THE RIEMANN PROBLEM WITH DELTA INITIAL DATA FOR THE ONE-DIMENSIONAL CHAPLYGIN GAS EQUATIONS

In this article, we study the Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations. Under the generalized Rankine-Hugoniot conditions and the entropy condition, we constructively obtain the global existence of generalized solutions that explicitly exhibit four kinds of different structures. Moreover, we obtain the stability of generalized solutions by making use of the perturbation of the initial data.

Generalized δ-entropy condition to Riemann solution for Chaplygin gas in traffic flow

The Riemann problem for the Aw-Rascle model in the traffic flow with the Delta initial data for the Chaplygin gas is studied. The solutions are constructed globally under the generalized Rankine-Hugoniot relations, the δ-entropy conditions, and the generalized δ-entropy conditions. A new Delta wave, which is called a primary Delta wave, is defined in some solutions. The primary Delta wave satisfies the generalized Rankine- Hugoniot relations and the generalized δ-entropy conditions. It generates initially from the Delta initial data, which either evaluates a Delta wave, whose weight becomes stronger and stronger, or disappears at a finite time.

Axisymmetric solutions of the Chaplygin gas for initial negative radial velocity

In this paper, a three-parameter family of self-similar and weak solutions are constructed rigorously in two space dimensions for all positive time to the Euler equations with axisymmetric and radial negative initial velocity for the Chaplygin gas. Under the axisymmetry and self-similar assumptions, the equations are reduced to a system of three ordinary differential equations, from which we obtain structures of solutions besides their existence. The solutions exhibit some phenomena, such as formation and evolution of black hole, expansion and explosive expansion, in the evolution of universe.

Observational constraints on the new generalized Chaplygin gas model

We use the latest data to investigate observational constraints on the new generalized Chaplygin gas （NGCG） model. Using the Markov Chain Monte Carlo method, we constrain the NGCG model with type Ia supernovae from the Union2 set （557 data）, the usual baryonic acoustic oscillation （BAO） observation from the spectroscopic Sloan Digital Sky Survey data release 7 galaxy sample, the cosmic mi- crowave background observation from the 7-year Wilkinson Microwave Anisotropy Probe results, newly revised data on H（z）, as well as a value of θBAO （Z = 0.55） = （3.90° ±0.38°） for the angular BAO scale. The constraint results for the NGCG model are ωX=-1.0510＋0.1563-0.1685（1σ）＋0.2226-0.2398（2σ）,η=1.0117＋0.0469-0.0502（1σ）＋0.0693-0.0716（2σ）and ΩX=0.7297＋0.0229-0.0276（1σ）＋0.0329-0.0402（2σ）, which give a rather stringent constraint. From the results, we can see that a phantom model is slightly favored and the proba- bility that energy transfers from dark matter to dark energy is a little larger than the inverse.

Constraining the Generalized and Superfluid Chaplygin Gas Models with the Sandage-Loeb Test

The Sandage Loeb （SL） test is a direct measurement of the cosmic expansion by probing the redshift drifts of quasi-stellar objects in the ＇redshift desert＇ of 2 〈 z 〈 5. In this work, we investigate its constraints on the unified dark energy and dark matter models including the generalized Chaplygin gas and the superfluid Chaplygin gas. In addition, type Ia supernovae （SNIa） data and the distance ratios derived from the cosmic microwave background radiation and baryon acoustic oscillation observations （CMB/BAO） are also used. We find that the mock SL data gives the tightest constraints on the model parameters and it can help to reduce the parameter regions allowed by the present SNIa＋CMB/BAO by about 75% when all datasets considered are combined. Thus the SL test is a worthy and long awaited measurement to probe effectively the cosmic expanding history and the properties of dark energy.

Wave Interactions of the Aw-Rascle Model for Generalized Chaplygin Gas

In this paper, we investigate the elementary wave interactions of the Aw-Rascle model for the generalized Chaplygin gas. We construct the unique solution by the characteristic analysis method and obtain the stability of the corresponding Riemann solutions under such small perturbations on the initial values. We find that the elementary wave interactions have a much more simple structure for Temple class than general systems of conservation laws. It is important to study the elementary waves interactions of the traffic flow system for the generalized Chaplygin gas not only because of their significance in practical applications in the traffic flow system, but also because of their basic role for the general mathematical theory.

Wave Interactions for Chaplygin Gas with Combustion

The elementary wave interactions for the Chapman-Jouguet model with combustion are investigated. We obtain the unique solution of the initial value problem under the global entropy conditions. We analyze the elementary wave interactions in the phase plane and construct uniquely the solution of this initial value problem. It is found that the combustion wave solution of the corresponding Riemann may be extinguished after perturbation which shows that the unburnt gas is unstable.

