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 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.     

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.     

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

We study an initial boundary value problem for the three-dimensional Navier–Stokes equations of viscous heat-conductive fluids in a bounded smooth domain. We establish a blow-up criterion for the local strong solutions in terms of the temperature and the gradient of velocity only, similar to the Beale–Kato–Majda criterion for ideal incompressible flows.     

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.     

Effective velocity in compressible Navier-Stokes equations with third-order derivatives

A formulation of certain barotropic compressible Navier-Stokes equations with third-order derivatives as a viscous Euler system is proposed by using an effective velocity variable. The equations model, for instance, viscous Korteweg or quantum Navier-Stokes flows. The formulation in the new variable allows for the derivation of an entropy identity, which is known as the BD (Bresch-Desjardins) entropy equation. As a consequence of this estimate, a new global-in-time existence result for the one-dimensional quantum Navier-Stokes equations with strictly positive particle densities is proved.     

Blowup Criteria for Full Compressible Navier-Stokes Equations with Vacuum State

This paper deals with the global strong solution to the three-dimensional(3D) full compressible Navier-Stokes systems with vacuum.The authors provide a sufficient condition which requires that the Sobolev norm of the temperature and some norm of the divergence of the velocity are bounded,for the global regularity of strong solution to the 3D compressible Navier-Stokes equations.This result indicates that the divergence of velocity fields plays a dominant role in the blowup mechanism for the full compressible Navier-Stokes equations in three dimensions.     

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]).     

Stationary solutions to the equations of dynamics of mixtures of heat-conductive compressible viscous fluids

We consider the equations describing the three-dimensional steady motions of binary mixtures of heat-conductive compressible viscous fluids. An existence theorem for the boundary value problem that corresponds to flows in a bounded domain is proved in the class of weak generalized solutions.     

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.     

Blowup criterion for viscous, compressible micropolar fluids with vacuum

We prove that the maximum norm of velocity gradients controls the possible breakdown of smooth (strong) solutions for the 3-dimensional viscous, compressible micropolar fluids. More precisely, if a solution of the system is initially regular and loses its regularity at some later time, then the loss of regularity implies the growth without the bound of the velocity gradients as the critical time approaches. Our result is a generalization of Huang et al. (2011) [13] from viscous barotropic flows to the viscous, compressible micropolar fluids. In addition, initial vacuum states are also allowed in our result.     

An effective model of the dynamics of a barotropic gas with fast oscillating initial data

Under study are the classical three-dimensional Navier-Stokes equations of a compressible inhomogeneous viscous fluid in a smooth bounded domain endowed with no-slip conditions on the boundary of the domain and fast oscillating initial density distributions. The state equation of the medium is the state equation for a barotropic gas. We assume that the adiabatic constant is greater than 3. We give a rigorous derivation of the homogenization procedure as the frequencies of fast oscillations tend to infinity and obtain a limit effective model of the dynamics of a compressible viscous gas with fast oscillating initial data.     

A note on barotropic compressible quantum Navier-Stokes equations

The global-in-time existence of weak solutions to the barotropic compressible quantum Navier-Stokes equations has been proved very recently, by Jüngel (2009) [1], if the viscosity constant is smaller than the scaled Plank constant. This paper extends the results to the case that the viscosity constant equals the scaled Plank constant. By using a new estimate on the square root of the solution, apparently not available in [1], the semiclassical limit for the viscous quantum Euler equations (which are equivalent to the barotropic compressible quantum Navier-Stokes equations) can be performed; then the results of this paper are obtained easily.     

A Serrin criterion for compressible nematic liquid crystal flows

This paper is concerned with a simplified system, proposed by Ericksen and Leslie, modeling the flow of nematic liquid crystals. We establish a blowup criterion for three‐dimensional compressible nematic liquid crystal flows, which is analogous to the well‐known Serrin's blowup criterion for three‐dimensional incompressible viscous flows. Copyright © 2012 John Wiley & Sons, Ltd.     

Existence of Weak Solutions to the Problem on Three-Dimensional Steady Heat-Conductive Motions of Compressible Viscous Multicomponent Mixtures

Siberian Mathematical Journal - We consider the equations describing the three-dimensional steady heat-conductive motions of compressible viscous multicomponent mixtures. We also prove the...     

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

In this paper, we prove a blow-up criterion of strong solutions to the 3D viscous and non-resistive isentropic compressible magnetohydrodynamic equations with initial vacuum. This blow-up criterion depends only on the gradient of velocity, which is analogous to the one for the compressible Navier–Stokes equations (cf. Huang et al. (2010) [40]).     

Global regularity for the Cauchy problem of three-dimensional compressible magnetohydrodynamics equations

In this paper, we study the Cauchy problem for the three-dimensional compressible magnetohydrodynamics equations. We establish a blowup criterion for global regularity of strong solutions, which depends only on density and magnetic field. In addition, the initial data can be arbitrarily large and contain vacuum states. The proof is based on the new a priori estimates for three-dimensional compressible magnetohydrodynamics equations.     

On local strong and classical solutions to the three-dimensional barotropic compressible Navier-Stokes equations with vacuum

We consider the local well-posedness of strong and classical solutions to the three-dimensional barotropic compressible Navier-Stokes equations with density containing vacuum initially. We first prove the local existence and uniqueness of the strong solutions, where the initial compatibility condition proposed by Cho et al.(2004), Cho and Kim(2006) and Choe and Kim(2003) is removed in a suitable sense. Then the continuous dependence of strong solutions on the initial data is derived under an additional compatibility condition. Moreover, for the initial data satisfying some additional regularity and the compatibility condition,the strong solution is proved to be a classical one.     

Numerical computation of viscous profiles for hyperbolic conservation laws

Viscous profiles of shock waves in systems of conservation laws can be viewed as heteroclinic orbits in associated systems of ordinary differential equations (ODE). In the case of overcompressive shock waves, these orbits occur in multi-parameter families. We propose a numerical method to compute families of heteroclinic orbits in general systems of ODE. The key point is a special parameterization of the heteroclinic manifold which can be understood as a generalized phase condition; in the case of shock profiles, this phase condition has a natural interpretation regarding their stability. We prove that our method converges and present numerical results for several systems of conservation laws. These examples include traveling waves for the Navier-Stokes equations for compressible viscous, heat-conductive fluids and for the magnetohydrodynamics equations for viscous, heat-conductive, electrically resistive fluids that correspond to shock wave solutions of the associated ideal models, i.e., the Euler, resp. Lundquist, equations.

     

Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density

We present a sufficient condition on the blowup of smooth solutions to the compressible Navier-Stokes equations in arbitrary space dimensions with initial density of compact support. As an immediate application, it is shown that any smooth solutions to the compressible Navier-Stokes equations for polytropic fluids in the absence of heat conduction will blow up in finite time as long as the initial densities have compact support, and an upper bound, which depends only on the initial data, on the blowup time follows from our elementary analysis immediately. Another implication is that there is no global small (decay in time) or even bounded (in the case that all the viscosity coefficients are positive) smooth solutions to the compressible Navier-Stokes equations for polytropic fluids, no matter how small the initial data are, as long as the initial density is of compact support. This is in contrast to the classical theory of global existence of small solutions to the same system with initial data being a small perturbation of a constant state that is not a vacuum. The blowup of smooth solutions to the compressible Euler system with initial density and velocity of compact support is a simple consequence of our argument. © 1998 John Wiley & Sons, Inc.