多维ZND爆轰波的高频稳定性
High-frequency stability of multidimensional ZND detonations
摘要：
Erpenbeck在20世纪60年代开始对强爆轰的光谱稳定性进行严格的研究。通过Zeldovitch-von Neumann-Döring（ZND）模型，他确定了两类基本的爆轰剖面，称为递减（D）型和递增（I）型，它们在高频扰动下表现出非常不同的行为。Erpenbeck综合运用了严格和非严格的论证，得出结论认为，I型爆轰对某些振荡扰动是不稳定的，对于这些振荡扰动，（矢量）频率是任意大的幅度，而D型爆轰只要频率幅度足够高，则是稳定的。对于D型爆轰，Erpenbeck的方法不允许他获得相对于频率方向为\emph{uniform}的稳定性截止幅度。因此，他留下了一个问题，当某些频率方向接近时，稳定性的截止震级是否可能接近$+\infty$。本文用完全不同的方法证明了D型爆轰存在一个与频率方向无关的稳定性一致的截止振幅。通过减少对不稳定频率的搜索到一个有界的频率集，这里得到的均匀截止是一个关键步骤，严格验证的计算爆轰文献中的一些结果。分析中的主要困难是如何处理“无穷大转折点”。
英文摘要：
The rigorous study of spectral stability of strong detonations was begun by Erpenbeck in the 1960s. Working with the Zeldovitch-von Neumann-Döring (ZND) model, he identified two fundamental classes of detonation profiles, referred to as those of decreasing (D) and increasing (I) type, which appeared to exhibit very different behavior with respect to high-frequency perturbations. Using a combination of rigorous and non-rigorous arguments, Erpenbeck concluded that type I detonations were unstable to some oscillatory perturbations for which the (vector) frequency was of arbitrarily large magnitude, while type D detonations were stable provided the frequency magnitude was sufficiently high. For type D detonations Erpenbeck's methods did not allow him to obtain a cutoff magnitude for stability that was \emph{uniform} with respect to frequency direction. Thus, he left open the question whether the cutoff magnitude for stability might approach $+\infty$ as certain frequency directions were approached. In this paper we show by quite different methods that for type D detonations there exists a uniform cutoff magnitude for stability independent of frequency direction. By reducing the search for unstable frequencies to a bounded frequency set, the uniform cutoff obtained here is a key step toward the rigorous validation of a number of results in the computational detonation literature. The main difficulty in the analysis is to treat "turning points at infinity".
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