On the existence of local classical solutions to the Navier-Stokes equations with degenerate viscosities

Abstract

In this paper, we consider the three-dimensional Cauchy problem of compressible Navier-Stokes equations with degenerate viscosities depending on density. With the initial data containing vacuum at far fields, we prove the local existence of regular solutions with some compatibility conditions. The regular solution is classical due to the smoothing effect of the velocity u.

On the existence of global strong solutions to 2D compressible Navier–Stokes–Smoluchowski equations with large initial data

“…Li, Pan and Zhu investigated the local existence of regular solutions for compressible barotropic Navier-Stokes equations with density-dependent viscosities in [28,29]. Luo and Zhou extended the result in [32]. Recently, Xin and Zhu [44] proved the global-in-time well-posedness of regular solutions for a class of smooth initial data for Cauchy problem.…”

Non-existence of global classical solutions to barotropic compressible Navier-Stokes equations with degenerate viscosity and vacuum

“…By Hölder's inequality, we get due to(20). This relation combined with(23) and(22) gives(21).With the estimates above at hand, we can start proving Theorem 2.3.…”

“…Li, Pan and Zhu investigated the local existence of regular solutions for compressible barotropic Navier-Stokes equations with density-dependent viscosities in [19,20]. Luo and Zhou extended the result in [23]. Recently, Xin and Zhu [33] proved the global-in-time well-posedness of regular solutions for a class of smooth initial data for Cauchy problem.…”