Stability of contact discontinuities to 1-D piston problem for the compressible Euler equations

We consider 1-D piston problem for the compressible Euler equations when the piston is static relatively to the gas in the tube. By a modified wave front tracking method, we prove that a contact discontinuity is structurally stable under the assumptions that the total variation of the initial data and the perturbation of the piston velocity are both sufficiently small. Meanwhile, we study the asymptotic behavior of the solutions by the generalized characteristic method and approximate conservation law theory as $t\to +\infty$.     

Global entropy solutions to exothermically reacting, compressible Euler equations

The global existence of entropy solutions is established for the compressible Euler equations for one-dimensional or plane-wave flow of an ideal gas, which undergoes a one-step exothermic chemical reaction under Arrhenius-type kinetics. We assume that the reaction rate is bounded away from zero and the total variation of the initial data is bounded by a parameter that grows arbitrarily large as the equation of state converges to that of an isothermal gas. The heat released by the reaction causes the spatial total variation of the solution to increase. However, the increase in total variation is proved to be bounded in t>0 as a result of the uniform and exponential decay of the reactant to zero as t approaches infinity.     

Characteristic decompositions and interactions of rarefaction waves of 2-D Euler equations

This paper is concerned with classical solutions to the interaction of two arbitrary planar rarefaction waves for the self-similar Euler equations in two space dimensions. We develop the direct approach, started in Chen and Zheng (in press) [3], to the problem to recover all the properties of the solutions obtained via the hodograph transformation of Li and Zheng (2009) [14]. The direct approach, as opposed to the hodograph transformation, is straightforward and avoids the common difficulties of the hodograph transformation associated with simple waves and boundaries. The approach is made up of various characteristic decompositions of the self-similar Euler equations for the speed of sound and inclination angles of characteristics.     

Stability of boundary layer and rarefaction wave to an outflow problem for compressible Navier-Stokes equations under large perturbation

In this paper, we investigate the large-time behavior of solutions to an outflow problem for compressible Navier-Stokes equations. In 2003, Kawashima, Nishibata and Zhu [S. Kawashima, S. Nishibata, P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys. 240 (2003) 483-500] showed there exists a boundary layer (i.e., stationary solution) to the outflow problem and the boundary layer is nonlinearly stable under small initial perturbation. In the present paper, we show that not only the boundary layer above but also the superposition of a boundary layer and a rarefaction wave are stable under large initial perturbation. The proofs are given by an elementary energy method.     

Stability of rarefaction wave for compressible Navier-Stokes equations on the half line

This paper is concerned with the large-time behavior of solutions to an initial-boundary-value problem for full compressible Navier-Stokes equations on the half line (0,∞), which is named impermeable wall problem. It is shown that the 3-rarefaction wave is stable under partially large initial perturbation if the adiabatic exponent γ is close to 1. Here partially large initial perturbation means that the perturbation of absolute temperature is small, while the perturbations of specific volume and velocity can be large. The proof is given by the elementary energy method.     

Stability of the planar rarefaction wave to two-dimensional Navier-Stokes-Korteweg equations of compressible fluids

This study is concerned with the large time behavior of the two-dimensional compressible Navier-Stokes-Korteweg equations, which are used to model compressible fluids with internal capillarity. Based on the fact that the rarefaction wave, one of the basic wave patterns to the hyperbolic conservation laws is nonlinearly stable to the one-dimensional compressible Navier-Stokes-Korteweg equations, the planar rarefaction wave to the two-dimensional compressible Navier-Stokes-Korteweg equations is first derived. Then, it is shown that the planar rarefaction wave is asymptotically stable in the case that the initial data are suitably small perturbations of the planar rarefaction wave. The proof is based on the delicate energy method. This is the first stability result of the planar rarefaction wave to the multi-dimensional viscous fluids with internal capillarity.     

Global existence of shock front solutions in 1-dimensional piston problem in the relativistic Euler equations

The paper studies the 1-D piston problem of the relativistic Euler equations when the speed of the piston is a perturbation of a constant. A sequence of approximate solutions constructed by a modified Glimm scheme is proved to be convergent to the weak solution (which includes a strong leading shock) to the piston problem. In particular, we give the precise estimates on the reflection of the perturbed waves on the piston and the leading shock.     

Global existence of shock front solutions in 1-dimensional piston problem in the relativistic Euler equations

The paper studies the 1-D piston problem of the relativistic Euler equations when the speed of the piston is a perturbation of a constant. A sequence of approximate solutions constructed by a modified Glimm scheme is proved to be convergent to the weak solution (which includes a strong leading shock) to the piston problem. In particular, we give the precise estimates on the reflection of the perturbed waves on the piston and the leading shock. The paper is supported by the National Natural Science Foundation of China (Grant 10626034) and the Special Research Fund for Selecting Excellent Young Teachers of the Universities in Shanghai.     

