Vanishing Pressure Limit Research Articles

From the Highly Compressible Navier–Stokes Equations to Fast Diffusion and Porous Media Equations, Existence of Global Weak Solution for the Quasi-Solutions

We consider the compressible Navier–Stokes equations for viscous and barotropic fluids with density dependent viscosity. The aim is to investigate mathematical properties of solutions of the Navier–Stokes equations using solutions of the pressureless Navier–Stokes equations, that we call quasi solutions. This regime corresponds to the limit of highly compressible flows. In this paper we are interested in proving the announced result in Haspot (Proceedings of the 14th international conference on hyperbolic problems held in Padova, pp 667–674, 2014) concerning the existence of global weak solution for the quasi-solutions, we also observe that for some choice of initial data (irrotationnal) the quasi solutions verify the porous media, the heat equation or the fast diffusion equations in function of the structure of the viscosity coefficients. In particular it implies that it exists classical quasi-solutions in the sense that they are $${C^{\infty}}$$ on $${(0,T)\times \mathbb{R}^{N}}$$ for any $${T > 0}$$ . Finally we show the convergence of the global weak solution of compressible Navier–Stokes equations to the quasi solutions in the case of a vanishing pressure limit process. In particular for highly compressible equations the speed of propagation of the density is quasi finite when the viscosity corresponds to $${\mu(\rho)=\rho^{\alpha}}$$ with $${\alpha > 1}$$ . Furthermore the density is not far from converging asymptotically in time to the Barrenblatt solution of mass the initial density $${\rho_{0}}$$ .

Delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations for modified Chaplygin gas

The phenomena of concentration and cavitation and the formation of δ-shocks and vacuum states in solutions to the isentropic Euler equations for a modified Chaplygin gas are analyzed as the double parameter pressure vanishes. Firstly, the Riemann problem of the isentropic Euler equations for a modified Chaplygin gas is solved analytically. Secondly, it is rigorously shown that, as the pressure vanishes, any two-shock Riemann solution to the isentropic Euler equations for a modified Chaplygin gas tends to a δ-shock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted δ-measure that forms the δ-shock; any two-rarefaction-wave Riemann solution to the isentropic Euler equations for a modified Chaplygin gas tends to a two-contact-discontinuity solution to the transport equations, the nonvacuum intermediate state between the two rarefaction waves tends to a vacuum state. Finally, some numerical results exhibiting the formation of δ-shocks and vacuum states are presented as the pressure decreases.

Vanishing pressure limits of Riemann solutions to the isentropic relativistic Euler system for Chaplygin gas

The Riemann solutions to the isentropic relativistic Euler system for Chaplygin gas with a small parameter are considered. Unlike the polytropic or barotropic gas cases, we find that firstly, as the parameter decreases to a certain critical number, the two-shock solution converges to a delta shock wave solution of the same system. Moreover, as the parameter goes to zero, that is, the pressure vanishes, the solution is nothing but the delta shock wave solution to the zero-pressure relativistic Euler system. Meanwhile, the two-rarefaction wave solution tends to the vacuum solution to the zero-pressure relativistic system, and the solution containing one rarefaction wave and one shock wave tends to the contact discontinuity solution to the zero-pressure relativistic system as pressure vanishes.

Delta wave formation and vacuum state in vanishing pressure limit for system of conservation laws to relativistic fluid dynamics

The Riemann solutions to the system of conservation laws to relativistic fluid dynamics with a scaled pressure are shown. Using the method of vanishing pressure limit, it is rigorously proved that the Riemann solution involving two shocks and possibly one contact discontinuity to the system of relativistic fluid dynamics tends to a delta wave solution of pressureless relativistic Euler equations. The intermediate density between the two shocks tends to a weighted δ-measure which forms the delta wave. While the Riemann solution containing two rarefaction waves and perhaps one contact discontinuity tends to a double-contact-discontinuity solution to pressureless relativistic Euler equations. The intermediate state between the two contact discontinuities is a vacuum state.

Limits of Riemann Solutions to the Relativistic Euler Systems for Chaplygin Gas as Pressure Vanishes

Vanishing pressure limits of Riemann solutions to relativistic Euler system for Chaplygin gas are identified and analyzed in detail. Unlike the polytropic or barotropic gas case, as the parameter decreases to a critical value, the two-shock solution converges firstly to a delta shock wave solution to the same system. It is shown that, as the parameter decreases, the strength of the delta shock increases. Then as the pressure vanishes ultimately, the solution is nothing but the delta shock wave solution to the zero pressure relativistic Euler system. Meanwhile, the two-rarefaction wave solution and the solution containing one-rarefaction wave and one-shock wave tend to the vacuum solution and the contact discontinuity solution to the zero pressure relativistic Euler system, respectively.

Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw–Rascle model

A traffic flow model describing the formation and dynamics of traffic jams was introduced by Berthelin et al., which consists of a constrained pressureless gas dynamics system and can be derived from the Aw–Rascle model under the constraint condition ρ ⩽ ρ ⁎ by letting the traffic pressure vanish. In this paper, we give up this constraint condition and consider the following form { ρ t + ( ρ u ) x = 0 , ( ρ u + ε p ( ρ ) ) t + ( ρ u 2 + ε u p ( ρ ) ) x = 0 , in which p ( ρ ) = ρ γ with γ > 1 . The formal limit of the above system is the pressureless gas dynamics system in which the density develops delta-measure concentration in the Riemann solution. However, the propagation speed and the strength of the delta shock wave in the limit situation are different from the classical results of the pressureless gas dynamics system with the same Riemann initial data. In order to solve it, the perturbed Aw–Rascle model is proposed as { ρ t + ( ρ u ) x = 0 , ( ρ u + ε γ p ( ρ ) ) t + ( ρ u 2 + ε u p ( ρ ) ) x = 0 , whose behavior is different from that of the Aw–Rascle model. It is proved that the limits of the Riemann solutions of the perturbed Aw–Rascle model are exactly those of the pressureless gas dynamics model.

Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for polytropic gases

The Riemann solutions for the Euler system of conservation laws of energy and momentum in special relativity for polytropic gases are considered. It is rigorously proved that, as pressure vanishes, they tend to the two kinds of Riemann solutions to the corresponding pressureless relativistic Euler equations: the one includes a delta shock, which is formed by a weighted δ -measure, and the other involves vacuum state.

Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations

The Riemann problems for the Euler system of conservation laws of energy and momentum in special relativity as pressure vanishes are considered. The Riemann solutions for the pressureless relativistic Euler equations are obtained constructively. There are two kinds of solutions, the one involves delta shock wave and the other involves vacuum. The authors prove that these two kinds of solutions are the limits of the solutions as pressure vanishes in the Euler system of conservation laws of energy and momentum in special relativity.

Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids

Numerical simulations [2-D Riemann problem in gas dynamics and formation of spiral, in: Nonlinear Problems in Engineering and Science—Numerical and Analytical Approach (Beijing, 1991), Science Press, Beijing, 1992, pp. 167–179; Discrete Contin. Dyn. Syst. 1 (1995) 555–584; 6 (2000) 419–430] for the Euler equations for gas dynamics in the regime of small pressure showed that, for one case, the particles seem to be more sticky and tend to concentrate near some shock locations, and for the other case, in the region of rarefaction waves, the particles seem to be far apart and tend to form cavitation in the region. In this paper we identify and analyze the phenomena of concentration and cavitation by studying the vanishing pressure limit of solutions of the full Euler equations for nonisentropic compressible fluids with a scaled pressure. It is rigorously shown that any Riemann solution containing two shocks and possibly one-contact-discontinuity to the Euler equations for nonisentropic fluids tends to a δ-shock solution to the corresponding transport equations, and the intermediate densities between the two shocks tend to a weighted δ-measure that, along with the two shocks and possibly contact-discontinuity, forms the δ-shock as the pressure vanishes. By contrast, it is also shown that any Riemann solution containing two rarefaction waves and possibly one-contact-discontinuity to the Euler equations for nonisentropic fluids tends to a two-contact-discontinuity solution to the transport equations, and the nonvacuum intermediate states between the two rarefaction waves tend to a vacuum state as the pressure vanishes. Some numerical results exhibiting the processes of concentration and cavitation are presented as the pressure decreases.

Formation of $\delta$-Shocks and Vacuum States in the Vanishing Pressure Limit of Solutions to the Euler Equations for Isentropic Fluids

The phenomena of concentration and cavitation and the formation of $\delta$-shocks and vacuum states in solutions to the Euler equations for isentropic fluids are identified and analyzed as the pressure vanishes. It is shown that, as the pressure vanishes, any two-shock Riemann solution to the Euler equations for isentropic fluids tends to a $\delta$-shock solution to the Euler equations for pressureless fluids, and the intermediate density between the two shocks tends to a weighted $\delta$-measure that forms the $\delta$-shock. By contrast, any two-rarefaction-wave Riemann solution of the Euler equations for isentropic fluids is shown to tend to a two-contact-discontinuity solution to the Euler equations for pressureless fluids, whose intermediate state between the two contact discontinuities is a vacuum state, even when the initial data stays away from the vacuum. Some numerical results exhibiting the formation process of $\delta$-shocks are also presented.