In this paper, we establish a weak Serrin‐type blowup criterion for the Cauchy problem of the three‐dimensional (3D) compressible barotropic Navier–Stokes equations in the whole space. It shows that the strong or smooth solution exists globally if the velocity satisfies the weak Serrin's condition and the $L^{\infty }(0,T;L^{{\alpha }_0})$‐norm of the density is bounded, where ${\alpha }_0$ is positive constant. Therefore, if the weak Serrin norm of the velocity remains bounded, it is not possible for other kinds of singularities (such as vacuum states vanish or vacuum appears in the non‐vacuum region or even milder singularities) to form before the density becomes unbounded. Furthermore, the initial data can be arbitrarily large and contain vacuum states. The proof is based on the new a priori estimates for 3D compressible Navier–Stokes equations. In particular, this extends the corresponding Huang et al.'s results (SIAM J. Math. Anal. 43 (2011) 1872–1886).

中文翻译：

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三维粘性可压缩Navier–Stokes系统的弱Serrin型判据

在本文中，我们为整个空间中的三维（3D）可压缩正压Navier-Stokes方程的Cauchy问题建立了弱Serrin型爆破准则。它表明，如果速度满足弱Serrin条件，且速度满足弱Serrin条件，则强解或光滑解整体存在。${\mathrm{大号}}^{\infty }（0，Ť;{\mathrm{大号}}^{{\alpha }_0}）$密度的范数有界 ${\alpha }_0$是正常数。因此，如果弱的Serrin范数速度仍然是有界的，则在密度变为无界之前，不可能形成其他种类的奇点（例如真空状态消失或在非真空区域出现真空或什至更温和的奇点）。此外，初始数据可以任意大并且包含真空状态。该证明基于对3D可压缩Navier–Stokes方程的新的先验估计。特别是，这扩展了相应的Huang等人的结果（SIAM J. Math。Anal。43（2011）1872-1886）。