We establish a blow up criterion for the two-dimensional $k$-$\varepsilon$ model equations for turbulent flows in a bounded smooth domain $\Omega$. It is shown that for the initial-boundary value problem of the 2D $k$-$\varepsilon$ model equations in a bounded smooth domain, if $\|\nabla u\|_{L^{1}(0, T; L^{\infty})}+\|\nabla\rho\|_{L^{2}(0, T; L^{\infty})} +\|\varepsilon\|_{L^{2}(0, T; L^{\infty})}$ $<\infty$, then the strong solution $(\rho, u, h,k, \varepsilon)$ can be extended beyond $T$.

中文翻译：

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湍流二维 $k$-$\varepsilon$ 模型方程的膨胀准则

我们为有界光滑域 $\Omega$ 中湍流的二维 $k$-$\varepsilon$ 模型方程建立了一个爆炸标准。证明对于有界光滑域中二维$k$-$\varepsilon$模型方程的初边值问题，如果$\|\nabla u\|_{L^{1}(0, T ; L^{\infty})}+\|\nabla\rho\|_{L^{2}(0, T; L^{\infty})} +\|\varepsilon\|_{L^{ 2}(0, T; L^{\infty})}$ $<\infty$，那么强解 $(\rho, u, h,k, \varepsilon)$ 可以扩展到 $T$ 之外。