arXiv 1204.43972012-04-19math.APmath-phmath.MPnlin.SI

混合型${p}$ -方程组的光滑解

Smooth solutions for a ${p}$-system of mixed type

Michael, Bialy

摘要：

本文分析了\textit{mixed}型$p$ -系统的光滑解.激励的例子，这是一个2分量减少的本尼时刻链，这似乎是连接到理论的可积系统。我们并不事先假定所讨论的解在双曲区域内。我们的主要结果表明，该系统在$x$中周期的唯一光滑解必然是常数。对于初值问题，我们证明了在$x$中，如果初值是严格双曲的和周期的，则解不能扩展到$[t_0;+\infty)$，必然产生冲击。

英文摘要：

In this note we analyze smooth solutions of a $p$-system of the \textit{mixed} type. Motivating example for this is a 2-components reduction of the Benney moments chain which appears to be connected to theory of integrable systems. We don't assume a-priory that the solutions in question are in the Hyperbolic region. Our main result states that the only smooth solutions of the system which are periodic in $x$ are necessarily constants. As for initial value problem we prove that if the initial data is strictly hyperbolic and periodic in $x$ then the solution can not extend to $[t_0;+\infty)$ and shocks are necessarily created.

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