Global solvability and boundedness to a coupled chemotaxis-fluid model with arbitrary porous medium diffusion

Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source

“…Liu et al [17] showed that under the conditions m ≥ 1 3 and α > 6 5 − m, and proper regularity hypotheses on the initial data, the corresponding initail-boundary problem possesses at least one global bounded weak solution for the Keller-Segel-Stokes system with nonlinear diffusion and logistic source in the three-dimensional bounded domains. Jin [9] improved the results in [17], and established the global existence and boundedness of weak solutions for any m > 0 and α > 0. When the system controlled by a given external force g, Zheng [58] revealed that m > max{ 6 5 − α, 1 3 } and α > 0, for all reasonably regular initial data, a corresponding no-flux Neumann initial-boundary value problem possesses at least one global weak solution.…”

“…Tao et al [27] assured global solvability within the larger range m > 8 7 , but only in a class of weak solutions locally bounded in Ω × [0, ∞). Based on energy-based arguments and maximal Sobolev regularity theory, Winkler's result in [48], which allows for the construction of global weak solution to an associated initial-boundary value problem under the milder assumption that m > 9 8 . Moreover, it is shown that such solution stabilizes to spatially homogeneous state ( 1|Ω| Ω n 0 , 0, 0) in the large time limit.…”

Boundedness in a two‐dimensional self‐consistent chemotaxis–Navier–Stokes system with nonlinear diffusion

“…Afterwords, Winkler supplemented the uniform boundedness of solutions for the case $$$ m>\frac{7}{6} $$$ when $$$ N=3 $$$ in a previous study 26 and improved the result for the case $$$ m>\frac{9}{8} $$$ when $$$ N=3 $$$ in the literature 27 . Recently, in a three‐dimensional bounded domain, the global boundedness of solution has been proved for the case $$$ m>1 $$$ 28 . On the other hand, under $$$ \kappa \ne 0 $$$ and some additional appropriate conditions on $$$ \phi $$$, it is proved that the existence of global weak solutions when $$$ m\ge \frac{2}{3} $$$ in the three‐dimensional case (see Zhang and Li 29 ).…”

Periodic pattern formation in the coupled chemotaxis-(Navier–)Stokes system with mixed nonhomogeneous boundary conditions

“…While, the study for the 3-D case is much more difficult. In 2010, Di Francesco [3] obtained the existence of a global bounded weak solution for m ∈ ((7 + √ 217)/12, 2]; a locally bounded global weak solution was then obtained for m ∈ ( 8 7 , +∞) in 2013 [23]; the uniform boundedness of solutions was subsequently supplemented for m ∈ ( 7 6 , +∞) [29]; further extension was made by Winkler for m > 9 8 in [31] to a convex domain; recently, we also [10] improved the results to the case m > 11 4 − √ 3 (approximating to 56 55 ). However, if a logistic term reflecting the cell proliferation is added to this model, Jin [7] established the existence of global bounded weak solutions for any m > 1 to the fluid-free case in dimension 3.…”