A blow-up criterion for 3D compressible viscoelastic flow with large initial data

In this paper, we consider a Cauchy problem for the three-dimensional compressible viscoelastic flow with large initial data. We establish a blow-up criterion for the strong solutions in terms of the gradient of velocity only, which is similar to the Beale-Kato-Majda criterion for ideal incompressible flow (cf. Beale et al. (1984) [20]) and the blow-up criterion for the compressible Navier-Stokes equations (cf. Huang et al. (2011) [21]).     

A blow-up criterion for 3-D non-resistive compressible heat-conductive magnetohydrodynamic equations with initial vacuum

In this paper, we prove a blow-up criterion of strong solutions to the 3-D viscous and non-resistive magnetohydrodynamic equations for compressible heat-conducting flows with initial vacuum. This blow-up criterion depends only on the gradient of velocity and the temperature, which is similar to the one for compressible Navier-Stokes equations.     

可压缩液晶系统强解的破裂准则

研究可压缩液晶方程组强解的破裂准则,建立了一种仅依据于速度梯度的破裂准则,此种准则类似于理想可压缩流情形的Beale-Kato-Majda准则和由Huang和Xin得到的可压缩Navier-Stokes方程组情形的准则.证明用到能量不等式和高阶能量不等式.主要困难是初始密度含有真空.     

A blow-up criterion for classical solutions to the compressible Navier-Stokes equations

In this paper, we obtain a blow-up criterion for classical solutions to the 3-D compressible Navier-Stokes equations just in terms of the gradient of the velocity, analogous to the Beal-Kato-Majda criterion for the ideal incompressible flow. In addition, the initial vacuum is allowed in our case.     

A Blow-up Criterion of Strong Solutions to the Compressible Viscous Heat-Conductive Flows with Zero Heat Conductivity

We study the compressible Navier-Stokes equations of viscous heat-conductive fluids in a periodic domain \mathbbT³\mathbb{T}^{3} with zero heat conductivity k=0. We prove a blow-up criterion for the local strong solutions in terms of the temperature and positive density, similar to the Beale-Kato-Majda criterion for ideal incompressible flows.     

Blow-up criterion for three-dimensional compressible viscoelastic fluids with vacuum

In this paper, we study a Cauchy problem for the equations of 3D compressible viscoelastic fluids with vacuum. We establish a blow-up criterion for the local strong solutions in terms of the upper bound of the density and deformation gradient.     

A blow-up criterion for compressible viscous heat-conductive flows

We study an initial boundary value problem for the Navier-Stokes equations of compressible viscous heat-conductive fluids in a 2-D periodic domain or the unit square domain. We establish a blow-up criterion for the local strong solutions in terms of the gradient of the velocity only, which coincides with the famous Beale-Kato-Majda criterion for ideal incompressible flows.     

A BLOW-UP CRITERION OF SPHERICALLY SYMMETRIC STRONG SOLUTIONS TO 3D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH FREE BOUNDARY

In this article, we consider the free boundary value problem of 3D isentropic compressible Navier-Stokes equations. A blow-up criterion in terms of the maximum norm of gradients of velocity is obtained for the spherically symmetric strong solution in terms of the regularity estimates near the symmetric center and the free boundary respectively.     

A Beale–Kato–Majda blow-up criterion for the 3-D compressible Navier–Stokes equations

We prove a blow-up criterion in terms of the upper bound of the density for the strong solution to the 3-D compressible Navier–Stokes equations. The initial vacuum is allowed. The main ingredient of the proof is a priori estimate for an important quantity under the assumption that the density is upper bounded, whose divergence can be viewed as the effective viscous flux.     

On well-posedness and blow-up for the full compressible Hall-MHD system

In this paper, we establish the local well-posedness and a blow-up criterion of strong solutions to the 3D compressible full Hall-MHD system with positive density.     

