Linear stability of the Couette flow in the 3D isentropic compressible Navier-Stokes equations

Abstract

Consider the linear stability of the three dimensional isentropic compressible Navier-Stokes equations on T × R × T. We prove the enhanced dissipation phenomenon for the linearized isentropic compressible Navier-Stokes equations around the Couette flow (y, 0, 0) ⊤ . Moreover, the lift-up phenomenon is also shown in this paper. Compared with the 3D incompressible Navier-Stokes equations [Ann. of Math.,185(2017), 541-608], the lift-up effect here is stronger due to the loss of the incompressible condition.

Linear stability of the Couette flow for the non-isentropic compressible fluid

“…The equations (1.1) then express respectively the conservation of mass, the balance of momentum, and the balance of energy under internal pressure, viscosity forces, and the conduction of thermal energy. A comprehensive understanding of the stability of compressible or incompressible shear flows is a fundamental problem in fluid mechanics and has been the subject of both theoretical and practical interest in astrophysics and engineering, see [1], [2], [6]- [9], [14]- [25], [27], [29]- [33], [38], [39] for the compressible fluid and [3]- [5], [10]- [13], [26], [28], [34]- [37] for incompressible fluid. The aim of the present paper is to study the long-time asymptotic behaviour of the linearized non-isentropic compressible Navier-Stokes equations around the Couette flow.…”

“…In the viscous case, they obtained the enhanced dissipation phenomenon. Zeng et al [38] considered the linear stability of the three dimensional isentropic compressible Navier-Stokes equations on T × R × T. They proved the enhanced dissipation phenomenon and the lift-up phenomenon around the Couette flow (y, 0, 0) ⊤ . The motivation of the present paper is to generalize the results obtained by Antonelli et al [1], [2] to the non-isentropic compressible Navier-Stokes equations with vanished shear viscosity.…”

Asymptotic stability of two-dimensional Couette flow in a viscous fluid

“…See also [8,45,32] for similar depletion phenomena in various systems. Such damping caused by mixing is a common phenomenon in fluid dynamics, see [45,54,53,70] for the plasma fluid, see [6,66,46] for the stratified fluid, see [3,4,69] for the compressible fluid, and see [59,65] for the geophysical fluid.…”