The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes System

Abstract

We consider initial-boundary value problems for the κ-dependent family of chemotaxis-(Navier-)Stokes systemsin a bounded domain Ω ⊂ R 3 with smooth boundary and given potential function φ ∈ C 1+β Ω for some β > 0. It is known that for fixed κ ∈ R an associated initial-boundary value problem possesses at least one global weak solution (n (κ) , c (κ) , u (κ) ), which after some waiting time becomes a classical solution of the system. In this work we will show that upon letting κ → 0 the solutions (n (κ) , c (κ) , u (κ) ) converge towards a weak solution of the Stokes variant (κ = 0) of the systems above with respect to the strong topology in certain Lebesgue and Sobolev spaces. We thereby extend the recently obtained result on the Stokes limit process for classical solutions in the two-dimensional setting to the more intricate three-dimensional case.

Global generalized solutions to a forager–exploiter model with superlinear degradation and their eventual regularity properties

“…Reasoning along these lines has previously been utilized in similar settings and can e.g. be found in [41,Lemma 7.13], [20,Lemma 3.12] and [6,Lemma 7.5].…”

Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary

“…Likewise, to derive (6.9) we observe that since the inequality n ≥ 3 warrants that 1 + (n+1)(n−2) 3n+2 = n(n+2) 3n+2 ≥ 4 3 , and since W 1, n(n+2) 3n+2 (Ω) ֒→ L n+2 2 (Ω) due to the fact that 1− 3n+2 n+2 = − 2n n+2 , with some c 2 > 0 we have ∇ψ…”

“…as documented for the Neumann problem for (1.1) in [26, (3.1)]. Replacing homogeneous Neumann by inhomogeneous Dirichlet boundary conditions for v, in the derivation of (1.2) additional boundary integrals arise, destroying the (quasi-)Lyapunov structure of (1.2) or relatives thereof on which existence results in, e.g., [18], [27], [35], [4], [40], [26], [33] or also [15] essentially relied.…”

The small‐convection limit in a two‐dimensional chemotaxis‐Navier‐Stokes system with singular sensitivity

“…In a recent paper, Wang et al 33 proved the solutions ( n κ , c κ , u κ ) of the chemotaxis‐Navier‐Stokes system () with χ ( n , c ) ≡ 1 stabilize to the solution ( n 0 , c 0 , u 0 ) of the corresponding chemotaxis‐Stokes system uniformly with respect to the time as κ → 0, which corresponds to Re → 0 ( is the Reynolds number ) due to the constant viscosity ζ of the fluid. The more recent paper 34 investigated the three‐dimensional version of this problem. Later, Wu‐Xiang 35 proved the solutions ( n κ , c κ , u κ ) of the chemotaxis‐Navier‐Stokes system () stabilize to the solution ( n 0 , c 0 , u 0 ) of the corresponding chemotaxis‐Stokes system uniformly with respect to the time as κ → 0.…”