Blow-up of smooth solutions to the compressible magnetohydrodynamic flows

In this paper, we prove the blow-up phenomena of smooth solutions to the Cauchy problem for the full compressible magnetohydrodynamic equations and isentropic compressible magnetohydrodynamic equations with constant and degenerate viscosities under some restrictions on the initial data. In particular, our results do not require that the initial data have compact support or contain vacuum in any finite region.     

A blow-up criterion for two dimensional compressible viscous heat-conductive flows

We establish a blow-up criterion in terms of the upper bound of the density and temperature for the strong solution to 2D compressible viscous heat-conductive flows. The initial vacuum is allowed.     

Notes on the Incompressible Euler and Related Equations on $\BR^N$\

The author reviews briefly some of the recent results on the blow-up problem for the incompressible Euler equations on R^N and also presents Liouville type theorems for the incompressible and compressible fluid equations.     

Blow-up of smooth solutions to the compressible fluid models of Korteweg type

We show the blow-up of smooth solutions to a non-isothermal model of capillary compressible fluids in arbitrary space dimensions with initial density of compact support. This is an extension of Xin’s result [Xin, Z.: Blow-up of smooth solutions to the compressible Navier-Stokes equations with compact density. Comm. Pure Appl. Math., 51, 229–240 (1998)] to the capillary case but we do not need the condition that the entropy is bounded below. Moreover, from the proof of Theorem 1.2, we also obtain the exact relationship between the size of support of the initial density and the life span of the solutions. We also present a sufficient condition on the blow-up of smooth solutions to the compressible fluid models of Korteweg type when the initial density is positive but has a decay at infinity.     

可压缩磁流体动力方程解的正则性

该文研究三维等熵磁流体动力方程和二维带正密度的热传导磁流体动力方程解的正则性.给出了局部强解爆破的条件.     

A BLOW-UP CRITERION OF STRONG SOLUTIONS TO THE QUANTUM HYDRODYNAMIC MODEL

In this article, we focus on the short time strong solution to a compressible quantum hydrodynamic model. We establish a blow-up criterion about the solutions of the compressible quantum hydrodynamic model in terms of the gradient of the velocity, the second spacial derivative of the square root of the density, and the first order time derivative and first order spacial derivative of the square root of the density.     

理想磁流体动力学方程组经典解的破裂

本文给出了理想磁流体动力学方程组的经典解在初始扰动适当大的情况下破裂的结果.文[1]证明了描述多方理想可压缩气体运动的欧拉系统的经典解在初始扰动适当大的情况下破裂的结果.本文将利用和文[1]相似的方法证明所得定理.     

Blow-up of viscous compressible reactive self-gravitating gas

In this paper,we prove some results concerning blow-up of viscous compressible reactive (selfgravitating) flows when the initial density is compactly supported and the other initial value satisfy proper conditions.It extends the work of Xin and Cho to the case of viscous compressible reactive self-gravitating flows equations.We control the lower bound of second moment by total energy and obtain the precise relationship between the size of the support of initial density and the existence time.     

The Blow-up of Solutions for Two-Dimensional Irrotational Compressible Euler Equations

For two-dimensional irrotational compressible Euler equations with initial data where that is a small perturbation from a constant state, we prove that the first-order derivatives of ρ, υ blow-up at the blow-up time, while ρ, υ remain continuous. In particular, in the irrotational case we prove S. Alinhac's statement. Project supported bythe Tianyuan Foundation of China and Lab. of Math. for Nonlinear Problems. Fudan. Univ.     

A Blow-up Criterion for 2D Compressible Nematic Liquid Crystal Flows in Terms of Density

This paper is concerned with the Cauchy problem for compressible nematic liquid crystal flows in the two-dimensional space (2D). We establish a blow-up criterion in terms of the density only, provided the macroscopic average of the nematic liquid crystal orientation field satisfies a geometric condition. In particular, the initial vacuum is allowed and the compatibility condition is removed.