Maximal L1 regularity for solutions to inhomogeneous incompressible Navier-Stokes equations

This paper is devoted to the maximal L1 regularity and asymptotic behavior for solutions inhomogeneous incompressible Navier-Stokes equations under a scaling-invariant smallness assumption on initial velocity. We obtain new global L1-in-time estimate Lipschitz seminorm of velocity field without any density fluctuation. In derivation this estimate, we study linear Stokes system with variable coefficients. The analysis tools are use semigroup generated by generalized operator characterize some Besov norms gradient class second-order elliptic divergence form. Our method can be used other issues arising from or compressible viscous fluids.