On the global well-posedness for the incompressible four-component chemotaxis-Navier–Stokes equations with gradient-dependent flux limitation in R2

We consider the four-component chemotaxis-Navier–Stokes system in $R^2$: $\left(\begin{array}{c}{n}_t+u\cdot \nabla n=\Delta n-\nabla \cdot (nf\left({|\nabla c|}^2\right)\nabla c)-nm,\\ c_t+u\cdot \nabla c=\Delta c-c+m,\\ m_t+u\cdot \nabla m=\Delta m-nm,\\ u_t+(u\cdot \nabla )u+\nabla P=\Delta u+(n+m)\nabla \varphi ,\\ \nabla \cdot u=0.\end{array}\right)$Utilizing the Fourier localization technique alongside the inherent structure of the equations, we achieve global well-posedness for a class of rough initial data in the context of the 2D incompressible four-component chemotaxis-Navier–Stokes equations with gradient-dependent flux limitation $f\left(\zeta \right)=K_f\cdot {(1+\zeta )}^{-\frac{\alpha }{2}}$ for $\alpha >0$.