Long-term behaviour in a chemotaxis-fluid system with logistic source

By Johannes Lankeit. 2016

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We consider the coupled chemotaxis Navier–Stokes model with logistic source terms: nt + u ⋅∇n = Δn − χ∇⋅ (n∇c) + κn − μn2, ct + u ⋅∇c = Δc − nc, ut + (u ⋅∇)u = Δu + ∇P + n∇Φ + f,∇⋅ u = 0 in a bounded, smooth domain Ω ⊂ ℝ3 under homogeneous Neumann boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u and with given functions f ∈ L∞(Ω × (0,∞)) satisfying certain decay conditions and Φ ∈ C1+β(Ω) for some β ∈ (0, 1). We construct weak solutions and prove that after some waiting time they become smooth and finally converge to the semi-trivial steady state (κ μ, 0, 0).

Subfields: Cell Biology, Computational Theory and Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/cc50b366df79b1670b19041fb69b4113b8faa57b

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Citations in the corpus

Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement (2017, methodology): “1 Basic decay information Next addressing the large time asymptotics of our solutions, as in several previous studies on qualitative behavior in related chemotaxis-fluid systems with signal absorption ([45], [24], [48], [46]) we shall rely on the following elementary information indicating a certain”
Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion (2016, background): “(The interested reader can consult the introduction of [16].”
Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity (2018, background): “2) the solutions constructed so far are all weak only ([5], [48], [32]) and may possibly develop singularities within finite time (see [6] for a detailed discussion on refined extensibility criteria), after all the dissipative effect of the signal absorption mechanism addressed in (1.”
Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes system with rotational flux components (2018, background): “The knowledge in this respect has become quite comprehensive in the spatially two-dimensional setting in which even Navier-Stokes fluid evolution may lead to global existence of uniquely determined classical solutions ([23], [39]); in the case N = 3, after all, certain global weak and at least event”
Boundedness and stabilization in a three-dimensional two-species chemotaxis-Navier--Stokes system with competitive kinetics (2017, background): “11) for all ε > 0, which together with arguments in [5] implies that there exist a sequence (εj)j∈N such that εj ↘ 0 as j →∞ and functions n1, n2, c, u such that n1, n2 ∈ L(2)loc([0,∞);L(2)(Ω)) ∩ L 4 3 loc([0,∞);W 1, 43 (Ω)), c ∈ L(2)loc([0,∞);W (1,2)(Ω)), u ∈ L(2)loc([0,∞);W 1,2 0,σ (Ω)) and that n”
A quantitative strong parabolic maximum principle and application to a taxis-type migration–consumption model involving signal-dependent degenerate diffusion (2022, background): “[32], [19], [39] for some small selection of examples).”
Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption (2016, background): “In a three-dimensional setting and in the presence of a Navier-Stokes fluid, in [11] it was recently possible to construct global weak solutions for any positive μ, which moreover eventually become classical and uniformly converge to the constant equilibrium in the large-time limit.”