Global Classical Solutions for Partially Dissipative Hyperbolic System of Balance Laws

Research output: Contribution to journalArticle

19 Citations (Scopus)

Abstract

The basic existence theory of Kato and Majda enables us to obtain local-in-time classical solutions to generally quasilinear hyperbolic systems in the framework of Sobolev spaces (in x) with higher regularity. However, it remains a challenging open problem whether classical solutions still preserve well-posedness in the case of critical regularity. This paper is concerned with partially dissipative hyperbolic system of balance laws. Under the entropy dissipative assumption, we establish the local well-posedness and blow-up criterion of classical solutions in the framework of Besov spaces with critical regularity with the aid of the standard iteration argument and Friedrichs' regularization method. Then we explore the theory of function spaces and develop an elementary fact that indicates the relation between homogeneous and inhomogeneous Chemin-Lerner spaces (mixed space-time Besov spaces). This fact allows us to capture the dissipation rates generated from the partial dissipative source term and further obtain the global well-posedness and stability by assuming at all times the Shizuta-Kawashima algebraic condition. As a direct application, the corresponding well-posedness and stability of classical solutions to the compressible Euler equations with damping are also obtained.

Original languageEnglish
Pages (from-to)513-553
Number of pages41
JournalArchive for Rational Mechanics and Analysis
Volume211
Issue number2
DOIs
Publication statusPublished - 2014 Feb 1
Externally publishedYes

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Global Classical Solution
Balance Laws
Dissipative Systems
Hyperbolic Systems
Classical Solution
Sobolev spaces
Regularity
Euler equations
Besov Spaces
Well-posedness
Entropy
Damping
Blow-up Criterion
Quasilinear Hyperbolic System
Compressible Euler Equations
Existence Theory
Local Well-posedness
Global Well-posedness
Regularization Method
Source Terms

ASJC Scopus subject areas

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

Cite this

Global Classical Solutions for Partially Dissipative Hyperbolic System of Balance Laws. / Xu, Jiang; Kawashima, Shuichi.

In: Archive for Rational Mechanics and Analysis, Vol. 211, No. 2, 01.02.2014, p. 513-553.

Research output: Contribution to journalArticle

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