Global stability of solutions with discontinuous initial data containing vacuum states for the isentropic Euler equations

The global stability of Lipschitz continuous solutions with discontinuous initial data is established in a broad class of entropy solutions in containing vacuum states. In particular, the uniqueness of Lipschitz solutions with discontinuous initial data is obtained in the broad class of entropy solutions in .     

Riemann problem with delta initial data for the isentropic relativistic Chaplygin Euler equations

In this paper, we study the Riemann problem with the initial data containing the Dirac delta function for the isentropic relativistic Chaplygin Euler equations. Under suitably generalized Rankine–Hugoniot relation and entropy condition, we constructively obtain the global existence of generalized solutions including delta shock waves that explicitly exhibit four kinds of different structures. Moreover, it can be found that the solutions constructed here are stable for the perturbation of the initial data.     

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.     

Global solutions of the Navier-Stokes equations for compressible flow with density-dependent viscosity and discontinuous initial data

In this paper, we study the evolutions of the interfaces between the gas and the vacuum for viscous one-dimensional isentropic gas motions. We prove the global existence and uniqueness for discontinuous solutions of the Navier-Stokes equations for compressible flow with density-dependent viscosity coefficient. Precisely, the viscosity coefficient μ is proportional to ρ^θ with 0<θ<1. Specifically, we require that the initial density be piecewise smooth with arbitrarily large jump discontinuities, bounded above and below away from zero, in the interior of gas. We show that the discontinuities in the density persist for all time, and give a decay result for the density as t→+∞.     

Global entropy solutions to the relativistic Euler equations for a class of large initial data

We are concerned with global entropy solutions to the relativistic Euler equations for a class of large initial data which involve the interaction of shock waves and rarefaction waves. We first carefully analyze the global behavior of the shock curves, the rarefaction wave curves, and their corresponding inverse curves in the phase plane. Based on these analyses, we use the Glimm scheme to construct global entropy solutions to the relativistic Euler equations for the class of large discontinuous initial data.     

Global entropy solutions to the relativistic Euler equations for a class of large initial data

We are concerned with global entropy solutions to the relativistic Euler equations for a class of large initial data which involve the interaction of shock waves and rarefaction waves. We first carefully analyze the global behavior of the shock curves, the rarefaction wave curves, and their corresponding inverse curves in the phase plane. Based on these analyses, we use the Glimm scheme to construct global entropy solutions to the relativistic Euler equations for the class of large discontinuous initial data.Received: May 23, 2004     

Global existence of smooth solutions to two-dimensional compressible isentropic Euler equations for Chaplygin gases

In this paper we investigate the two-dimensional compressible isentropic Euler equations for Chaplygin gases. Under the assumption that the initial data is close to a constant state and the vorticity of the initial velocity vanishes, we prove the global existence of the smooth solution to the Cauchy problem for twodimensional flow of Chaplygin gases.     

Global classical solution to the three-dimensional isentropic compressible Navier-Stokes equations with general initial data

We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the isentropic compressible Navier-Stokes equations in the three space dimensions with general initial data which could be either vacuum or non-vacuum under the assumption that the viscosity coefficient μ is large enough.     

Global singularity structure of weak solutions to 3-D semilinear dispersive wave equations with discontinuous initial data

We study the global singularity structure of solutions to 3-D semilinear wave equations with discontinuous initial data. More precisely, using Strichartz’ inequality we show that the solutions stay conormal after nonlinear interaction if the Cauchy data are conormal along a circle.     

Relativistic Euler equations for isentropic fluids: Stability of Riemann solutions with large oscillation

We analyze global entropy solutions of the 2 × 2 relativistic Euler equations for isentropic fluids in special relativity and establish the uniqueness of Riemann solutions in the class of entropy solutions in L     BV_loc with arbitrarily large oscillation. The uniqueness result does not require specific reference to any particular method for constructing the entropy solutions. Then the uniqueness of Riemann solutions implies their inviscid time-asymptotic stability under arbitrarily large L¹     L     BV_loc perturbation of the Riemann initial data, provided that the corresponding solutions are in L and have local bounded total variation that allows the linear growth in time. This approach is also extended to deal with the stability of Riemann solutions containing vacuum in the class of entropy solutions in L with arbitrarily large oscillation.     

Relativistic Euler equations for isentropic fluids: Stability of Riemann solutions with large oscillation

We analyze global entropy solutions of the 2 × 2 relativistic Euler equations for isentropic fluids in special relativity and establish the uniqueness of Riemann solutions in the class of entropy solutions in L BV_loc with arbitrarily large oscillation. The uniqueness result does not require specific reference to any particular method for constructing the entropy solutions. Then the uniqueness of Riemann solutions implies their inviscid time-asymptotic stability under arbitrarily large L¹ L BV_loc perturbation of the Riemann initial data, provided that the corresponding solutions are in L and have local bounded total variation that allows the linear growth in time. This approach is also extended to deal with the stability of Riemann solutions containing vacuum in the class of entropy solutions in L with arbitrarily large oscillation.Received: October 21, 2003     

Global axisymmetric classical solutions of full compressible magnetohydrodynamic equations with vacuum free boundary and large initial data

In this paper,the global existence of the classical solution to the vacuum free boundary problem of full compressible magnetohydrodynamic equations with large initial data and axial symmetry is studied.The solutions to the system(1.6)–(1.8) are in the class of radius-dependent solutions,i.e.,independent of the axial variable and the angular variable.In particular,the expanding rate of the moving boundary is obtained.The main difficulty of this problem lies in the strong coupling of the magnetic field,velocity,temperature and the degenerate density near the free boundary.We overcome the obstacle by establishing the lower bound of the temperature by using different Lagrangian coordinates,and deriving the uniform-in-time upper and lower bounds of the Lagrangian deformation variable r;by weighted estimates,and also the uniform-in-time weighted estimates of the higher-order derivatives of solutions by delicate analysis.     

Non-relativistic global limits of entropy solutions to the isentropic relativistic Euler equations

We consider in this paper the relativistic Euler equations in isentropic fluids with the equation of state p = κ²ρ, where κ, the sound speed, is a constant less than the speed of light c. We discuss the convergence of the entropy solutions as c→∞. The analysis is based on the geometric properties of nonlinear wave curves and the Glimm’s method.     

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.     

Riemann problem with Delta initial data for the relativistic Chaplygin Euler equations

In this paper, we study the Riemann problem with the initial data containing the Dirac delta function for the relativistic Chaplygin Euler equations. Under the generalized Rankine-Hugoniot conditions and entropy condition, we constructively obtain the global existence of generalized solutions including delta shock waves 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     

Global existence of solutions for compressible Navier-Stokes equations with vacuum

In this paper, we will investigate the global existence of solutions for the one-dimensional compressible Navier-Stokes equations when the density is in contact with vacuum continuously. More precisely, the viscosity coefficient is assumed to be a power function of density, i.e., μ(ρ)=Aρθ, where A and θ are positive constants. New global existence result is established for 0<θ<1 when the initial density appears vacuum in the interior of the gas, which is the novelty of the presentation.