Stability of Planar Stationary Solutions to the Compressible Navier-Stokes Equation on the Half Space

Arch Rational Mech Anal

2012

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The initial boundary value problem for the compressible Navier-Stokes equation is considered in an infinite layer of R 2 . It is proved that if the Reynolds and Mach numbers are sufficiently small, then strong solutions to the compressible Navier-Stokes equation around parallel flows exist globally in time for sufficiently small initial perturbations. The large time behavior of the solution is described by a solution of a one-dimensional viscous Burgers equation. The proof is given by a combination of spectral analysis of the linearized operator and a variant of the Matsumura-Nishida energy method.

“…As in [9,13,14], Theorem 3.1 is proved by showing the local existence and the a priori estimates. The local existence is proved by applying the local solvability result in [8].…”

Section: Resultsmentioning

confidence: 99%

“…m−2 CẼ(t) + C R (9) (t), (7.41) where R (9) (t) = Q F ∞ (t) 2 m−2 . It then follows from (7.40) and (7.41) that…”

Section: Proposition 74mentioning

confidence: 99%

Asymptotic Behavior of Solutions to the Compressible Navier–Stokes Equation Around a Parallel Flow

Arch Rational Mech Anal

2012

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“…6. Section 7 is devoted to the nonlinear problem and we here give an outline of the proof of (1.5) and (1.6) only, since the proof for the nonlinear problem is similar to those in [4,6,11,12]. In the appendix we give an estimate for the Fourier transform of solutions to the stationary Stokes problem on the half space, which is used in the analysis in Sect.…”

Section: Introductionmentioning

confidence: 99%

“…The conditions ν ≥ c 1 and (2ν + ν )/γ 2 ≤ c 2 appear in a similar manner to the proof of Proposition 6.8. The decay property u(t) ∞ → 0 as t → ∞ can be proved in the same way as in [6]. The proof of Theorem 3.3 (ii) is based on the H s -energy bound (7.1) and Theorem 4.1.…”

mentioning

confidence: 98%

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Asymptotic Behavior of Solutions of the Compressible Navier–Stokes Equation Around the Plane Couette Flow

2010

J. Math. Fluid Mech.

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Asymptotic behavior of solutions to the compressible Navier-Stokes equation around the plane Couette flow is investigated. It is shown that the plane Couette flow is asymptotically stable for initial disturbances sufficiently small in some L 2 Sobolev space if the Reynolds and Mach numbers are sufficiently small. Furthermore, the disturbances behave in large time in L 2 norm as solutions of an n − 1 dimensional linear heat equation with a convective term. (2000). 35Q30, 76N15. Mathematics Subject Classification

Asymptotic behavior of solutions of the compressible Navier‐Stokes equations in a cylinder under the slip boundary condition

Aihaiti

Math Methods in App Sciences

2019

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The large time behavior of solutions to the compressible Navier‐Stokes equations around the motionless state is considered in a cylinder under the slip boundary condition. It is shown that if the initial data are sufficiently small, the global solution uniquely exists and the large time behavior of the solution is described by a superposition of one‐dimensional nonlinear diffusion waves and a diffusive rigid rotation.