Zero dissipation limit to rarefaction wave with vacuum for the one-dimensional non-isentropic micropolar equations
Section snippets
Introduction and main result
In the present paper, we consider the zero dissipation limit of the one-dimensional non-isentropic micropolar equations here, , the unknowns , , and are the mass density, fluid velocity, microrotational velocity and absolute temperature respectively. Besides, is pressure and represents the specific total energy, where is the specific
Rarefaction wave
As we all know, there is no exact rarefaction wave profile for the micropolar equations (1.1), and the following approximate rarefaction wave profile satisfying the Euler equations was motivated by Matsumura–Nishihara [28] and Xin [29].
Consider the Riemann problem for the inviscid Burgers equation: If , the Riemann problem (2.1) admits a rarefaction wave solution given by However, is only Lipschitz
Reformulation of the problem
To prove Theorem 1.1, we consider the Cauchy problem (1.1) with the smooth initial data and treat the global smooth solution of system (1.1), (3.1) as the perturbation around the approximate rarefaction wave . For convenience, we reformulate the system by introducing a scaling for the independent variables Then we denote the perturbation For simplicity of notation, without
Proof of Theorem 1.1
In this section, we prove Theorem 1.1. Actually, there only left (1.15) with given in (1.16)–(1.18) need to be proved. By Lemmas 2.2, 2.3, Proposition 3.1 and and , it holds that for any given positive constant there exists a positive constant which is independent of such that similarly we have
Acknowledgments
The work was supported by the grants from the National Natural Science Foundation of China under contracts 11731008, 11671309 and 11971359. We would like to express our thanks to the anonymous referees for their valuable comments, which lead to substantial improvements of the original manuscript. Last but not least, the author would like to thank Professor Huijiang Zhao for his support and encouragement.
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