Viscous Modified Chaplygin Gas in Classical and Loop Quantum Cosmology

We investigate the cosmological model of viscous modified Chaplygin gas （VMCG） in classical and loop quantum cosmology （LQC）. Firstly, we constrain its equation of state parameters in the framework of standard cosmology from Union 2.1 SNe Ia data. Then, we probe the dynamical stability of this model in a universe filled with VMCG and baryonic fluid in LQC background. It is found that the model is very suitable with （χ2/d.o.f = 0.974） and gives a good prediction of the current values of the deceleration parameter q0 =∈ （-0.60, -0.57） and the effective state parameter ωeff∈ （-0.76, -0.74） that is consistent with the recent observational data. The model can also predict the time crossing when （ρDE ≈ Pmatter） at z = 0.75 and can solve the coincidence problem. In LQC background, the Big Bang singularity found in classical cosmology ceases to exist and is replaced by a bounce when the Hubble parameter vanishes at ρtot≈ρc.

Universe Filled with Generalized Cosmic Chaplygin Gas and Barotropic Fluid

In this paper we have considered a model of the universe filled with Generalized Cosmic Chaplygin Gas and another fluid with barotropic equation of state. We observe its role in accelerating phase of the universe by considering the mixture of these two fluid models valid from the radiation era to for and the radiation era to quintessence model for . The statefinder parameters describe the evolution of the universe in different phases for these two fluid models.

Constraining the variable generalized Chaplygin gas model in matter creation cosmology

We explore the variable generalized Chaplygin gas(VGCG)model in the theory of matter creation cosmology within the framework of a spatially homogeneous and isotropic flat Friedmann—Lemai tre—Robertson—Walker space-time.Matter creation cosmology is based on reinterpretation of the energy–momentum tensor in Einstein's field equations.This creation corresponds to an irreversible energy flow from the gravitational field to the created matter constituents.The variable Chaplygin gas(VCG)is also studied as a particular solution.We use the Markov chain Monte Carlo method to constrain the free parameters of three models,namely,Lambda-Cold-Dark matter(ΛCDM),VGCG and VCG models with and without matter creation from the latest observational data from baryon acoustic oscillations,cosmic chronometer,type Ia supernovae(Pantheon)including gamma-ray bursts,quasars and the local measurement of H_(0) from R21 data.Two different combinations of dataset provide a fairly tight constraint on the parameters of theΛCDM,VGCG and VCG models.The present values of various cosmological parameters are obtained,which are very close to theΛCDM model.Furthermore,we perform stability analysis,Bayesian evidence analysis and information criteria analysis for these models through studying the sound speed,Bayes factor,and Akaike information criteria(AIC)and Bayesian information criteria(BIC)selection criteria.The values of sound speed for VGCG and VCG models shows that both the models are stable.According to AIC,it is observed that VGCG and VCG models with matter creation are supported considerably less by current observations,while BIC shows that these models are not favoured by observational data.

Riemann boundary value problems and reflection of shock for the Chaplygin gas

In this paper,we study two-dimensional Riemann boundary value problems of Euler system for the isentropic and irrotational Chaplygin gas with initial data being two constant states given in two sectors respectively,where one sector is a quadrant and the other one has an acute vertex angle.We prove that the Riemann boundary value problem admits a global self-similar solution,if either the initial states are close,or the smaller sector is also near a quadrant.Our result can be applied to solving the problem of shock reflection by a ramp.

Supersonic flow of a Chaplygin gas past a delta wing

We consider the problem of supersonic flow of a Chaplygin gas past a delta wing with a shock or a rarefaction wave attached to the leading edges.The flow under study is described by the three-dimensional steady Euler system.In conical coordinates,this problem can be reformulated as a boundary value problem for a nonlinear equation of mixed type.The type of this equation depends fully on the solutions of the problem itself,and thus it cannot be determined in advance.We overcome the difficulty by establishing a crucial Lipschitz estimate,and finally prove the unique existence of the solution via the method of continuity.

The Riemann Problem for Chaplygin Gas Flow in a Duct with Discontinuous Cross-Section

The fluid flows in a variable cross-section duct are nonconservative because of the source term.Recently,the Riemann problem and the interactions of the elementary waves for the compressible isentropic gas in a variable cross-section duct were studied.In this paper,the Riemann problem for Chaplygin gas flow in a duct with discontinuous cross-section is studied.The elementary waves include rarefaction waves,shock waves,delta waves and stationary waves.

