Global solvability in a three-dimensional Keller-Segel-Stokes system involving arbitrary superlinear logistic degradation
Yulan Wang, Michael Winkler, Zhaoyin Xiang
Abstract
Abstract The Keller-Segel-Stokes system (*) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mtable columnalign="right center left" rowspacing="3pt" columnspacing="thickmathspace"> <m:mtr> <m:mtd> <m:mfenced open="{" close=""> <m:mtable columnalign="left center left left" rowspacing="0.683em 0.683em 0.4em" columnspacing="1em"> <m:mtr> <m:mtd> <m:msub> <m:mi>n</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo>+</m:mo> <m:mi>u</m:mi> <m:mo>⋅</m:mo> <m:mi mathvariant="normal">∇</m:mi> <m:mi>n</m:mi> </m:mtd> <m:mtd> <m:mo>=</m:mo> </m:mtd> <m:mtd> <m:mi>Δ</m:mi> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mi mathvariant="normal">∇</m:mi> <m:mo>⋅</m:mo> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mi mathvariant="normal">∇</m:mi> <m:mi>c</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>+</m:mo> <m:mi>ρ</m:mi> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mi>μ</m:mi> <m:msup> <m:mi>n</m:mi> <m:mi>α</m:mi> </m:msup> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:msub> <m:mi>c</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo>+</m:mo> <m:mi>u</m:mi> <m:mo>⋅</m:mo> <m:mi mathvariant="normal">∇</m:mi> <m:mi>c</m:mi> </m:mtd> <m:mtd> <m:mo>=</m:mo> </m:mtd> <m:mtd> <m:mi>Δ</m:mi> <m:mi>c</m:mi> <m:mo>−</m:mo> <m:mi>c</m:mi> <m:mo>+</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:msub> <m:mi>u</m:mi> <m:mi>t</m:mi> </m:msub> </m:mtd> <m:mtd> <m:mo>=</m:mo> </m:mtd> <m:mtd> <m:mi>Δ</m:mi> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi mathvariant="normal">∇</m:mi> <m:mi>P</m:mi> <m:mo>−</m:mo> <m:mi>n</m:mi> <m:mi mathvariant="normal">∇</m:mi> <m:mi>Λ</m:mi> <m:mo>,</m:mo> <m:mspace width="2em"/> <m:mi mathvariant="normal">∇</m:mi> <m:mo>⋅</m:mo> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mfenced> </m:mtd> </m:mtr> </m:mtable> </m:math> $$\begin{eqnarray*} \left\{ \begin{array}{lcll} n_t + u\cdot\nabla n &=& \it\Delta n - \nabla \cdot (n\nabla c) + \rho n - \mu n^\alpha, \\[1mm] c_t + u\cdot\nabla c &=& \it\Delta c-c+n, \\[1mm] u_t &=& \it\Delta u + \nabla P - n\nabla \it\Lambda, \qquad \nabla\cdot u =0, \end{array} \right. \end{eqnarray*}$$ is considered in a bounded domain Ω ⊂ ℝ 3 with smooth boundary, with parameters ρ ≥ 0, μ > 0 and α > 1, and with a given gravitational potential Λ ∈ W 2,∞ ( Ω ). It is shown that in this general setting, when posed under no-flux boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u , and for any suitably regular initial data, an associated initial value problem possesses at least one globally defined solution in an appropriate generalized sense. Since it is well-known that in the absence of absorption, already the corresponding fluid-free subsystem with u ≡ 0 and μ = 0 admits some solutions blowing up in finite time, this particularly indicates that any power-type superlinear degradation of the form in (*) goes along with some significant regularizing effect.