We are concerned with entropy solutions of the 2×2 relativistic Euler equations for perfect fluids in special relativity. We establish the uniqueness of Riemann solutions in the class of entropy solutions in L∞⌒BVloc with arbitrarily large oscillation. Our proof for solutions with large oscillation is based on a detailed analysis of global behavior of shock curves in the phase space and on special features of centered rarefaction waves in the physical plane for this system. The uniqueness result does not require specific reference to any particular method for constructingthe entropy solutions. Then the uniqueness of Riemann solutions yields their inviscid large-time stability under arbitrarily large L1⌒L∞⌒BVloc perturbation of the Riemann initial data, as longas the corresponding solutions are in LN and have local bounded total variation that allows the linear growth in time. We also extend our approach to deal with the uniqueness and stability of Riemann solutions containingvacuum in the class of entropy solutions in LN with arbitrarily large oscillation.

本文讨论了两根非线性弹性杆的耦合问题，利用沿特征线积分与能量估计相结合的方法，证明了在该系统一端及两杆之间用某一粘弹性单元连接时，具小初值的初边值问题存在整体经典解。

We prove the uniqueness of Riemann solutions in the class of entropy solutions in L∞ ∩ BVloc with arbitrarily large oscillation for the 3×3 system of Euler equations in gas dynamics. The proof for solutions with large oscillation is based on a detailed analysis of the global behavior of shock curves in the phase space and the singularity of centered rarefaction waves near the center in the physical plane. The uniqueness of Riemann solutions yields their inviscid large-time stability under arbitrarily large L1 ∩ L∞ ∩ BVloc perturbation of the Riemann initial data, as long as the corresponding solutions are in L∞ and have local bounded total variation satisfying a natural condition on its growth with time. No specific reference to any particular method for constructing the entropy solutions is needed. The uniqueness result for Riemann solutions can easily be extended to entropy solutions U(x, t), piecewise Lipschitz in x, for any t>0, with arbitrarily large oscillation.

We consider in this paper the relativistic Euler equations in isentropic °uids with the equation of state p=к2p, whereк, 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.

本文利用整体迭代法讨论了具耗散项的完全非线性波动方程具小初值的柯西问题的经典解的存在性及生命跨度的下界估计。

The global stability of Lipschitz continuous solutions with discontinuous initial data for the relativistic Euler equations is established in a broad class of entropy solutions in L∞ containing vacuum states. As a corollary, the uniqueness of Lipschitz solutions with discontinuous initial data is obtained in the broad class of entropy solutions in L∞.

本文采用整体迭代法证明了带小初值的一阶非线性耗偏微分方程组的Cauchy问题的整体经典解的存在性及指数衰减性质。

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.

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∞ ∩BVloc 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∞ ∩ BVloc 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.

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