions.…”

Existence of global solutions for a chemotaxis-fluid system with nonlinear diffusion

“…[1], [2], [4], [9] and [15] for the case that α = 0 and references therein). It was proved in [5] that weak solutions of (KSS) for bounded domains exist in case that α ∈ (1/2, 1] in two dimensions or in case that α ∈ ( −5+ √…”

Global existence and boundedness in a higher-dimensional quasilinear chemotaxis system

“…This system can be used to describe the dynamics of oxygen, swimming bacteria and viscous incompressible fluids. In the past several years, numerous analytical approaches have addressed issues of well-posedness for corresponding initial-value problems of system (1.2) in either bounded or unbounded domains, with various assumptions on the scalar functions S, f and φ (see e.g., [1,2,4,17,31,[33][34][35]38]). A considerable literature also addresses related models with nonlinear diffusion (see e.g., [3,5,16,25,26,28]).…”

“…A pair (u, v) of nonnegative functions defined on Ω × (0, T ) is called a weak solution to system (1.1) if (1). u ∈ L ∞ (0, T ); L ∞ (Ω) and ∇ u 0 D(σ)dσ ∈ L 2 (0, T ); L 2 (Ω) , (2). v ∈ L ∞ (0, T ); W 1,∞ (Ω) , (3).…”