Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion

Shuichi Kawashima, Akitaka Matsumura

Research output: Contribution to journalArticle

238 Citations (Scopus)

Abstract

The asymptotic stability of traveling wave solutions with shock profile is investigated for several systems in gas dynamics. 1) The solution of a scalar conservation law with viscosity approaches the traveling wave solution at the rate t (for some γ>0) as t→∞, provided that the initial disturbance is small and of integral zero, and in addition decays at an algebraic rate for |x|→∞. 2) The traveling wave solution with Nishida and Smoller's condition of the system of a viscous heat-conductive ideal gas is asymptotically stable, provided the initial disturbance is small and of integral zero. 3) The traveling wave solution with weak shock profile of the Broadwell model system of the Boltzmann equation is asymptotically stable, provided the initial disturbance is small and its hydrodynamical moments are of integral zero. Each proof is given by applying an elementary energy method to the integrated system of the conservation form of the original one. The property of integral zero of the initial disturbance plays a crucial role in this procedure.

Original languageEnglish
Pages (from-to)97-127
Number of pages31
JournalCommunications in Mathematical Physics
Volume101
Issue number1
DOIs
Publication statusPublished - 1985 Mar 1
Externally publishedYes

Fingerprint

Traveling Wave Solutions
traveling waves
Asymptotic Stability
disturbances
Disturbance
Motion
Zero
Asymptotically Stable
gases
Shock
shock
energy methods
Scalar Conservation Laws
Ideal Gas
gas dynamics
Gas Dynamics
Energy Method
ideal gas
Integrated System
profiles

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. / Kawashima, Shuichi; Matsumura, Akitaka.

In: Communications in Mathematical Physics, Vol. 101, No. 1, 01.03.1985, p. 97-127.

Research output: Contribution to journalArticle

@article{ee87c0afa12d4057bef438bf4cef8a9a,
title = "Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion",
abstract = "The asymptotic stability of traveling wave solutions with shock profile is investigated for several systems in gas dynamics. 1) The solution of a scalar conservation law with viscosity approaches the traveling wave solution at the rate t-γ (for some γ>0) as t→∞, provided that the initial disturbance is small and of integral zero, and in addition decays at an algebraic rate for |x|→∞. 2) The traveling wave solution with Nishida and Smoller's condition of the system of a viscous heat-conductive ideal gas is asymptotically stable, provided the initial disturbance is small and of integral zero. 3) The traveling wave solution with weak shock profile of the Broadwell model system of the Boltzmann equation is asymptotically stable, provided the initial disturbance is small and its hydrodynamical moments are of integral zero. Each proof is given by applying an elementary energy method to the integrated system of the conservation form of the original one. The property of integral zero of the initial disturbance plays a crucial role in this procedure.",
author = "Shuichi Kawashima and Akitaka Matsumura",
year = "1985",
month = "3",
day = "1",
doi = "10.1007/BF01212358",
language = "English",
volume = "101",
pages = "97--127",
journal = "Communications in Mathematical Physics",
issn = "0010-3616",
publisher = "Springer New York",
number = "1",

}

TY - JOUR

T1 - Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion

AU - Kawashima, Shuichi

AU - Matsumura, Akitaka

PY - 1985/3/1

Y1 - 1985/3/1

N2 - The asymptotic stability of traveling wave solutions with shock profile is investigated for several systems in gas dynamics. 1) The solution of a scalar conservation law with viscosity approaches the traveling wave solution at the rate t-γ (for some γ>0) as t→∞, provided that the initial disturbance is small and of integral zero, and in addition decays at an algebraic rate for |x|→∞. 2) The traveling wave solution with Nishida and Smoller's condition of the system of a viscous heat-conductive ideal gas is asymptotically stable, provided the initial disturbance is small and of integral zero. 3) The traveling wave solution with weak shock profile of the Broadwell model system of the Boltzmann equation is asymptotically stable, provided the initial disturbance is small and its hydrodynamical moments are of integral zero. Each proof is given by applying an elementary energy method to the integrated system of the conservation form of the original one. The property of integral zero of the initial disturbance plays a crucial role in this procedure.

AB - The asymptotic stability of traveling wave solutions with shock profile is investigated for several systems in gas dynamics. 1) The solution of a scalar conservation law with viscosity approaches the traveling wave solution at the rate t-γ (for some γ>0) as t→∞, provided that the initial disturbance is small and of integral zero, and in addition decays at an algebraic rate for |x|→∞. 2) The traveling wave solution with Nishida and Smoller's condition of the system of a viscous heat-conductive ideal gas is asymptotically stable, provided the initial disturbance is small and of integral zero. 3) The traveling wave solution with weak shock profile of the Broadwell model system of the Boltzmann equation is asymptotically stable, provided the initial disturbance is small and its hydrodynamical moments are of integral zero. Each proof is given by applying an elementary energy method to the integrated system of the conservation form of the original one. The property of integral zero of the initial disturbance plays a crucial role in this procedure.

UR - http://www.scopus.com/inward/record.url?scp=34250115323&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34250115323&partnerID=8YFLogxK

U2 - 10.1007/BF01212358

DO - 10.1007/BF01212358

M3 - Article

AN - SCOPUS:34250115323

VL - 101

SP - 97

EP - 127

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -