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

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