二维无压气体中激波的非哈密顿性character
On the nonhamiltonian character of shocks in 2-D pressureless gas
摘要：
本文讨论了Ya.Zeldovich于1970年提出的描述宇宙大尺度结构形成的二维无压气体动力学系统。这类方程组是介于常微分方程组和偏微分方程双曲组之间的一个中间对象。它的主要特点是奇异性的出现：速度的不连续性和密度的各种类型的δ函数。引入了关于Radon测度的广义解的严格概念，得到了Rankine-Hugoniot条件的推广.在此基础上，证明了在一维情况下适用的广义解的变分表示在二维情况下一般不成立。一个非平凡的非严格双曲型的1-D系统也得到了描述的冲击内部的演变。
英文摘要：
The paper deals with the 2-D system of gas dynamics without pressure which was introduced in 1970 by Ya.Zeldovich to describe the formation of large-scale structure of the Universe. Such system occurs to be an intermediate object between the systems of ordinary differential equations and hyperbolic systems of PDE. The main its feature is the arising of singularities: discontinuities for velocity and delta functions of various types for density. The rigorous notion of generalized solutions in terms of Radon measures is introduced and the generalization of Rankine-Hugoniot conditions is obtained. On the basis of such conditions it is shown that the variational representation for the generalized solutions, which is valid for 1-D case, in 2-D case generally speaking does not take place. A nontrivial 1-D system of nonstrictly hyperbolic type is also obtained to describe the evolution inside the shock.
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