Singularity formation for one dimensional full Euler equations

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• 作者：Ronghua Pan^a ; ^{panrh@math.gatech.edu" class="auth_mail" title="E-mail the corresponding author} ; Yi Zhu^b ; ^{y_zhu12@fudan.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：76N15 ; 35L65 ; 35L67
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 December 2016
• 年：2016
• 卷：261
• 期：12
• 页码：7132-7144
• 全文大小：250 K

文摘

We investigate the basic open question on the global existence v.s. finite time blow-up phenomena of classical solutions for the one-dimensional compressible Euler equations of adiabatic flow. For isentropic flows, it is well-known that the solutions develop singularity if and only if initial data contain any compression (the Riemann variables have negative spatial derivative). The situation for non-isentropic flow is not quite clear so far, due to the presence of non-constant entropy. In [4], it is shown that initial weak compressions do not necessarily develop singularity in finite time, unless the compression is strong enough for general data. In this paper, we identify a class of solutions of the full (non-isentropic) Euler equations, developing singularity in finite time even though their initial data do not contain any compression. This is in sharp contrast to the isentropic flow.