In this paper, the three-dimensional (3D) isentropic compressible Navier–Stokes equations with degenerate viscosities (ICND) is considered in both the whole space and the periodic domain. First, for the corresponding Cauchy problem, when shear and bulk viscosity coefficients are both given as a constant multiple of the density’s power (\(\rho ^\delta \) with \(0<\delta <1\)), based on some elaborate analysis of this system’s intrinsic singular structures, we show that the \(L^\infty \) norm of the deformation tensor D(u) and the \(L^6\) norm of \(\nabla \rho ^{\delta -1}\) control the possible breakdown of regular solutions with far field vacuum. This conclusion means that if a solution with far field vacuum of the ICND system is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of D(u) or \(\nabla \rho ^{\delta -1}\) as the critical time approaches. Second, when \(0<\delta \le 1\), under the additional assumption that the shear and second viscosities (respectively \(\mu (\rho )\) and \(\lambda (\rho )\)) satisfy the BD relation \(\lambda (\rho )=2(\mu '(\rho )\rho -\mu (\rho ))\), if we consider the corresponding problem in some periodic domain and the initial density is away from the vacuum, it can be proved that the possible breakdown of classical solutions can be controlled only by the \(L^\infty \) norm of D(u). It is worth pointing out that, except the conclusions mentioned above, another purpose of the current paper is to show how to understand the intrinsic singular structures of the fluid system considered now, and then how to develop the corresponding nonlinear energy estimates in the specially designed energy space with singular weights for the unique regular solution with finite energy.

在本文中，在整个空间和周期域中都考虑了具有退化粘度（ICND）的三维（3D）等熵可压缩Navier-Stokes方程。首先，对于相应的柯西问题，当剪切力和体积粘度系数均以密度乘方的常数（\（\ rho ^ \ delta \）与\（0 <\ delta <1 \））给出时，这个系统的内在奇异结构的一些复杂的分析，我们表明，\（L ^ \ infty \）变形张量的规范d（ü）和\（L ^ 6 \）的规范\（\ nabla \ RHO ^ { \ delta -1} \）通过远场真空控制常规溶液的可能分解。该结论意味着，如果ICND系统具有远场真空的解决方案最初是规则的，而在以后的某个时间失去其规则性，则奇异性的形成必定是由于D（u）或\（\ nabla \临界时间临近时，rho ^ {\ delta -1} \）。第二，当\（0 <\ delta \ le 1 \）时，在剪切和第二粘度（分别为\（\ mu（\ rho）\）和\（\ lambda（\ rho）\））满足的附加假设下BD关系\（\ lambda（\ rho）= 2（\ mu'（\ rho）\ rho-\ mu（\ rho））\），如果我们考虑某个周期域中的相应问题，并且初始密度远离真空，则可以证明经典解的可能分解只能由D（\（L ^ \ infty \）范数来控制你）。值得指出的是，除上述结论外，本文的另一个目的是展示如何理解现在考虑的流体系统的固有奇异结构，然后如何在经过特殊设计的情况下得出相应的非线性能量估计。具有奇异权重的能量空间，用于具有有限能量的唯一正则解。