On spherically symmetric solutions of the relativistic Euler equation

This work investigates the spherical symmetric solutions of more realistic equation of states. We generalize the method of Hsu et al. (Methods Appl. Anal. 8 (2001) 159) to show the existence of spherical symmetric weak solution of the relativistic Euler equation with initial data containing the vacuum state.     

Exact solutions for the Cauchy problem to the 3D spherically symmetric incompressible Navier-Stokes equations

In this article, we establish exact solutions to the Cauchy problem for the 3D spherically symmetric incompressible Navier-Stokes equations and further study the existence and asymptotic behavior of solution.     

On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities

We study the Navier-Stokes equations for compressible barotropic fluids in a bounded or unbounded domain Ω of R³. We first prove the local existence of solutions (ρ,u) in C([0,T_*]; (ρ^∞ +H³(Ω)) × under the assumption that the data satisfies a natural compatibility condition. Then deriving the smoothing effect of the velocity u in t>0, we conclude that (ρ,u) is a classical solution in (0,T_**)×Ω for some T_** ∈ (0,T_*]. For these results, the initial density needs not be bounded below away from zero and may vanish in an open subset (vacuum) of Ω.     

From the dynamics of gaseous stars to the incompressible Euler equations

A model for the dynamics of gaseous stars is introduced and formulated by the Navier-Stokes-Poisson system for compressible, reacting gases. The combined quasineutral and inviscid limit of the Navier-Stokes-Poisson system in the torus Tn is investigated. The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations is proven for the global weak solution and for the case of general initial data.     

Existence and asymptotic behavior of C solutions to the multi-dimensional compressible Euler equations with damping

In this paper, the existence and asymptotic behavior of C¹$C^1$ solutions to the multi-dimensional compressible Euler equations with damping on the framework of Besov space are considered. Comparing with the well-posedness results of Sideris–Thomases–Wang [T. Sideris, B. Thomases, D.H. Wang, Long time behavior of solutions to the three-dimensional compressible Euler with damping, Comm. Partial Differential Equations 28 (2003) 953–978], we weaken the regularity assumptions on the initial data. The global existence lies on a crucial a-priori estimate which is obtained by the spectral localization method. The main analytic tools are the Littlewood–Paley decomposition and Bony’s paraproduct formula.     

Uniform stability at permanently acting disturbances of solutions of Navier-Stokes equations for compressible isothermic fluids

We prove that the uniform stability at permanently acting disturbances of a given solution of the Navier-Stokes equations for viscous compressible isothermic fluid is a consequence of the uniform exponential stability of the zero solution of so-called linearized equations.The research was supported by the grant No. 201/93/2177 of Grant Agency of Czech Republic.     

Behaviour of weak solutions of compressible Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor

The aim of this paper is to study the behaviour of a weak solution to Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor for time going to infinity. In an analogous way as in [18], we construct a suitable function which approximates the density for time going to infinity. Using properties of this function, we can prove the strong convergence of the density to its limit state. The behaviour of the velocity field and kinetic energy is mentioned as well.     

Asymptotic stability of viscous contact discontinuity to an inflow problem for compressible Navier-Stokes equations

This paper is concerned with an initial-boundary value problem for one-dimensional full compressible Navier-Stokes equations with inflow boundary conditions in the half space R₊=(0,+∞). The asymptotic stability of viscous contact discontinuity is established under the conditions that the initial perturbations and the strength of contact discontinuity are suitably small. Compared with the free-boundary and the initial value problems, the inflow problem is more complicated due to the additional boundary effects and the different structure of viscous contact discontinuity. The proofs are given by the elementary energy method.     

On center singularity for compressible spherically symmetric nematic liquid crystal flows

The paper is concerned with a simplified system, proposed by Ericksen [6] and Leslie [20], modeling the flow of nematic liquid crystals. In the first part, we give a new Serrin's continuation principle for strong solutions of general compressible liquid crystal flows. Based on new observations, we establish a localized Serrin's regularity criterion for the 3D compressible spherically symmetric flows. It is proved that the classical solution loses its regularity in finite time if and only if, either the concentration or vanishing of mass forms or the norm inflammation of gradient of orientation field occurs around the center.     

Free boundary problem for the equations of spherically symmetric motion of compressible gas with density-dependent viscosity

We consider a free boundary problem for the equations of spherically symmetric motion of a isentropic gas with a density-dependent viscosity , where and λ are positive constants. We prove that the problem admits a weak solution provided that 0 < λ < 1/4.      

On the Riemann problem for 2D compressible Euler equations in three pieces

The Riemann problem for two-dimensional isentropic Euler equations is considered. The initial data are three constants in three fan domains forming different angles. Under the assumption that only a rarefaction wave, shock wave or contact discontinuity connects two neighboring constant initial states, it is proved that the cases involving three shock or rarefaction waves are impossible. For the cases involving one rarefaction (shock) wave and two shock (rarefaction) waves, only the combinations when the three elementary waves have the same sign are possible (impossible).     

Self-similar solutions of the Euler equations with spherical symmetry

We consider self-similar flows arising from the uniform expansion of a spherical piston and preceded by a shock wave front. With appropriate boundary conditions imposed on the piston surface and the spherical shock, the isentropic compressible Euler system is transformed into a nonlinear ODE system. We formulate the problem in a simple form in order to present the analytic proof of the global existence of positive smooth solutions.     

On the uniqueness in critical spaces for compressible Navier-Stokes equations

We show some new uniqueness results for compressible flows with data having critical regularity. In the barotropic case, uniqueness is stated whenever the space dimension N satisfies N ≥ 2, and in the polytropic case, whenever N ≥ 3. The endpoints N = 2 in the barotropic case and N = 3 in the polytropic case were left open in [4], [5] and [6].     

Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum

The asymptotic behavior of solutions of the damped compressible Euler equations is conjectured to obey to the famous porous media equations (PMES). The previous works on this topic concern the case away from vacuum where the system is strictly hyperbolic. In present paper, we prove that the L^∞ entropy weak solution with vacuum, obtained by the compensated compactness theory, converges strongly in space to the unique similarity solution of the related PME, as time goes to infinity.     

Global existence of shock front solution to axially symmetric piston problem in compressible flow

In this paper we study the global existence of BV solution to two-dimensional piston problem in fluid dynamics. Different from previous results on related problems we remove the restriction on the strength of the leading shock and require the velocity of the piston is rather fast or the density is quite small instead. The main tool in our proof is Glimm Scheme with some improvement. To define the Glimm functional we derive more precise estimates for the interaction of elementary waves, particularly in the region near the leading shock. The paper is partially supported by National Natural Science Foundation of China 10531020, the National Basic Research Program of China 2006CB805902 and the Doctorial Foundation of National Educational Ministry 20050246001.     

On the zero-velocity-limit solutions to the Navier–Stokes equations of compressible flow

We consider the zero-velocity stationary problem of the Navier–Stokes equations of compressible isentropic flow describing the distribution of the density ϱ of a fluid in a spatial domain Ω⊂ℝ ^N driven by a time-independent potential external force b=∇F. A sharp condition in terms of F is given for the problem to possess a unique nonnegative solution ϱ having a prescribed mass m > 0. Received: 20 October 1997     

On the singular limit for slightly compressible fluids

In this paper we study the motion of slightly compressible inviscid fluids. We prove that the solution of the corresponding system of nonlinear partial differential equations converges (uniformly) in the strong norm (that of the data space) to the solution of the incompressible equations, as the Mach number goes to zero (see Theorem 1.2). Actually, our proof applies to a large class of singular limit problems as shown in the Theorem 2.2.