Convergence rate of solutions towards the stationary solutions to symmetric hyperbolic-parabolic systems in half space

Abstract

In the present paper, we study a system of viscous conservation laws, which is rewritten to a symmetric hyperbolic-parabolic system, in onedimensional half space. For this system, we derive a convergence rate of the solutions towards the corresponding stationary solution with/without the stability condition. The essential ingredient in the proof is to obtain the a priori estimate in the weighted Sobolev space. In the case that all characteristic speeds are negative, we show the solution converges to the stationary solution exponentially if an initial perturbation belongs to the exponential weighted Sobolev space. The algebraic convergence is also obtained in the similar way. In the case that one characteristic speed is zero and the other characteristic speeds are negative, we show the algebraic convergence of solution provided that the initial perturbation belongs to the algebraic weighted Sobolev space. The Hardy type inequality with the best possible constant plays an essential role in deriving the optimal upper bound of the convergence rate. Since these results hold without the stability condition, they immediately mean the asymptotic stability of the stationary solution even though the stability condition does not hold.

Asymptotic stability for the inflow problem of the heat-conductive ideal gas without viscosity

“…Our main analysis is on the stability of combination of boundary layer solution and rarefaction wave, which extended the result of [18]. There are also other interesting works for symmetric hyperbolic-parabolic system, see ( [25]- [27]).…”

Existence and Nonlinear Stability of Stationary Solutions to the Viscous Two-Phase Flow Model in a Half Line

“…Since the pressure is a nonlinear function of two densities, it is difficult to obtain the stability of the stationary solution by the ideas used in compressible Navier-Stokes equation for the outflow/inflow problems in [6,7,11,12,14,[16][17][18][19]21]. Inspired by the idea dealing with two-phase flow model in [5,23,28,29], we first take a similar nonlinear variable transformation to divide the two mass variables from each other.…”

“…This paper will use the following Hardy type inequality to get the upper 5 for the index λ as in [18]. For a rigorous proof of this lemma the reader is referred to [9].…”

Existence and Nonlinear Stability of Steady-States to Outflow Problem for the Full Two-Phase Flow

“…The existence and time decay rates of the solution to the outflow problem for Navier-Stokes equations in a half line were shown in [27,28] for supersonic and sonic cases. The existence and nonlinear stability of steady-states to the outflow problem for full compressible Navier-Stokes equations [2,18] were shown for supersonic, sonic and subsonic cases, and the time decay rates for the supersonic case or sonic case were obtained.…”

“…in basic energy estimates for both supersonic and sonic cases. For the sonic case, inspired by [27,28], we take the linear coordinate transformation…”