Nishida-Smoller type large solutions and exponential growth for the compressible Navier-Stokes equations with slip boundary conditions in 3D bounded domain

TL;DR

This paper proves the global existence of large-energy solutions with vacuum for 3D compressible Navier-Stokes equations under slip boundary conditions, showing exponential decay or growth depending on initial conditions and parameters.

Contribution

It extends Nishida-Smoller's 1973 results to 3D with slip boundaries, establishing global solutions with large initial energy and vacuum, including exponential growth of density gradients.

Findings

Global existence of large-energy solutions with vacuum in 3D

Exponential decay of solutions for nearly isothermal fluids

Unbounded exponential growth of density gradient with initial vacuum

Abstract

This paper concerns the isentropic compressible Navier-Stokes equations in a three-dimensional (3D) bounded domain with slip boundary conditions and vacuum. It is shown that the classical solutions to the initial-boundary-value problem of this system with large initial energy and vacuum exist globally in time and have an exponential decay rate which is decreasing with respect to the adiabatic exponent $\gamma>1$ provided that the fluid is nearly isothermal (namely, the adiabatic exponent is close enough to 1). This constitutes an extension of the celebrated result for the one-dimensional Cauchy problem of the isentropic Euler equations that has been established in 1973 by Nishida and Smoller (Comm. Pure Appl. Math. 26 (1973), 183-200). In addition, it is also shown that the gradient of the density will grow unboundedly with an exponential rate when the initial vacuum appears (even at a point). In contrast to previous related works, where either small initial energy are required or boundary effects are ignored, this establishes the first result on the global existence and exponential growth of large-energy solutions with vacuum to the 3D isentropic compressible Navier-Stokes equations with slip boundary conditions.