Reaction-Driven Relaxation in Three-Dimensional Keller–Segel–Navier–Stokes Interaction

Abstract

The Keller–Segel–Navier–Stokes system $$\begin{aligned} \left\{ \begin{array}{rcll} n_t + u\cdot \nabla n &{}=&{} \Delta n - \chi \nabla \cdot (n\nabla c) + \rho n-\mu n^2,\\ c_t + u\cdot \nabla c &{}=&{} \Delta c-c+n, \\ u_t + (u\cdot \nabla )u &{}=&{} \Delta u + \nabla P + n \nabla \phi + f(x,t), \qquad \nabla \cdot u=0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$ n t + u · ∇ n = Δ n - χ ∇ · ( n ∇ c ) + ρ n - μ n 2 , c t + u · ∇ c = Δ c - c + n , u t + ( u · ∇ ) u = Δ u + ∇ P + n ∇ ϕ + f ( x , t ) , ∇ · u = 0 , ( ⋆ ) is considered in a smoothly bounded convex domain $$\Omega \subset \mathbb {R}^3$$ Ω ⊂ R 3 , with $$\phi \in W^{2,\infty }(\Omega )$$ ϕ ∈ W 2 , ∞ ( Ω ) and $$f\in C^1(\bar{\Omega }\times [0,\infty );\mathbb {R}^3)$$ f ∈ C 1 ( Ω ¯ × [ 0 , ∞ ) ; R 3 ) , and with $$\chi >0, \rho \in \mathbb {R}$$ χ > 0 , ρ ∈ R and $$\mu >0$$ μ > 0 . As recent literature has shown, for all reasonably mild initial data a corresponding no-flux/no-flux/Dirichlet initial-boundary value problem possesses a global generalized solution, but the knowledge on its regularity properties has not yet exceeded some information on fairly basic integrability features. The present study reveals that whenever $$\omega >0$$ ω > 0 , requiring that $$\begin{aligned} \frac{\rho }{\min \{\mu ,\mu ^{\frac{3}{2}+\omega }\}} < \eta \end{aligned}$$ ρ min { μ , μ 3 2 + ω } < η with some $$\eta =\eta (\omega )>0$$ η = η ( ω ) > 0 , and that f satisfies a suitable assumption on ultimate smallness, is sufficient to ensure that each of these generalized solutions becomes eventually smooth and classical. Furthermore, under these hypotheses ($$\star $$ ⋆ ) is seen to admit an absorbing set with respect to the topology in $$L^\infty (\Omega )$$ L ∞ ( Ω ) . By trivially applying to the case when $$\mu >0$$ μ > 0 is arbitrary and $$\rho \le 0$$ ρ ≤ 0 , these results especially assert essentially unconditional statements on eventual regularity in taxis-reaction systems interacting with liquid environments, such as arising in contexts of models for broadcast spawning discussed in recent literature.

Stabilization in a two‐dimensional fractional chemotaxis‐Navier–Stokes system with logistic source

“…He further proved that each of these generalized solutions becomes eventually smooth and classical under some specific conditions in Winkler. 46 Concerning the whole space R N , there are also some results. If g(n) = 0, Liu and Lors 24 proved the global existence of weak solutions to the two-dimensional chemotaxis-Navier-Stokes system for large data.…”

“…Recently, Winkler 40 proved that the three‐dimensional chemotaxis‐Navier–Stokes system with logistic source admits at least one globally defined solution in an appropriate generalized sense and obtained a spatially homogeneous state under the hypothesis that $$$ b>\frac{\chi \sqrt{a_{+}}}{4} $$$. He further proved that each of these generalized solutions becomes eventually smooth and classical under some specific conditions in Winkler 46 …”

Singular limit problem for the Keller–Segel–Navier–Stokes system on torus

“…Nowadays, people have derived many kinds of Keller–Segel systems, see [9, 13, 14, 18, 21, 27‐32, 35‐41], where [9] investigated a degenerate $$$ p $$$‐Laplacian Keller–Segel model, [14] proved the well‐posedness of a full Keller–Segel model on nonsmooth domains, and [35, 37] obtained the global existence of the Keller–Segel–Navier–Stokes system in the weak sense and derived the boundedness and asymptotic stabilization of the solution. Furthermore, [40] established a singular limit result for a Keller–Segel–Navier–Stokes system on total space $$$ {\mathrm{\mathbb{R}}}^n $$$.…”

On a three‐dimensional chemotaxis‐Stokes system with nonlinear sensitivity modeling coral fertilization

“…When $$N=3$$, $$\rho =\mu =\kappa =0$$, $$S(n)=n(n+1)^{-\alpha }$$, Winkler [8] proved that under the condition that $$\alpha >\frac{1}{3}$$, for all sufficiently regular initial data a corresponding Neumann‐Neumann‐Dirichlet initial‐boundary value problem possesses a globally bounded classical solution. When $$\kappa =1$$, $$S(n)=\chi n$$ with $$\chi >0$$, it is shown in [9] that whenever $$\omega >0$$, requiring that $$\frac{\rho }{\min \lbrace \mu ,\mu ^{\frac{3}{2}+\omega }\rbrace }<\eta$$ with some $$\eta =\eta (\omega )>0$$, and that f satisfies a suitable assumption on ultimate smallness, which is sufficient to ensure that each of these generalized solutions becomes eventually smooth and classical in three dimensional case. Furthermore, under these hypotheses, it admits an absorbing set with respect to the topology in $$L^{\infty }(\Omega )$$.…”

“…When 𝑁 = 3, 𝜌 = 𝜇 = 𝜅 = 0, 𝑆(𝑛) = 𝑛(𝑛 + 1) −𝛼 , Winkler [8] proved that under the condition that 𝛼 > , for all sufficiently regular initial data a corresponding Neumann-Neumann-Dirichlet initial-boundary value problem possesses a globally bounded classical solution. When 𝜅 = 1, 𝑆(𝑛) = 𝜒𝑛 with 𝜒 > 0, it is shown in [9] that whenever 𝜔 > 0, requiring that 𝜌 min{𝜇,𝜇 < 𝜂 with some 𝜂 = 𝜂(𝜔) > 0, and that 𝑓 satisfies a suitable assumption on ultimate smallness, which is sufficient to ensure that each of these generalized solutions becomes eventually smooth and classical in three dimensional case. Furthermore, under these hypotheses, it admits an absorbing set with respect to the topology in 𝐿 ∞ (Ω).…”