Warm Standard Scalar Field Modified Chaplygin Gas Inflation Inspired by Generalized Dissipative Coefficient on the Brane

We discuss the warm inflation in the presence of standard scalar field model and modified Chaplyggin gas in brahe-world scenario. We consider weak and strong dissipative regimes with generalized dissipative coefficient. We extract various inflationary parameters. For example, we analyze the behavior of different ratios （ratio of dissipative co-efficient and Hubble parameter Г/3H, ratio of temperature and Hubble parameter T/ H, scalar-to-tensor ratio ＇r） with respect to spectral index ns for the weak and strong dissipative regimes through parametric plotting. It is found that T/H and Г/3H satisfied the required conditions in both dissipative regimes. It is also noted that the spectral index （ns） ns=0.96＋0.10-0.10 It is remarked here that our results are consistence with observational data WMAP7, WMAP9, and recent Planck data.

Comparison of Riemann Solutions for Non-isentropic Modified and Pure Chaplygin Gas Dynamics

In this paper,we investigate the Riemann solutions of the non-isentropic Euler equations for the modified Chaplygin gas and the pure Chaplygin gas,which are the set of all rarefaction waves,shock waves,contact discontinuity andδ-shock waves.Under some appropriate conditions,by studing the limiting behavior,we find that the Riemann solutions of modified Chaplygin gas is the same as pure Chaplygin gas including theδ-shock waves.

On the Irrotational Approximation to Adiabatic Chaplygin Gas Dynamic System

In this paper we mainly study the difference of the weak solutions generated by a wave front tracking algorithm for the steady adiabatic Chaplygin gas dynamic system and the steady irrotational system. Under the hypothesis that the initial data are of sufficiently small total variation, we prove that the difference between the solutions to these two systems can be bounded by the cube of the total variation of the initial perturbation.

Anisotropic Inflationary Scenario Via Generalized Chaplygin Gas Model

We investigate generalized chaplygin gas for warm inflationary scenario in the context of locally rotationally symmetric Bianchi type I universe model.We assume two different cases of dissipative coefficient,i.e.,constant as well as function of scalar field.We construct dynamical equations as well as a relationship between scalar and radiation energy densities under slow-roll approximation.We also derive slow-roll parameters,scalar and tensor power spectra,scalar spectral index,tensor to scalar ratio for analyzing inflationary background during high dissipative regime.We also use the WMAP7 data for the discussion of our parameters.

Anisotropic Bianchi Type Ⅰ Cosmological Models with Generalized Chaplygin Gas and Dynamical Gravitational and Cosmological Constants

This paper deals with study of generalized Chaplygin gas model with dynamical gravitational and cosmological constants. In this paper a new set of exact solutions of Einstein field equations for spatially homogeneous and anisotropic Bianchi type Ⅰ space-time have been obtained. The solutions of the Einstein's field equations are obtained by considering(i) the power law relation between Hubble parameter H and scale factor R and(ii) scale factor of the form R =-1/t + t^2, t > 1. The assumptions lead to constant and variable deceleration parameter respectively. The physical and dynamical behaviors of the models have been discussed with the help of graphical representations. Also we have discussed the stability and physical acceptability of solutions for solution type-Ⅰ and solution type-Ⅱ.

Constraints on the unified model of dark matter and dark energy

Using recently observed data： the Constitution dataset of type supernovae Ia （SNIa）, the observational Hubble data （OHD）, the measurement results of baryon acoustic oscillation （BAO） from the Sloan Digital Sky Survey （SDSS） and the Two Degree Field Galaxy Redshift Survey （2dFGRS）, and the current cosmic microwave background （CMB） data from the five-year Wilkinson Microwave Anisotropy Probe （WMAP）, we apply the Markov Chain Monte Carlo method to investigate the observational constraints on the generalized Chaplygin gas （GCG） model as the unification of dark matter and dark energy. For this unified model, the constraints on GCG mixture are discussed by considering the different expressions of current matter density or considering constraints as being independent of the matter quantity Ωm.

Comparing the VGCG model as the unification of dark sectors with observations

Current observations indicate that 95% of the energy density in the universe is the unknown dark component.The dark component is considered composed of two fluids:dark matter and dark energy.Or it is a mixture of these two dark components,i.e.,one can consider it an exotic unknown dark fluid.With this consideration,the variable generalized Chaplygin gas(VGCG)model is studied with not dividing the unknown fluid into dark matter and dark energy parts in this paper.By using the Markov Chain Monte Carlo method,the VGCG model as the unification of dark sectors is constrained,and the constraint results on the VGCG model parameters are,n=0.00057+0.0001+0.0009-0.0006-0.0006,α=0.0015+0.0003+0.0017-0.0015-0.0015and B s=0.778+0.016+0.030-0.016-0.035,obtained by the cosmic microwave background data from the 7-year WMAP full data points,the baryon acoustic oscillation data from Sloan Digital Sky Survey(SDSS)and 2-degree Field Galaxy Redshift(2dFGRS)survey,and the Union2 type Ia supernova data with systematic errors.At last,according to the evolution of deceleration parameter it is shown that an expanded universe from deceleration to acceleration can be obtained in VGCG cosmology.