Fujita–Kato theorem for the 3-D inhomogeneous Navier–Stokes equations

In this paper, we prove the global existence and uniqueness of the solution with discontinuous density for the 3-D inhomogeneous Navier–Stokes equations if the initial data (ρ0,u0)∈L∞(R3)×Hs(R3)(ρ0,u0)∈L∞(R3)×Hs(R3) with s>12 satisfies00ε>0 depending only on c0c0, C0C0. Furthermore, we introduce the dual method to show that if u0∈Lp(R3)u0∈Lp(R3) for p∈[65,2], the velocity satisfies the decay estimate‖∇ku(t)‖L2≤C(1+t)−k2−α(p)fort≥1,k=0,1, with α(p)=32(1p−12).