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

Abstract

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.