Serrin Type Criterion for the Three-Dimensional Viscous Compressible Flows

We extend the well-known Serrin's blowup criterion for the three-dimensional (3D) incompressible Navier-Stokes equations to the 3D viscous compressible cases. It is shown that for the Cauchy problem of the 3D compressible Navier-Stokes system in the whole space, the strong or smooth solution exists globally if the velocity satisfies the Serrin's condition and either the supernorm of the density or the $L^1(0,T;L^\infty)$-norm of the divergence of the velocity is bounded. Furthermore, in the case that either the shear viscosity coefficient is suitably large or there is no vacuum, the Serrin's condition on the velocity can be removed in this criteria.Comment: 16 page

No evidence for the evolution of mass density power-law index γ\gamma from strong gravitational lensing observation

In this paper, we consider the singular isothermal sphere lensing model that has a spherically symmetric power-law mass distribution $\rho_{tot}(r)\sim r^{-\gamma}$. We investigate whether the mass density power-law index $\gamma$ is cosmologically evolutionary by using the strong gravitational lensing (SGL) observation, in combination with other cosmological observations. We also check whether the constraint result of $\gamma$ is affected by the cosmological model, by considering several simple dynamical dark energy models. We find that the constraint on $\gamma$ is mainly decided by the SGL observation and independent of the cosmological model, and we find no evidence for the evolution of $\gamma$ from the SGL observation.Comment: 7 pages, 3 figure

Global Well-Posedness of Classical Solutions with Large Oscillations and Vacuum to the Three-Dimensional Isentropic Compressible Navier-Stokes Equations

We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the isentropic compressible Navier-Stokes equations in three spatial dimensions with smooth initial data which are of small energy but possibly large oscillations with constant state as far field which could be either vacuum or non-vacuum. The initial density is allowed to vanish and the spatial measure of the set of vacuum can be arbitrarily large, in particular, the initial density can even have compact support. These results generalize previous results on classical solutions for initial densities being strictly away from vacuum, and are the first for global classical solutions which may have large oscillations and can contain vacuum states.Comment: 30 page

A probabilistic method for gradient estimates of some geometric flows

In general, gradient estimates are very important and necessary for deriving convergence results in different geometric flows, and most of them are obtained by analytic methods. In this paper, we will apply a stochastic approach to systematically give gradient estimates for some important geometric quantities under the Ricci flow, the mean curvature flow, the forced mean curvature flow and the Yamabe flow respectively. Our conclusion gives another example that probabilistic tools can be used to simplify proofs for some problems in geometric analysis.Comment: 22 pages. Minor revision to v1. Accepted for publication in Stochastic Processes and their Application

Reexploration of interacting holographic dark energy model: Cases of interaction term excluding the Hubble parameter

In this paper, we make a deep analysis for the five typical interacting holographic dark energy models with the interaction terms $Q=3\beta H_{0}\rho_{\rm{de}}$, $Q=3\beta H_{0}\rho_{\rm{c}}$, $Q=3\beta H_{0}(\rho_{\rm{de}}+\rho_{\rm c})$, $Q=3\beta H_{0}\sqrt{\rho_{\rm{de}}\rho_{\rm c}}$, and $Q=3\beta H_{0}\frac{\rho_{\rm{de}}\rho_{c}}{\rho_{\rm{de}}+\rho_{\rm c}}$, respectively. We obtain observational constraints on these models by using the type Ia supernova data (the Joint Light-curve Analysis sample), the cosmic microwave background data (Planck 2015 distance priors), the baryon acoustic oscillations data, and the direct measurement of the Hubble constant. We find that the values of $\chi_{\rm min}^2$ for all the five models are almost equal (around~699), indicating that the current observational data equally favor these IHDE models. In addition, a comparison with the cases of interaction term involving the Hubble parameter $H$ is also made.Comment: 14 pages, 6 figures. arXiv admin note: text overlap with arXiv:1710.0306