arXiv subjects

Shinya Nishibata

Publications and source records attributed to Shinya Nishibata.

4 recordsLinked to original sources

arXiv2024

Asymptotic Stability of 3D Out-flowing Compressible Viscous Fluid under Non-Spherical Perturbation

We study an outflow problem for the $3$-dimensional isentropic compressible Navier-Stokes equations. The fluid under consideration occupies the exterior domain of the unit ball $\Omega=\{x\in\mathbb{R}^3\,\vert\, |x|\ge 1\}$ and it is flowing out from the unit ball $\Omega$ at a constant speed $|u_b|$, in the normal direction to the boundary surface $\partial\Omega$. The existence of a unique spherically symmetric stationary solution $(\tilde{\rho},\tilde{u})$ is obtained by I.~Hashimoto and A.~Matsumura in 2021, provided that the fluid velocity at the far-field is assumed to be zero, and $|u_b|$ is sufficiently small. Subsequently, authors of the present article prove in 2024 that $(\tilde{\rho},\tilde{u})$ is time-asymptotically stable under large spherically symmetric initial perturbations in the suitable Sobolev norm. The main purpose of the present paper is to investigate the case when the initial perturbations are possibly non-spherically symmetric. We show that $(\tilde{\rho},\tilde{u})$ remains asymptotically stable in time, under general small initial perturbations in the $H^3$-norm.

math.AP↗

arXiv2024

On the stability of the spherically symmetric solution to an inflow problem for an isentropic model of compressible viscous fluid

We investigate an inflow problem for the multi-dimensional isentropic compressible Navier-Stokes equations. The fluid under consideration occupies the exterior domain of unit ball, $\Omega=\{x\in\mathbb{R}^n\,\vert\, |x|\ge 1\}$, and a constant stream of mass is flowing into the domain from the boundary $\partial\Omega=\{|x|=1\}$. It is shown in Hashimoto-Matsumura(2021) that if the fluid velocity at the far-field is assumed to be zero, then there exists a unique spherically symmetric stationary solution, denoted as $(\tilde{\rho},\tilde{u})(r)$ with $r\equiv |x|$. In this paper, we show that either $\tilde{\rho}$ is monotone increasing or $\tilde{\rho}$ attains a unique global minimum, and this is classified by the boundary condition of density. In addition, we also derive a set of spatial decay rates for $(\tilde{\rho},\tilde{u})$ which allows us to prove the time-asymptotic stability of $(\tilde{\rho},\tilde{u})$ using the energy method. More specifically, we prove this under small initial perturbation on $(\tilde{\rho},\tilde{u})$, provided that the density at the far-field is supposed to be strictly positive but suitably small, in other words, the far-field state of the fluid is not vacuum but suitably rarefied. The main difficulty for the proof is the boundary terms that appears in the a-priori estimates. We resolve this issue by reformulating the problem in Lagrangian coordinate system.

math.AP↗

arXiv2023

Large-time behaviour of the spherically symmetric solution to an outflow problem for isentropic model of compressible viscous fluid

We study the large time behaviour of a spherically symmetric motion of out-flowing isentropic and compressible viscous gas. The fluid occupies an unbounded exterior domain in $\mathbb{R}^n \; (n \ge 2)$, and it flows out from an inner sphere centred at the origin of radius $r=1$. The unique existence of a stationary solution satisfying the outflow boundary condition has been obtained by I. Hashimoto and A. Matsumura in 2021. The main aim of present paper is to show that this stationary solution becomes a time asymptotic state to the initial boundary value problem with the same boundary and spatial asymptotic conditions. Here, the initial data is chosen arbitrarily large if it belongs to the suitable weighted Sobolev space. The main strategy is to approximate the unbounded exterior problem by solving a sequence of outflow-inflow initial boundary value problems posed in finite annular domain. Then the solution is obtained as a limit of these approximate solutions. The key argument for the stability theorem is based on the derivation of a-priori estimates in the weighted Sobolev space, executed under the Lagrangian coordinate. The essential step of the proof is to obtain the point-wise upper and lower bound for the density. It is derived through employing a representation formula of the density with the aid of the weighted energy method.

math.AP↗

arXiv2009

Stationary waves to viscous heat-conductive gases in half space: existence, stability and convergence rate

The present paper is concerned with large-time behavior of solutions to an outflow problem for an ideal polytropic model of compressible viscous gases in one-dimensional half space, and with a convergence rate of solutions toward a corresponding stationary solution. With the aid of center manifold theory, we prove the existence of the stationary solution, under a smallness condition on the boundary data. We also investigate, by employing an energy method, the time asymptotic stability of the stationary solution under suitable smallness assumptions, and estimate the convergence rate which coincides with the spatial decay rate of the initial perturbation. The proof is mainly based on a priori estimates, which are derived by a time and space weighted energy method.

math.AP↗