大振荡系数双曲边值问题：多重放大
Hyperbolic boundary problems with large oscillatory coefficients: multiple amplification
摘要：
我们研究弱稳定的双曲边界问题的高振荡系数是大的，$O(1)$，相比之下，小波长的振荡$\eps$。例如，在研究有关马赫数杆和可压缩涡面稳定性的经典问题时，就出现了这样的问题。对于这样的应用，人们试图证明能量估计是在一个适当的意义上“均匀”相对于小波长$\eps$，但大的振荡系数是一个可怕的障碍，以获得这样的估计。在本文中，我们分析了一个简化形式的线性化问题，是有关上述稳定性问题，并获得结果，既积极和消极的。一方面，我们确定有利的结构条件下，它是可能的，以证明一致的估计，然后这样做的一个新的方法。我们还构造的例子表明，大的振荡系数可以引起一个瞬时\emph{multiple}放大的振幅的解决方案相对于数据，例如，边界数据的一个给定的振幅$O(1)$可以\emph{immediately}引起一个解决方案的振幅$O(\frac{1}{\eps^K})$，其中$K>1$。\footnote{Examples of first-order amplification, where $K=1$, are well-known \cite{MA,CG}.}我们使用多重放大的例子来确认我们的一致估计的最优性时，有利的结构条件成立。当这些条件不成立时，我们解释了无限阶的多重放大如何排除有用的估计。
英文摘要：
We study weakly stable hyperbolic boundary problems with highly oscillatory coefficients that are large, $O(1)$, compared to the small wavelength $\eps$ of oscillations. Such problems arise, for example, in the study of classical questions concerning the stability of Mach stems and compressible vortex sheets. For such applications one seeks to prove energy estimates that are in an appropriate sense "uniform" with respect to the small wavelength $\eps$, but the large oscillatory coefficients are a formidable obstacle to obtaining such estimates. In this paper we analyze a simplified form of the linearized problems that are relevant to the above stability questions, and obtain results that are both positive and negative. On the one hand we identify favorable structural conditions under which it is possible to prove uniform estimates, and then do so by a new approach. We also construct examples showing that large oscillatory coefficients can give rise to an instantaneous \emph{multiple} amplification of the amplitude of solutions relative to data; for example, boundary data of a given amplitude $O(1)$ can \emph{immediately} give rise to a solution of amplitude $O(\frac{1}{\eps^K})$, where $K>1$.\footnote{Examples of first-order amplification, where $K=1$, are well-known \cite{MA,CG}.} We use the examples of multiple amplification to confirm the optimality of our uniform estimates when the favorable structural conditions hold. When those conditions do not hold, we explain how multiple amplification of infinite order may rule out useful estimates.
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