Global Classical Solutions for Partially Dissipative Hyperbolic System of Balance Laws

By Jiang Xu, Shuichi Kawashima. 2012

76 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/405c86074e85db3e280613546c5fe09ca01e1889

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Jiang Xu
2. Shuichi Kawashima
3. Yongqing Liu

In Computational Mechanics

1. Jiang Xu
2. Shuichi Kawashima

In Mathematical Physics

1. Jiang Xu
2. Shuichi Kawashima
3. Yongqing Liu

Citations in the corpus

• The Optimal Decay Estimates on the Framework of Besov Spaces for Generally Dissipative Systems (2014, methodology): “How does the large-time behavior of solutions impact upon the framework of spatially critical Besov spaces? Following from the Gagliardo–Nirenberg–Sobolev inequality and the complex interpolation inequality related to Besov spaces, the asymptotics of solutions U and (I − P)U have been shown (see [29”