Global symmetric classical and strong solutions of the full compressible Navier-Stokes equations with vacuum and large initial data

Abstract

First of all, we get the global existence of classical and strong solutions of the full compressible Navier-Stokes equations in three space dimensions with initial data which is large and spherically or cylindrically symmetric. The appearance of vacuum is allowed. In particular, if the initial data is spherically symmetric, the space dimension can be taken not less than two. The analysis is based on some delicate {\it a priori} estimates globally in time which depend on the assumption $\kappa =O(1+{\theta }^q)$ where $q>r$ ($r$ can be zero), which relaxes the condition $q\ge 2+2r$ in [14,29,42]. This could be viewed as an extensive work of [18] where the equations hold in the sense of distributions in the set where the density is positive with initial data which is large, discontinuous, and spherically or cylindrically symmetric in three space dimension. Finally, with the assumptions that vacuum may appear and that the solutions are not necessarily symmetric, we establish a blow-up criterion in terms of $\Vert \rho {\Vert }_{L_t^{\infty }{L}_x^{\infty }}$ and $\Vert \rho \theta {\Vert }_{L_t^4{L}^{(}12/5{)}_x}$ for strong solutions.

arXiv admin note: text overlap with arXiv:1103.1421

References in corpus (4)

• Global Classical Large Solutions to Navier-Stokes Equations for Viscous Compressible and Heat Conducting Fluids with Vacuum
• Global Well-Posedness of Classical Solutions with Large Oscillations and Vacuum to the Three-Dimensional Isentropic Compressible Navier-Stokes Equations
• Blow-up criterions of strong solutions to 3D compressible Navier-Stokes equations with vacuum
• A Blow-up Criterion for Two Dimensional Compressible Viscous Heat-Conductive Flows

Cited by in corpus (4)

• Long-time behavior for three dimensional compressible viscous and heat-conductive gases
• Global classical solution to 3D isentropic compressible Navier-Stokes equations with large initial data and vacuum
• Blow-up criterions of strong solutions to 3D compressible Navier-Stokes equations with vacuum
• Global solutions to the three-dimensional full compressible Navier-Stokes equations with vacuum at infinity in some classes of large data