This paper deals with the self‐consistent chemotaxis–Navier–Stokes system with porous medium diffusion
{left left leftarrayarraynt+u·∇n=Δnm−∇·(n∇c)+∇·(n∇ϕ),arrayx∈Ω,t>0,arrayarrayct+u·∇c=Δc−nc,arrayx∈Ω,t>0,arrayarrayut+(u·∇)u+∇P=Δu−n∇ϕ+n∇c,arrayx∈Ω,t>0,arrayarray∇·u=0,arrayx∈Ω,t>0$$ \left\{\begin{array}{lll}& {n}_t+u\cdotp \nabla n=\Delta {n}^m-\nabla \cdotp \left(n\nabla c\right)+\nabla \cdotp \left(n\nabla \phi \right),\kern2em & x\in \Omega, t>0,\\ {}& {c}_t+u\cdotp \nabla c=\Delta c- nc,\kern2em & x\in \Omega, t>0,\\ {}& {u}_t+\left(u\cdotp \nabla \right)u+\nabla P=\Delta u-n\nabla \phi +n\nabla c,\kern2em & x\in \Omega, t>0,\\ {}& \nabla \cdotp u=0,\kern2em & x\in \Omega, t>0\end{array}\right. $$
in the two‐dimensional smoothly bounded domain, where the gravitational potential
ϕ∈W1,∞false(normalΩfalse)$$ \phi \in {W}^{1,\infty}\left(\Omega \right) $$ is given function. Systems of this type involve the effect of gravity on cells (
∇·false(n∇ϕfalse)$$ \nabla \cdotp \left(n\nabla \phi \right) $$) and the effect of the chemotactic force on fluid (
n∇c$$ n\nabla c $$), which leads to a stronger coupling than usual chemotaxis‐fluid system that proposed by Tuval et al. (Proc. Nat. Acad. Sci. 102 (2005), 2277‐2282) and studied widely in the most existing literatures. Under no‐flux boundary conditions for
n$$ n $$ and
c$$ c $$ and no‐slip boundary condition for
u$$ u $$, it is proved that the system admits at least one global weak solution which is uniformly bounded if
m>1$$ m>1 $$. In particular, our result improves that established by Wang (Discr. Cont. Dyn. Syst. S 13 (2020), 329‐349), which asserts the global existence of weak solutions under the constraint
m>1$$ m>1 $$.
