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

Abstract

We consider chemotaxis-Navier–Stokes systems with logistic proliferation and signal consumption of the form for parameter choices kappa ge 0 and mu >0. Herein, we moreover impose a nonnegative and time-constant prescribed concentration c_star in C^2({overline{Omega }}) for the signal chemical on the boundary of the domain Omega subset {mathbb {R}}^{mathcal {N}} with {mathcal {N}}in {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 being also constant in space. We show that sufficiently strong logistic influence, in the sense that for omega >0 and mu _0>0 there is some eta =eta (omega ,mu _0,c_star )>0 with the property that whenever μ0≤μandκmin{μ,μN+66+ω}<η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mu _0\\le \\mu \\quad \ ext {and}\\quad \\frac{\\kappa }{\\min \\{\\mu ,\\mu ^{\\frac{{\\mathcal {N}}+6}{6}+\\omega }\\}}<\\eta \\end{aligned}$$\\end{document}are satisfied the global weak solution eventually becomes a smooth and classical solution with waiting time depending on omega ,mu _0,eta ,c_star and the initial data.