On Isentropic Approximations for Compressible Euler Equations

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 B2,1d2+1(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{\frac{d}{2}+1}_{2,1}(\mathbb {R}^d)$$\end{document} 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 R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^3$$\end{document}. 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.