Asymptotic Stability of the Stationary Solution to the Compressible Navier–Stokes Equations in the Half Space

Kawashima, Shuichi; Nishibata, Shinya; Zhu, Ping

doi:10.1007/s00220-003-0909-2

Cited by 129 publications

(134 citation statements)

References 8 publications

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“…We shall investigate the asymptotic stability of the rarefaction wave, the boundary layer solution and their combination. Unlike the outflow problem, the velocity u − on the boundary for the inflow problem is positive which is not good for the analysis, we have to estimate more boundary terms than those of [16].…”

Section: Case 3 (Positive Velocity On the Boundary)mentioning

confidence: 99%

“…Comparing with isentropic gas, the boundary conditions for the full compressible Navier-Stokes system become more complicated and analytically it is more difficult to control the terms coming from the boundary. More recently, Kawashima and Zhu [16] investigated the existence of the boundary layer solution and its asymptotic stability for the outflow problem of (1.1) in which the sign of the velocity u − on the boundary is negative, which plays a key role in the a priori estimate. In this paper we focus our attention on the impermeable wall problem and inflow problem.…”

Section: Case 3 (Positive Velocity On the Boundary)mentioning

confidence: 99%

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Asymptotic behavior of solutions to the full compressible Navier-Stokes equations in the half space

Huang¹,

Li²,

Shi³

2010

Communications in Mathematical Sciences

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Abstract. The one-dimensional motion of compressible viscous and heat-conductive fluid is investigated in the half space. By examining the sign of fluid velocity prescribed on the boundary, initial boundary value problems with Dirichlet type boundary conditions are classified into three cases: impermeable wall problem, inflow problem and outflow problem. In this paper, the asymptotic stability of the rarefaction wave, boundary layer solution, and their combination is established for both the impermeable wall problem and the inflow problem under some smallness conditions. The proof is given by an elementary energy method.

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Section: Case 3 (Positive Velocity On the Boundary)mentioning

confidence: 99%

Section: Case 3 (Positive Velocity On the Boundary)mentioning

confidence: 99%

Asymptotic behavior of solutions to the full compressible Navier-Stokes equations in the half space

Huang¹,

Li²,

Shi³

2010

Communications in Mathematical Sciences

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“…Only the results rigorously proved concerning above asymptotic behaviors are given by Kawashima & Nishibata [1]. They proved that if w+ G Qj U p er i W-G BL(w+), and \w+ -w-\ is suitably small, then the boundary layer solution W BL (x',W-,w+) is asymptotically stable.…”

Section: Suppose U-> 0 and W+ G Bl(w-) I) If W-gmentioning

confidence: 99%

“…which is simply equivalent to u+ < U-in our cases, we can define the shock curves S\(w-) and S2(w-) for any fixed left state W-G fi by =1 , that is, RR(w-) with the boundary Ri(w-) and /^(w-), RS(w-) with the boundary Ri(w-) and 52(^_), SR{w-) with the boundary Si(w-) and R2(w-) 1 and SS(w-) with the boundary Si^-) and 82 (11) …”

Section: U4 -V-mentioning

confidence: 99%

“…They proved that if w+ G Qj U p er i W-G BL(w+), and \w+ -w-\ is suitably small, then the boundary layer solution W BL (x',W-,w+) is asymptotically stable. It should be noted that they employ a method where the monotonicity of the boundary layer solution is not used, so they can treat the both cases W-< w+ and w+ < W-(see [1] for details). All other cases are open problems!…”

Section: Suppose U-> 0 and W+ G Bl(w-) I) If W-gmentioning

confidence: 99%

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Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas

Matsumura¹

2001

Methods and Applications of Analysis

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Abstract. We consider asymptotic behaviors in time of solutions to the initial boundary value problems in the half space for a one-dimensional isentropic model system of compressible viscous gas. In particular, we focus our attention on inflow(or outflow) problems where the velocity on the boundary is given as a constant inward (or outward) flow, and try to classify all asymptotic behaviors of the solutions. It turns out that depending on the data both on the boundary and at far field (especially depending on whether the state is subsonic, transonic, or supersonic), the asymptotic state variously consists of rarefaction waves, viscous shock waves, and also stationary boundary layer. Moreover, we give a survey of our recent results on some particular cases which justify our classification.

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Stability of stationary solutions for the unipolar isentropic compressible Navier–Stokes–Poisson system

Hong

Kim

2021

Math Methods in App Sciences

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The stationary solutions to the outflow problem for unipolar isentropic Navier–Stokes–Poisson system in a half line (0, ∞) have recently been shown to be asymptotically stable in13 and26, provided that all the L2 norms of initial perturbations and their derivatives are small. The main purpose of this paper is to study the asymptotic stability of the stationary solutions under large initial perturbations. First, for the outflow problem, we show that the stationary solutions are asymptotically stable, provided that only the L2 norm of initial perturbation is small. Next, for the inflow problem, we show asymptotic stability under a large initial perturbation in the H1 norm. The main ingredient in the proofs is a careful analysis in order to control the growth induced by nonlinearities of the system in a priori estimates.

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Asymptotic Stability of the Stationary Solution to the Compressible Navier–Stokes Equations in the Half Space

Cited by 129 publications

References 8 publications

Asymptotic behavior of solutions to the full compressible Navier-Stokes equations in the half space

Asymptotic behavior of solutions to the full compressible Navier-Stokes equations in the half space

Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas

Stability of stationary solutions for the unipolar isentropic compressible Navier–Stokes–Poisson system

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