Blowup for $ {{\rm{C}}}^{1} $ solutions of Euler equations in $ {{\rm{R}}}^{N} $ with the second inertia functional of reference

Abstract

The compressible Euler equations are an elementary model in mathematical fluid mechanics. In this article, we combine the Sideris and Makino-Ukai-Kawashima's classical functional techniques to study the new second inertia functional of reference:

\begin{document}$ { H}_{ref}{ (t) = }\frac{1}{2}\int_{\Omega(t)}\left( { \rho-\bar{\rho}}\right) \left\vert { \vec{x} }\right\vert ^{2}dV{{ , }} $\end{document}

for the blowup phenomena of $ C^{1} $ solutions $ (\rho, \vec{u}) $ with the support of $ \left({ \rho-\bar{\rho}}, \vec{u}\right) $, and with a positive constant $ { \bar{\rho}} $ for the adiabatic index $ \gamma > 1 $. We find that if the total reference mass

\begin{document}$ M_{ref}(0) = { \int_{{\bf R}^{N}}} (\rho_{0}({ \vec{x}})-\bar{\rho})dV\geq0, $\end{document}

and the total reference energy

\begin{document}$ E_{ref}(0) = \int_{{\bf R}^{N}}\left( \frac{1}{2}\rho_{0}({ \vec {x}})\left\vert \vec{u}_{0}({ \vec{x}})\right\vert ^{2}+\frac {K}{\gamma-1}\left( \rho_{0}^{\gamma}({ \vec{x}})-\bar{\rho }^{\gamma}\right) \right) dV, $\end{document}

with a positive constant $ K $ is sufficiently large, then the corresponding solution blows up on or before any finite time $ T > 0 $.