Degenerate pullback attractors for the 3D Navier-Stokes equations

As in our previous paper, the 3D Navier-Stokes equations with a translationally bounded force contain pullback attractors in a weak sense. Moreover, those attractors consist of complete bounded trajectories. In this paper, we present a sufficient condition under which the pullback attractors are degenerate. That is, if the Grashof constant is small enough, the pullback attractor will be a single point on a unique, complete, bounded, strong solution. We then apply our results to provide a new proof of the existence of a unique, strong, periodic solution to the 3D Navier-Stokes with a small, periodic forcing term

Weak-Strong uniqueness for compressible Navier-Stokes system with degenerate viscosity coefficient and vacuum in one dimension

We prove weak-strong uniqueness results for the compressible Navier-Stokes system with degenerate viscosity coefficient and with vacuum in one dimension. In other words, we give conditions on the weak solution constructed in \cite{Jiu} so that it is unique. The novelty consists in dealing with initial density $\rho_0$ which contains vacuum. To do this we use the notion of relative entropy developed recently by Germain, Feireisl et al and Mellet and Vasseur (see \cite{PG,Fei,15}) combined with a new formulation of the compressible system (\cite{cras,CPAM,CPAM1,para}) (more precisely we introduce a new effective velocity which makes the system parabolic on the density and hyperbolic on this velocity).Comment: arXiv admin note: text overlap with arXiv:1411.550