Abstract
In this paper, we first generalize the classical results on Cauchy problem for positive symmetric quasilinear systems to more general Besov space. Through this generalization, we obtain the local well-posedness with initial data in the space \(B^{\frac{d}{2}+1}_{2,1}(\mathbb {R}^d)\) which has critical regularity index. We then apply these results to give an explicit characterization on the isentropic approximation for full compressible Euler equations in \(\mathbb {R}^3\). This characterization tells us that isentropic compressible Euler equations is a reasonable approximation to Non-isentropic compressible Euler equations in the regime of classical solutions. The failure of such characterization was illustrated when singularities occur in the solutions.
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Acknowledgments
The authors are very grateful to valuable suggestions of the anonymous referees improving the presentations of this paper. J. Jia would like to thank China Scholarship Council that has provided a scholarship for supporting his research in the United States. J. Jia’s research is supported partially by National Natural Science Foundation of China under the Grant No. 11131006, and by the National Basic Research Program of China under the Grant No. 2013CB329404. The research of R. Pan is supported by the National Science Foundation through the Grant DMS-1108994.
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Jia, J., Pan, R. On Isentropic Approximations for Compressible Euler Equations. J Sci Comput 64, 745–760 (2015). https://doi.org/10.1007/s10915-014-9843-z
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DOI: https://doi.org/10.1007/s10915-014-9843-z