Euler equations for isentropic gas dynamics with general pressure law (original) (raw)

Isothermal Limit of Entropy Solutions of the Euler Equations for Isentropic Gas Dynamics

2022

We are concerned with the isothermal limit of entropy solutions in L, containing the vacuum states, of the Euler equations for isentropic gas dynamics. We prove that the entropy solutions in L of the isentropic Euler equations converge strongly to the corresponding entropy solutions of the isothermal Euler equations, when the adiabatic exponent γ → 1. This is achieved by combining careful entropy analysis and refined kinetic formulation with compensated compactness argument to obtain the required uniform estimates for the limit. The entropy analysis involves careful estimates for the relation between the corresponding entropy pairs for the isentropic and isothermal Euler equations when the adiabatic exponent γ → 1. The kinetic formulation for the entropy solutions of the isentropic Euler equations with the uniformly bounded initial data is refined, so that the total variation of the dissipation measures in the formulation is locally uniformly bounded with respect to γ > 1.

The Incompressible Limit of the Non-Isentropic Euler Equations

Archive for Rational Mechanics and Analysis, 2001

We study the incompressible limit of classical solutions to the compressible Euler equations for non-isentropic fluids in a domain Ω ⊂ R d. We consider the case of general initial data. For a domain Ω, bounded or unbounded, we first prove the existence of classical solutions for a time independent of the small parameter. Then, in the exterior case, we prove that the solutions converge to the solution of the incompressible Euler equations.

Global Ill-Posedness of the Isentropic System of Gas Dynamics

Communications on Pure and Applied Mathematics, 2014

We consider the isentropic compressible Euler system in 2 space dimensions with pressure law p(ρ) = ρ 2 and we show the existence of classical Riemann data, i.e. pure jump discontinuities across a line, for which there are infinitely many admissible bounded weak solutions (bounded away from the void). We also show that some of these Riemann data are generated by a 1-dimensional compression wave: our theorem leads therefore to Lipschitz initial data for which there are infinitely many global bounded admissible weak solutions.

Existence Theory for the Isentropic Euler Equations

Archive for Rational Mechanics and Analysis, 2003

We establish an existence theorem for entropy solutions to the Euler equations modeling isentropic compressible fluids. We develop a new approach for constructing mathematical entropies for the Euler equations, which are singular near the vacuum. In particular, we identify the optimal assumption required on the singular behavior on the pressure law at the vacuum in order to validate the two-term asymptotic expansion of the entropy kernel we proposed earlier. For more general pressure laws, we introduce a new multiple-term expansion based on the Bessel functions with suitable exponents, and we also identify the optimal assumption needed to validate the multiple-term expansion and to establish the existence theory. Our results cover, as a special example, the density-pressure law p(ρ) = κ 1 ρ γ 1 + κ 2 ρ γ 2 where γ 1 , γ 2 ∈ (1, 3) and κ 1 , κ 2 > 0 are arbitrary constants.

THE RIEMANN PROBLEM FOR THE MULTI-PRESSURE EULER SYSTEM

Journal of Hyperbolic Differential Equations, 2005

We prove the existence and uniqueness of the Riemann solutions to the Euler equations closed by N independent constitutive pressure laws. This model stands as a natural asymptotic system for the multi-pressure Navier-Stokes equations in the regime of infinite Reynolds number. Due to the inherent lack of conservation form in the viscous regularization, the limit system exhibits measure-valued source terms concentrated on shock discontinuities. These non-positive bounded measures, called kinetic relations, are known to provide a suitable tool to encode the small-scale sensitivity in the singular limit. Considering N independent polytropic pressure laws, we show that these kinetic relations can be derived by solving a simple algebraic problem which governs the endpoints of the underlying viscous shock profiles, for any given but prescribed ratio of viscosity coefficient in the viscous perturbation. The analysis based on traveling wave solutions allows us to introduce the asymptotic Euler system in the setting of piecewise Lipschitz continuous functions and to study the Riemann problem.

Conservation laws of the one-dimensional isentropic gas dynamics equations in Lagrangian coordinates

MODERN TREATMENT OF SYMMETRIES, DIFFERENTIAL EQUATIONS AND APPLICATIONS (Symmetry 2019)

Isentropic flows of a one-dimensional gas dynamics equations in Lagrangian coordinates are studied in the paper. Equations describing these flows are reduced to the Euler-Lagrange equation. Using group classification and Noether's theorem, conservation laws are obtained. Their analogs in Eulerian coordinates are given.

Vanishing viscosity limit for Riemann solutions to zero-pressure gas dynamics with flux perturbation

Boundary Value Problems, 2018

In this paper, by the viscosity vanishing approach, we consider the Riemann problem for zero-pressure gas dynamics with flux perturbation. The Riemann solutions involve parameterized delta shock wave and constant density state. For the parameterized delta-shock solution, its generalized Rankine-Hugoniot and entropy condition are clarified. While for the constant density solution, the formation of it is rigorously analyzed. Moreover, all of their existence, uniqueness, and stability to reasonable viscous perturbations are shown.

Compressible Euler Equations¶with General Pressure Law

Archive for Rational Mechanics and Analysis, 2000

We study the hyperbolic system of Euler equations for an isentropic, compressible fluid governed by a general pressure law. The existence and regularity of the entropy kernel that generates the family of weak entropies is established by solving a new Euler-Poisson-Darboux equation, which is highly singular when the density of the fluid vanishes. New properties of cancellation of singularities in combinations of the entropy kernel and the associated entropy-flux kernel are found.

On regular solutions of the 333D compressible isentropic Euler-Boltzmann equations with vacuum

Discrete and Continuous Dynamical Systems, 2015

In this paper, we discuss the Cauchy problem for the compressible isentropic Euler-Boltzmann equations with vacuum in radiation hydrodynamics. We establish the existence of unique local regular solutions with vacuum by the theory of quasi-linear symmetric hyperbolic systems and the Minkowski's inequality under some physical assumptions for the radition quantities. Moreover, it is interesting to show the non-global existence of regular solutions caused by the effect of vacuum for polytropic gases with adiabatic exponent 1 < γ ≤ 3 via some observation on the propagation of the radiation field. Compared with [12][16][22], some new initial conditions that will lead to the finite time blow-up for classical solutions have been introduced. These blow-up results tell us that the radiation effect on the fluid is not strong enough to prevent the formation of singularities caused by the appearance of vacuum.