The lifespan of solutions of the 3D inhomogeneous incompressible Navier Stokes equations

The lifespan of solutions of 3D inhomogeneous incompressible Navier-Stokes system is investigated. In precisely, the lower estimate of the lifespan of solutions in Besov space is established if the bounded initial density possesses small perturbations near equilibrium, which is a generalization of the result of Zhang (2020) [14] in Sobolev spaces. By imposing additional regularity assumption that ${\rho }_0^{-1}-1\in B_{\lambda ,1}^{3/\lambda },1<\lambda <6$, the lifespan estimate of solution is also achieved without the small perturbations restriction on initial density.