Local existence and non-relativistic limits of shock solutions to a multidimensional piston problem for the relativistic Euler equations

The multidimensional piston problem is a special initial-boundary value problem. The boundary conditions are given in two conical surfaces: one is the boundary of the piston, and the other is the shock whose location is to be determined later. In this paper, we are concerned with spherically symmetric piston problem for the relativistic Euler equations. A local shock front solution with the state equation p = a ² ρ, a is a constant and has been established by the Newton iteration. To overcome the difficulty caused by the free boundary, we introduce a coordinate transformation to fix it and employ the linear iteration scheme to establish a sequence of approximate solutions to the auxiliary problems by iteration. In each step, the value of the solution of the previous problem is taken as the data to determine the solution of the next problem. We obtain the existence of the original problem by establishing the convergence of these sequences. Meanwhile, we establish the convergence of the local solution as c → ∞ to the corresponding solution of the classical non-relativistic Euler equations.     

Riemann problem for the relativistic Chaplygin Euler equations

The relativistic Euler equations for a Chaplygin gas are studied. The Riemann problem is solved constructively. There are five kinds of Riemann solutions, in which four only contain different contact discontinuities and the other involves delta shock waves. Under suitable generalized Rankine-Hugoniot relation and entropy condition, the existence and uniqueness of delta-shock solutions are established.     

Stability of contact discontinuities for the nonisentropic Euler equations

We study the stability of contact discontinuities for the nonisentropic Euler equations in two or three space dimensions. A simple criterion predicting neutral stability or violent instability is given.

---

Sunto Si studia la stabilità delle discontinuità di contatto per le equazioni di Eulero non isentropiche in dimensione di spazio 2 e 3. Viene presentato un criterio semplice per la stabilità neutrale e l’instabilità violenta.

     

Zero dissipation limit to rarefaction wave with vacuum for the one-dimensional non-isentropic micropolar equations

The zero dissipation limit of the one-dimensional non-isentropic micropolar equations is studied in this paper. If the given rarefaction wave which connects to vacuum at one side, a sequence of solution to the micropolar equations can be constructed which converge to the above rarefaction wave with vacuum as the viscosity and the heat conduction coefficient tend to zero. Moreover, the uniform convergence rate is obtained. The key point in our analysis is how to control the degeneracies in the vacuum region in the zero dissipation limit process.     

Nonconservative scheme with the isentropic condition in rarefaction waves as applied to the compressible Euler equations

A previously developed second-order accurate quasi-monotone scheme is tested using the Riemann problem with high initial pressure and density ratios. For shock waves, the scheme is conservative, while, in rarefaction waves, the isentropic condition along the trajectory of a Lagrangian particle is used instead of conservativeness in energy. It is shown that the shock front position produced by the scheme has no considerable errors typical of a representative set of conservative quasi-monotone schemes of various orders of accuracy. The numerical accuracy is significantly improved in the case of moving grids with a contact discontinuity explicitly introduced in the form of a grid node. It is shown how the method can be extended to cover the multidimensional case and the presence of additional terms in the original equations.     

Riemann problem for the isentropic relativistic Chaplygin Euler equations

This paper studies the Riemann problem of the isentropic relativistic Euler equations for a Chaplygin gas. The solutions exactly include five kinds. The first four consist of different contact discontinuities while the rest involves delta-shock waves. Under suitable generalized Rankine?CHugoniot relation and entropy condition, the existence and uniqueness of delta-shock solutions are established.     

Well-posedness of the IBVP for 2-D Euler equations with damping

In this paper we focus on the initial-boundary value problem of the 2-D isentropic Euler equations with damping. We prove the global-in-time existence of classical solution to the initial-boundary value problem for small smooth initial data by the method of local existence of solution combined with a priori energy estimates, where the appropriate boundary condition plays an important role.     

Stability of the rarefaction wave for a two-fluid plasma model with diffusion

We study the large-time asymptotics of solutions toward the weak rarefaction wave of the quasineutral Euler system for a two-fluid plasma model in the presence of diffusions of velocity and temperature under small perturbations of initial data and also under an extra assumption θ_i,+/θ_e,+=θ_i,-/θ_e,-≥m_i/2m_e,namely, the ratio of the thermal speeds of ions and electrons at both far fields is not less than one half. Meanwhile,we obtain the global existence of solutions based on energy method.     

Initial-boundary problem for the 1-D Euler-Boltzmann equations in radiation hydrodynamics

We study the initial-boundary value problem for the one dimensional EulerBoltzmann equation with reflection boundary condition. For initial data with small total variation, we use a modified Glimm scheme to construct the global approximate solutions(U_(△t,d), I_(△t,d)) and prove that there is a subsequence of the approximate solutions which is convergent to the global solution.