Global well-posedness to the 3D Cauchy problem of nonhomogeneous Navier–Stokes equations with density-dependent viscosity and large initial velocity

Abstract

We are concerned with the global well-posedness of strong solutions to the Cauchy problem of nonhomogeneous Navier–Stokes equations with density-dependent viscosity and vacuum in R3. With the help of energy method, we prove the global existence and uniqueness of strong solutions provided that the initial mass is properly small. In particular, the initial velocity can be arbitrarily large. This improves He, Li, and Lü’s work [Arch. Ration. Mech. Anal. 239, 1809–1835 (2021)]. Moreover, we also extend the result of Liu [Discrete Contin. Dyn. Syst. B 26, 1291–1303 (2021)] to the case that large oscillations of the solutions are allowed.