Narrow Results

We study cosmological scenarios of the polytropic gas (PG) dark matter-energy proposal in a Friedmann–Robertson–Walker (FRW) Universe. As a first step, we obtain an exact expression for the energy density of PG model by making use of the thermodynamics. Later, we investigate some cosmological quantities and perform neo-classical analyzes. Finally, we implement a connection between the PG and a homogenous minimally coupled scalar field by introducing its self-interacting potential.

Recent astrophysical datasets have implied that the universe has entered a speedy expansion phase. The Polytropic gas model, which describes a unified formulation of dark contents (matter plus energy), is one of the most reasonable definitions of this mysterious phenomenon. This interesting formulation allows to simulate the dark contents in the cosmic form of the perfect fluid and gives an interesting point of view in the discussion of fundamental theories of physics. In the first step of our investigation, we discuss the thermal equation-of-state (EoS henceforth) and obtain the EoS and deceleration parameters as explicit functions of temperature. Subsequently, we obtain a relation for the thermal efficiency of the Carnot heat engine which depends on free parameters given in the cosmological Polytropic gas description and the limits of maximal and minimal temperatures imposed on the Carnot cycle.

We focus on the thermodynamic behavior of Polytropic gas as a candidate for dark energy (DE). We use the general arguments of thermodynamics to investigate its properties and behavior. We find that a Polytropic gas can exhibit the DE-like behavior. It also may be used to simulate a fluid with zero pressure at the small volume and high temperature limits. Briefly, our study shows that this gas may be used to describe the universe expansion history from the matter dominated era to the current accelerating era. By applying some initial conditions to the system, we can establish a relation between the Polytropic gas parameters and the initial conditions. Relationships with related works have also been addressed.

We critically analyze the models of a main-sequence star based on polytropic gases. First, we put in evidence that the hypothesis of polytropic gas is compatible with the constitutive equation of an ideal gas if the transformations inside the star are adiabatic. Then, neither the mono- or the polyphasic models of an ideal gas can be applied inside the stellar core for the existence in this region of nuclear reactions. Moreover, we prove that polyphasic models doesn’t allow the existence of C¹ solutions of the stationary hydrodynamic equations.

We study the subsonic flows governed by full Euler equations in the half plane bounded below by a piecewise smooth curve asymptotically approaching the x₁-axis. Nonconstant conditions in the far field are prescribed to ensure the real Euler flows. The Euler system is reduced to a single elliptic equation for the stream function. The existence, uniqueness, and asymptotic behaviors of the solutions for the reduced equation are established by the Schauder fixed point argument and some delicate estimates. The existence of subsonic flows for the original Euler system is proved based on the results for the reduced equation, and their asymptotic behaviors in the far field are also obtained.

We study the global existence of spatially periodic solutions for certain models of gas flow in Lagrangian coordinates for which the pressure has the form , where v, as usual, is the specific volume, and , are smooth functions of the variable coefficient , which is assumed to satisfy suitable smoothness and decay properties, in particular, , uniformly, as t →0. One important feature of our analysis is that the initial total variation over one period may be taken as large as we wish as long as is sufficiently close to 1. We also prove a non-homogeneous entropy inequality which implies the decay of the solution to the mean value as t → ∞.