Prescribed signal concentration on the boundary: eventual smoothness in a chemotaxis-Navier–Stokes system with logistic proliferation

Abstract

AbstractWe consider chemotaxis-Navier–Stokes systems with logistic proliferation and signal consumption of the form "Equation missing"for parameter choices $$\kappa \ge 0$$ κ ≥ 0 and $$\mu >0$$ μ > 0 . Herein, we moreover impose a nonnegative and time-constant prescribed concentration $$c_\star \in C^2({\overline{\Omega }})$$ c ⋆ ∈ C 2 ( Ω ¯ ) for the signal chemical on the boundary of the domain $$\Omega \subset {\mathbb {R}}^{\mathcal {N}}$$ Ω ⊂ R N with $${\mathcal {N}}\in \{2,3\}$$ N ∈ { 2 , 3 } . After first extending the previously known result on time-global existence of weak solutions for the Stokes variant to the full Navier–Stokes setting, we proceed with an investigation of eventual regularity properties in the slightly more restrictive setting of $$c_\star $$ c ⋆ being also constant in space. We show that sufficiently strong logistic influence, in the sense that for $$\omega >0$$ ω > 0 and $$\mu _0>0$$ μ 0 > 0 there is some $$\eta =\eta (\omega ,\mu _0,c_\star )>0$$ η = η ( ω , μ 0 , c ⋆ ) > 0 with the property that whenever $$\begin{aligned} \mu _0\le \mu \quad \text {and}\quad \frac{\kappa }{\min \{\mu ,\mu ^{\frac{{\mathcal {N}}+6}{6}+\omega }\}}<\eta \end{aligned}$$ μ 0 ≤ μ and κ min { μ , μ N + 6 6 + ω } < η are satisfied the global weak solution eventually becomes a smooth and classical solution with waiting time depending on $$\omega ,\mu _0,\eta ,c_\star $$ ω , μ 0 , η , c ⋆ and the initial data.