Existence results for viscous polytropic fluids with degenerate viscosities and far field vacuum

In this paper, we considered the isentropic Navier-Stokes equations for compressible fluids with density-dependent viscosities in $\mathbb{R}^3$. These systems come from the Boltzmann equations through the Chapman-Enskog expansion to the second order, cf.\cite{tlt}, and are degenerate when vacuum appears. We firstly establish the existence of the unique local regular solution (see Definition \ref{d1} or \cite{sz3}) when the initial data are arbitrarily large with vacuum at least appearing in the far field. Moreover it is interesting to show that we could't obtain any global regular solution that the $L^\infty$ norm of $u$ decays to zero as time $t$ goes to infinity.Comment: 30page

On classical solutions of the compressible magnetohydrodynamic equations with vacuum

In this paper, we consider the 3-D compressible isentropic MHD equations with infinity electric conductivity. The existence of unique local classical solutions is firstly established when the initial data is arbitrarily large, contains vacuum and satisfies some initial layer compatibility condition. The initial mass density needs not be bounded away from zero and may vanish in some open set. Moreover, we prove that the $L^\infty$ norm of the deformation tensor of velocity gradients controls the possible blow-up (see \cite{olga}\cite{zx}) for classical (or strong) solutions, which means that if a solution of the compressible MHD equations is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by the losing the bound of the deformtion tensor as the critical time approches. Our result (see (1.12) is the same as Ponce's criterion for $3$-D incompressible Euler equations \cite{pc} and Huang-Li-Xin's blow-up criterion for the $3$-D compressible Navier-stokes equations \cite{hup}.Comment: 31page

Blow-up criterion for the 33D non-resistive compressible Magnetohydrodynamic equations

In this paper, we prove a blow-up criterion in terms of the magnetic field $H$ and the mass density $\rho$ for the strong solutions to the $3$D compressible isentropic MHD equations with zero magnetic diffusion and initial vacuum. More precisely, we show that the $L^\infty$ norms of $(H,\rho)$ control the possible blow-up (see \cite{olga}\cite{zx}) for strong solutions, which means that if a solution of the compressible isentropic non-resistive MHD equations is initially smooth and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of the $L^\infty$ norm of $H$ or $\rho$ as the critical time approaches.Comment: 22 pages. arXiv admin note: text overlap with arXiv:1401.270

On regular solutions of the 3-D compressible isentropic Euler-Boltzmann equations with vacuum

In this paper, we discuss the Cauchy Problem for the compressible isentropic Euler-Boltzmann equations with vacuum in radiation hydrodynamics. Firstly, we establish the local existence of regular solutions by the fundamental methods in the theory of quasi-linear symmetric hyperbolic systems under some physical assumptions. Then we give the non-global existence of regular solutions caused by the effect of vacuum for $1<\gamma\leq 3$. Finally, we extend our result to the initial-boundary value problem under some suitable boundary conditions. These blow-up results tell us that the radiation cannot prevent the formation of singularities caused by the appearance of vacuum.Comment: 28 page

Well-posedness of three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities and far field vacuum

In this paper, the Cauchy problem for the three-dimensional (3-D) isentropic compressible Navier-Stokes equations is considered. When viscosity coefficients are given as a constant multiple of the density's power ($\rho^\delta$ with $0<\delta<1$), based on some analysis of the nonlinear structure of this system, we identify the class of initial data admitting a local regular solution with far field vacuum and finite energy in some inhomogeneous Sobolev spaces by introducing some new variables and initial compatibility conditions, which solves an open problem of degenerate viscous flow partially mentioned by Bresh-Desjardins-Metivier [3], Jiu-Wang-Xin [11] and so on. Moreover, in contrast to the classical theory in the case of the constant viscosity, we show that one can not obtain any global regular solution whose $L^\infty$ norm of $u$ decays to zero as time $t$ goes to infinity.Comment: 51 Page

Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum

We study the Cauchy problem for multi-dimensional compressible radiation hydrodynamics equations with vacuum. First, we present some sufficient conditions on the blow-up of smooth solutions in multi-dimensional space. Then, we obtain the invariance of the support of density for the smooth solutions with compactly supported initial mass density by the property of the system under the vacuum state. Based on the above-mentioned results, we prove that we cannot get a global classical solution, no matter how small the initial data are, as long as the initial mass density is of compact support. Finally, we will see that some of the results that we obtained are still valid for the isentropic flows with degenerate viscosity coefficients as well as 1-D case.Comment: 31page

Formation of singularities for the Relativistic Euler equations

This paper contributes to the study of large data problems for $C^1$ solutions of the relativistic Euler equations. In the $(1+1)$-dimensional spacetime setting, if the initial data are away from vacuum, a key difficulty in proving the global well-posedness or finite time blow-up is coming up with a way to obtain sharp enough control on the lower bound of the mass-energy density function $\rho$. First, for $C^1$ solutions of the 1-dimensional classical isentropic compressible Euler equations in the Eulerian setting, we show a novel idea of obtaining a mass density time-dependent lower bound by studying the difference of the two Riemann invariants, along with certain weighted gradients of them. Furthermore, using an elaborate argument on a certain ODE inequality and introducing some key artificial (new) quantities, we apply this idea to obtain the lower bound estimate for the mass-energy density of the (1+1)-dimensional relativistic Euler equations. Ultimately, for $C^1$ solutions with uniformly positive initial mass-energy density of the (1+1)-dimensional relativistic Euler equations, we give a necessary and sufficient condition for the formation of singularity in finite time, which gives a complete picture for the ($C^1$) large data problem in dimension $(1+1)$. Moreover, for the (3+1)-dimensional relativistic fluids, under the assumption that the initial mass-energy density vanishes in some open domain, we give two sufficient conditions for $C^1$ solutions to blow up in finite time, no matter how small the initial data are. We also do some interesting studies on the asymptotic behavior of the relativistic velocity when vacuum appears at the far field, which tells us that one can not obtain any global regular solution whose $L^\infty$ norm of $u$ decays to zero as time $t$ goes to infinity.Comment: v

On classical solutions to 2D Shallow water equations with degenerate viscosities

In this paper, the $2$-D isentropic Navier-Stokes systems for compressible fluids with density-dependent viscosity coefficients are considered. In particular, we assume that the viscosity coefficients are proportional to density. These equations, including several models in $2$-D shallow water theory, are degenerate when vacuum appears. We introduce the notion of regular solutions and prove the local existence of solutions in this class allowing the initial vacuum in the far field. This solution is further shown to be stable with respect to initial data in $H^2$ sense. A Beal-Kato-Majda type blow-up criterion is also established.Comment: 43pages. arXiv admin note: substantial text overlap with arXiv:1407.782

Blow-up criterion for the compressible magnetohydrodynamic equations with vacuum

In this paper, the 3-D compressible MHD equations with initial vacuum or infinity electric conductivity is considered. We prove that the $L^\infty$ norms of the deformation tensor $D(u)$ and the absolute temperature $\theta$ control the possible blow-up (see [5][18][20]) for strong solutions, which means that if a solution of the compressible MHD equations is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of $D(u)$ and $\theta$ as the critical time approaches. The viscosity coefficients are only restricted by the physical conditions. Our criterion (see (\ref{eq:2.911})) is similar to [17] for $3$-D incompressible Euler equations, [10] for $3$-D compressible isentropic Navier-stokes equations and [22]for $3$-D compressible isentropic MHD equations.Comment: 21pages. arXiv admin note: substantial text overlap with arXiv:1401.270

On classical solutions for viscous polytropic fluids with degenerate viscosities and vacuum

In this paper, we consider the three-dimensional isentropic Navier-Stokes equations for compressible fluids with viscosities depending on density in a power law and allowing initial vacuum. We introduce the notion of regular solutions and prove the local-in-time well-posedness of solutions with arbitrarily large initial data and vacuum in this class, which is a long-standing open problem due to the very high degeneracy caused by vacuum. Moreover, for certain classes of initial data with local vacuum, we show that the regular solution that we obtained will break down in finite time, no matter how small and smooth the initial data are.Comment: 46page