A three-dimensional Keller-Segel-Navier–Stokes system involving subquadratic logistic degradation: global generalized solutions and eventual smoothness

Abstract

In this paper, we consider a Keller-Segel-Navier–Stokes system involving subquadratic logistic degradation: nt+u·∇n=Δn-∇·(n∇c)+ρn-μnα,ct+u·∇c=Δc-c+n,ut+(u·∇)u=Δu+∇P+n∇ϕ,∇·u=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ccl} n_t + {\ extbf{u}}\\cdot \ abla n & =& \\Delta n-\ abla \\cdot (n\ abla c)+\\rho n- \\mu n^{\\alpha }, \\\\ \\, c_t + {\ extbf{u}}\\cdot \ abla c & =& \\Delta c -c+n, \\\\ {\ extbf{u}}_t+ ({\ extbf{u}}\\cdot \ abla ) {\ extbf{u}}& =& \\Delta {\ extbf{u}}+ \ abla P + n\ abla \\phi , \\\\ \ abla \\cdot {\ extbf{u}}& = & 0 \\end{array} \\right. \\end{aligned}$$\\end{document}in a three-dimensional smoothly bounded domain along with reasonably mild initial conditions and no-flux/no-flux/Dirichlet boundary conditions, where ρ∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho \\in {\\mathbb {R}}$$\\end{document} and μ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu >0$$\\end{document}. The purpose of the present work is to firstly establish the generalized solvability for the model under the subquadratic exponent restriction α≥43\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\ge \\frac{4}{3}$$\\end{document}, which indicates that persistent Dirac-type singularities can be ruled out, and to secondly exhibit the eventual smoothness of these solutions under the stronger restriction α>53\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha > \\frac{5}{3}$$\\end{document} whenever ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document} is not too large in the sense of (ρ++1)α-1ρ+≤δ0μα,(ρ++1)min{1,α-1}ρ+max{1,3-α}≤δ0μ2,ρ+≤δ0μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\big (\\rho _++1\\big )^{\\alpha -1}{\\rho _+}\\le \\delta _0\\mu ^{\\alpha },\\qquad \\big (\\rho _++1\\big )^{\\min \\{1,\\,\\alpha -1\\}}{\\rho _+}^{\\max \\{1,\\,3-\\alpha \\}}\\le \\delta _0\\mu ^{2},\\qquad \\rho _+\\le \\delta _0\\mu \\end{aligned}$$\\end{document}for some δ0=δ0(α)>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta _0=\\delta _0(\\alpha )>0$$\\end{document}. These results especially extend the precedent works due to Winkler (J Functional Anal 276: 1339-1401, 2019; Comm Math Phys 367: 439–489, 2022), where, among other things, the corresponding studies focus on the case α=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =2$$\\end{document} of quadratic degradation.