Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity
Yu Tian, Zhaoyin Xiang
Abstract
Abstract In this article, we will develop an analytical approach to construct the global bounded weak solutions to the initial-boundary value problem of a three-dimensional chemotaxis-Stokes system with porous medium cell diffusion <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Δ</m:mi> <m:msup> <m:mrow> <m:mi>n</m:mi> </m:mrow> <m:mrow> <m:mi>m</m:mi> </m:mrow> </m:msup> </m:math> \Delta {n}^{m} for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>m</m:mi> <m:mo>≥</m:mo> <m:mfrac> <m:mrow> <m:mn>65</m:mn> </m:mrow> <m:mrow> <m:mn>63</m:mn> </m:mrow> </m:mfrac> </m:math> m\ge \frac{65}{63} and general sensitivity. In particular, this extended the precedent results which asserted global solvability within the larger range <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>m</m:mi> <m:mo>></m:mo> <m:mfrac> <m:mrow> <m:mn>7</m:mn> </m:mrow> <m:mrow> <m:mn>6</m:mn> </m:mrow> </m:mfrac> </m:math> m\gt \frac{7}{6} for general sensitivity (M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. 54 (2015), 3789–3828) or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>m</m:mi> <m:mo>></m:mo> <m:mfrac> <m:mrow> <m:mn>9</m:mn> </m:mrow> <m:mrow> <m:mn>8</m:mn> </m:mrow> </m:mfrac> </m:math> m\gt \frac{9}{8} for scalar sensitivity (M. Winkler, Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement, J. Differ. Equ. 264 (2018), 6109–6151). Our proof is based on a new observation on the quasi-energy-type functional and on an induction argument.