### Abstract

This paper is concerned with the time-asymptotic behavior toward strong rarefaction waves of solutions to one-dimensional compressible Navier-Stokes equations. Assume that the corresponding Riemann problem to the compressible Euler equations can be solved by rarefaction waves (V^{R},U ^{R},S^{R})(t, x). If the initial data (v_{0},u _{0},s_{0})(x) to the nonisentropic compressible Navier-Stokes equations is a small perturbation of an approximate rarefaction wave constructed as in [S. Kawashima, A. Matsumura, and K. Nishihara, Proc. Japan Acad. Ser. A, 62 (1986), pp. 249-252], then we show that, for the general gas, the Cauchy problem admits a unique global smooth solution (v, u, s)(t, x) which tends to (V^{R}, U^{R}, S^{R})(t, x) as t tends to infinity. A global stability result can also be established for the nonisentropic ideal polytropic gas, provided that the adiabatic exponent γ is close to 1. Furthermore, we show that for the isentropic compressible Navier-Stokes equations, the corresponding global stability result holds, provided that the resulting compressible Euler equations are strictly hyperbolic and both characteristical fields are genuinely nonlinear. Here, global stability means that the initial perturbation can be large. Since we do not require the strength of the rarefaction waves to be small, these results give the nonlinear stability of strong rarefaction waves for the one-dimensional compressible Navier-Stokes equations.

Original language | English |
---|---|

Pages (from-to) | 1561-1597 |

Number of pages | 37 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 35 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2004 |

### Fingerprint

### Keywords

- Compressible Navier-Stokes equations
- Global stability
- Strong rarefaction waves

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics
- Numerical Analysis

### Cite this

*SIAM Journal on Mathematical Analysis*,

*35*(6), 1561-1597. https://doi.org/10.1137/S003614100342735X

**Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations.** / Nishihara, Kenji; Yang, Tong; Zhao, Huijiang.

Research output: Contribution to journal › Article

*SIAM Journal on Mathematical Analysis*, vol. 35, no. 6, pp. 1561-1597. https://doi.org/10.1137/S003614100342735X

}

TY - JOUR

T1 - Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations

AU - Nishihara, Kenji

AU - Yang, Tong

AU - Zhao, Huijiang

PY - 2004

Y1 - 2004

N2 - This paper is concerned with the time-asymptotic behavior toward strong rarefaction waves of solutions to one-dimensional compressible Navier-Stokes equations. Assume that the corresponding Riemann problem to the compressible Euler equations can be solved by rarefaction waves (VR,U R,SR)(t, x). If the initial data (v0,u 0,s0)(x) to the nonisentropic compressible Navier-Stokes equations is a small perturbation of an approximate rarefaction wave constructed as in [S. Kawashima, A. Matsumura, and K. Nishihara, Proc. Japan Acad. Ser. A, 62 (1986), pp. 249-252], then we show that, for the general gas, the Cauchy problem admits a unique global smooth solution (v, u, s)(t, x) which tends to (VR, UR, SR)(t, x) as t tends to infinity. A global stability result can also be established for the nonisentropic ideal polytropic gas, provided that the adiabatic exponent γ is close to 1. Furthermore, we show that for the isentropic compressible Navier-Stokes equations, the corresponding global stability result holds, provided that the resulting compressible Euler equations are strictly hyperbolic and both characteristical fields are genuinely nonlinear. Here, global stability means that the initial perturbation can be large. Since we do not require the strength of the rarefaction waves to be small, these results give the nonlinear stability of strong rarefaction waves for the one-dimensional compressible Navier-Stokes equations.

AB - This paper is concerned with the time-asymptotic behavior toward strong rarefaction waves of solutions to one-dimensional compressible Navier-Stokes equations. Assume that the corresponding Riemann problem to the compressible Euler equations can be solved by rarefaction waves (VR,U R,SR)(t, x). If the initial data (v0,u 0,s0)(x) to the nonisentropic compressible Navier-Stokes equations is a small perturbation of an approximate rarefaction wave constructed as in [S. Kawashima, A. Matsumura, and K. Nishihara, Proc. Japan Acad. Ser. A, 62 (1986), pp. 249-252], then we show that, for the general gas, the Cauchy problem admits a unique global smooth solution (v, u, s)(t, x) which tends to (VR, UR, SR)(t, x) as t tends to infinity. A global stability result can also be established for the nonisentropic ideal polytropic gas, provided that the adiabatic exponent γ is close to 1. Furthermore, we show that for the isentropic compressible Navier-Stokes equations, the corresponding global stability result holds, provided that the resulting compressible Euler equations are strictly hyperbolic and both characteristical fields are genuinely nonlinear. Here, global stability means that the initial perturbation can be large. Since we do not require the strength of the rarefaction waves to be small, these results give the nonlinear stability of strong rarefaction waves for the one-dimensional compressible Navier-Stokes equations.

KW - Compressible Navier-Stokes equations

KW - Global stability

KW - Strong rarefaction waves

UR - http://www.scopus.com/inward/record.url?scp=9744244879&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=9744244879&partnerID=8YFLogxK

U2 - 10.1137/S003614100342735X

DO - 10.1137/S003614100342735X

M3 - Article

AN - SCOPUS:9744244879

VL - 35

SP - 1561

EP - 1597

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 6

ER -