### Abstract

This paper is concerned with the asymptotic stability of degenerate stationary waves for viscous gases in the half space. We discuss the following two cases: (1) viscous conservation laws and (2) damped wave equations with nonlinear convection. In each case, we prove that the solution converges to the corresponding degenerate stationary wave at the rate t^{-α/4} as t → ∞, provided that the initial perturbation is in the weighted space L^{2} _{α}=L^{2}(ℝ_{+};(1+x)^{α}dx). This convergence rate t^{-α/4} is weaker than the one for the non-degenerate case and requires the restriction α < α_{*}(q), where α_{*}(q) is the critical value depending only on the degeneracy exponent q. Such a restriction is reasonable because the corresponding linearized operator for viscous conservation laws cannot be dissipative in L^{2} _{α} for α > α ^{*}(q) with another critical value α^{*}(q). Our stability analysis is based on the space-time weighted energy method in which the spatial weight is chosen as a function of the degenerate stationary wave.

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

Pages (from-to) | 735-762 |

Number of pages | 28 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 198 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2010 Sep 17 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Analysis
- Mathematics (miscellaneous)
- Mechanical Engineering

### Cite this

*Archive for Rational Mechanics and Analysis*,

*198*(3), 735-762. https://doi.org/10.1007/s00205-010-0369-8

**Stability of Degenerate Stationary Waves for Viscous Gases.** / Ueda, Yoshihiro; Nakamura, Tohru; Kawashima, Shuichi.

Research output: Contribution to journal › Article

*Archive for Rational Mechanics and Analysis*, vol. 198, no. 3, pp. 735-762. https://doi.org/10.1007/s00205-010-0369-8

}

TY - JOUR

T1 - Stability of Degenerate Stationary Waves for Viscous Gases

AU - Ueda, Yoshihiro

AU - Nakamura, Tohru

AU - Kawashima, Shuichi

PY - 2010/9/17

Y1 - 2010/9/17

N2 - This paper is concerned with the asymptotic stability of degenerate stationary waves for viscous gases in the half space. We discuss the following two cases: (1) viscous conservation laws and (2) damped wave equations with nonlinear convection. In each case, we prove that the solution converges to the corresponding degenerate stationary wave at the rate t-α/4 as t → ∞, provided that the initial perturbation is in the weighted space L2 α=L2(ℝ+;(1+x)αdx). This convergence rate t-α/4 is weaker than the one for the non-degenerate case and requires the restriction α < α*(q), where α*(q) is the critical value depending only on the degeneracy exponent q. Such a restriction is reasonable because the corresponding linearized operator for viscous conservation laws cannot be dissipative in L2 α for α > α *(q) with another critical value α*(q). Our stability analysis is based on the space-time weighted energy method in which the spatial weight is chosen as a function of the degenerate stationary wave.

AB - This paper is concerned with the asymptotic stability of degenerate stationary waves for viscous gases in the half space. We discuss the following two cases: (1) viscous conservation laws and (2) damped wave equations with nonlinear convection. In each case, we prove that the solution converges to the corresponding degenerate stationary wave at the rate t-α/4 as t → ∞, provided that the initial perturbation is in the weighted space L2 α=L2(ℝ+;(1+x)αdx). This convergence rate t-α/4 is weaker than the one for the non-degenerate case and requires the restriction α < α*(q), where α*(q) is the critical value depending only on the degeneracy exponent q. Such a restriction is reasonable because the corresponding linearized operator for viscous conservation laws cannot be dissipative in L2 α for α > α *(q) with another critical value α*(q). Our stability analysis is based on the space-time weighted energy method in which the spatial weight is chosen as a function of the degenerate stationary wave.

UR - http://www.scopus.com/inward/record.url?scp=78049318899&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78049318899&partnerID=8YFLogxK

U2 - 10.1007/s00205-010-0369-8

DO - 10.1007/s00205-010-0369-8

M3 - Article

AN - SCOPUS:78049318899

VL - 198

SP - 735

EP - 762

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 3

ER -