On the Breakdown of Regular Solutions with Finite Energy for 3D Degenerate Compressible Navier–Stokes Equations

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