Global well-posedness of regular solutions to the three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities and vacuum

In this paper, the Cauchy problem for the three-dimensional (3-D) isentropic compressible Navier-Stokes equations with degenerate viscosities is considered. By introducing some new variables and making use of the quasi-symmetric hyperbolic-degenerate elliptic coupled structure to control the behavior of the fluid velocity, we prove the global-in-time well-posedness of regular solutions with vacuum for a class of smooth initial data that are of small density but possibly large velocities. Here the initial mass density is required to decay to zero in the far field, and the spectrum of the Jacobi matrix of the initial velocity are all positive. The result here applies to a class of degenerate density-dependent viscosity coefficients, is independent of the BD-entropy, and seems to be the first on the global existence of smooth solutions which have large velocities and contain vacuum state for such degenerate system in three space dimensions. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

This paper discusses the Cauchy problem for the three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities. By introducing new variables and utilizing a coupled structure, global-in-time well-posedness of solutions with vacuum is proven for initial data with small density but possibly large velocities. The result is the first on the global existence of smooth solutions with degenerate density-dependent viscosity coefficients in three space dimensions.