Global Existence of Classical Solutions to Full Compressible Navier-Stokes System with Large Oscillations and Vacuum in 3D Bounded Domains

TL;DR

This paper proves the global existence and exponential stability of classical solutions to the full compressible Navier-Stokes equations in 3D bounded domains, allowing large oscillations and initial vacuum, a first in the field.

Contribution

It establishes the first global existence result for classical solutions with vacuum in 3D bounded domains for the full compressible Navier-Stokes system.

Findings

Global existence of classical solutions with vacuum.

Exponential stability of density, velocity, and temperature.

Unbounded growth of density oscillation if vacuum appears initially.

Abstract

The full compressible Navier-Stokes system describing the motion of a viscous, compressible, heat-conductive, and Newtonian polytropic fluid is studied in a three-dimensional simply connected bounded domain with smooth boundary having a finite number of two-dimensional connected components. For the initial-boundary-value problem with slip boundary conditions on the velocity and Neumann boundary one on the temperature, the global existence of classical and weak solutions which are of small energy but possibly large oscillations is established. In particular, both the density and temperature are allowed to vanish initially. Finally, the exponential stability of the density, velocity, and temperature is also obtained. Moreover, it is shown that for the classical solutions, the oscillation of the density will grow unboundedly in the long run with an exponential rate provided vacuum appears (even at a point) initially. This is the first result concerning the global existence of classical solutions to the full compressible Navier-Stokes equations with vacuum in general three-dimensional bounded smooth domains.