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

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 R-3. 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 Lu'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.

Automated summary

In this paper, we address the global well-posedness of strong solutions to nonhomogeneous Navier-Stokes equations with density-dependent viscosity and vacuum. Using the energy method, we prove the existence and uniqueness of strong solutions globally, provided the initial mass is sufficiently small. Notably, the initial velocity can be arbitrarily large. This work builds upon the previous research by He, Li, and Lu (Arch. Ration. Mech. Anal. 239, 1809-1835, 2021) and extends the results of Liu (Discrete Contin. Dyn. Syst. B 26, 1291-1303, 2021) to allow for large oscillations of the solutions.