An optimal result for global existence and boundedness in a three-dimensional Keller-Segel(-Navier)-Stokes system (involving a tensor-valued sensitivity with saturation)
arXiv:1806.07067
Abstract
The coupled quasilinear Keller-Segel-Navier-Stokes system $$ \left\{ \begin{array}{l} n_t+u\cdot\nabla n=Δn-\nabla\cdot(nS(x,n,c)\nabla c),\quad x\in Ω, t>0, c_t+u\cdot\nabla c=Δc-c+n,\quad x\in Ω, t>0, u_t+κ(u \cdot \nabla)u+\nabla P=Δu+n\nabla ϕ,\quad x\in Ω, t>0, \nabla\cdot u=0,\quad x\in Ω, t>0 \end{array}\right.\eqno(KSNF) $$ is considered under Neumann boundary conditions for and and no-slip boundary conditions for in three-dimensional bounded domains with smooth boundary, where is given constant, , and the parameter . %For any small , If , then for all reasonably regular initial data, a corresponding initial-boundary value problem for possesses a globally defined weak solution. This result improves the result of Wang (Math. Models Methods Appl. Sci., 27(14):2745--2780, 2017), where the global {\bf very weak} solution for system is obtained. Moreover, if and , then the system exists at least one global classical solution which is bounded in . These results significantly improve or extend previous results of several authors. In comparison to the result for the corresponding fluid-free system, the {\bf optimal condition} on the parameter for global (weak) existence and boundedness is obtained. Our proofs rely on Maximal Sobolev regularity techniques and on a variant of the natural gradient-like energy functional.
48. arXiv admin note: text overlap with arXiv:1701.02060