Convergence rate of solutions toward stationary solutions to the compressible Navier–Stokes equation in a half line

Abstract

We study a large time behavior of a solution to the initial boundary value problem for an isentropic and compressible viscous fluid in a one-dimensional half space. The unique existence and the asymptotic stability of a stationary solution are proved by S. Kawashima, S. Nishibata and P. Zhu for an outflow problem where the fluid blows out through the boundary. The main concern of the present paper is to investigate a convergence rate of a solution toward the stationary solution. For the supersonic flow at spatial infinity, we obtain an algebraic or an exponential decay rate. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in the spatial asymptotic point, the solution converges to the corresponding stationary solution with the same rate in time as time tends to infinity. An algebraic convergence rate is also obtained for the transonic flow. These results are proved by the weighted energy method.

Convergence rate of solutions toward stationary solutions to a viscous liquid-gas two-phase flow model in a half line

“…Compared one order derivative estimate in our paper with in [29], the main difference is that we must use the weighted function to estimate one order derivative since the loss of L 1 (0, t)−norm of φ x 2 . Hence, we need to derived the weighted H 1 (R + ) a priori estimates in our paper.…”

“…Due to the similarity of the viscous liquid-gas two-phase model to the compressible Navier-Stokes equations, we can apply some ideas developed in proving the existence, stability and convergence rates of solutions to the compressible Navier-Stokes equations to deal with the two-phase flow model. One can refer to [7,19,21,23,29] and references therein. However, it is non-trivial to apply directly the ideas used in single-phase models into the two-phase models since the momentum equation is given only for the mixture and that the pressure involves the masses of two phases in a nonlinear way, which makes it rather difficult to get the uniform time-independent estimates of the masses and the mixed velocity (n, m, u).…”

Energy method in the partial Fourier space and application to stability problems in the half space

“…For compressible Navier-Stokes equations, the papers [10,7] discuss the convergence rate toward stationary waves in one-dimensional half space. Furthermore, [9] studied a generalization to the multi-dimensional case.…”

Hardy type inequality and application to the stability of degenerate stationary waves

“…See [4][5][6]14,16] for the details. See also [2,7,9,11] for the related stability results for stationary waves.…”