Stability of boundary layer and rarefaction wave to an outflow problem for compressible Navier–Stokes equations under large perturbation

Abstract

In this paper, we investigate the large-time behavior of solutions to an outflow problem for compressible Navier-Stokes equations. In 2003, Kawashima, Nishibata and Zhu [S. Kawashima, S. Nishibata, P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys. 240 (2003) 483-500] showed there exists a boundary layer (i.e., stationary solution) to the outflow problem and the boundary layer is nonlinearly stable under small initial perturbation. In the present paper, we show that not only the boundary layer above but also the superposition of a boundary layer and a rarefaction wave are stable under large initial perturbation. The proofs are given by an elementary energy method.

Inflow problem for the one-dimensional compressible Navier–Stokes equations under large initial perturbation

“…• For the outflow problem, Kawashima, Nishibata, and Zhu [4] and Kawashima and Zhu [5] showed that the boundary layer solution together with the superposition of the boundary layer solution and the rarefaction wave are asymptotically nonlinear stable under small initial perturbation, while Nakamura, Nishibata and Yuge [11] investigated the convergence rate toward the boundary layer solution. Recently, Huang and Qin [2] show that not only the boundary layer solution but also the superposition of the boundary layer solution and the rarefaction wave are still stable under large initial perturbation and improve the works of [4] and [5];…”

Viscous Shock Wave to an Inflow Problem for Compressible Viscous Gas with Large Density Oscillations

“…Throughout this manuscript, we will concerned with the inflow problem (1.1). For the corresponding impermeable wall problem and outflow problem, those interested are referred to [13,16] and [4,6,8,9,10,19,20,27] and the references cited therein, respectively.…”

Regularity of Milne problem with geometric correction in 3D

“…Since 1960s, people have discovered several methods to study the well-posedness of Milne problem, and apply them to asymptotic expansion. We refer to the references [15], [3], [4], [29], [20], [25], [13], [12], [1], [8], [11], [16], [19], [26], [5], [2], [10], [21], [22], [23], and [24] for more details. In 1979, diffusive limit of steady neutron transport equation was systematically investigated in [9] (see also [6] and [7]).…”