非自伴微分算子的不稳定指数计算
On computing the instability index of a non-selfadjoint differential operator associated with coating and rimming flows
摘要：
我们研究的问题，找到某些非自伴四阶微分算子的不稳定性指数出现的线性化的涂层和边缘流，其中一层薄薄的流体涂层水平旋转的圆柱体。主要结果减少了计算的不稳定性指数的三角多项式的有限维空间。该证明使用李雅普诺夫方法将微分算子与一个二次型联系起来，其最大正子空间的维数等于不稳定指数。该二次型由李雅普诺夫方程的解给出，该李雅普诺夫方程在这里采用两个变量的四阶线性偏微分方程的形式。该偏微分方程解的椭圆估计起着关键作用。我们包括一些数字的例子。
英文摘要：
We study the problem of finding the instability index of certain non-selfadjoint fourth order differential operators that appear as linearizations of coating and rimming flows, where a thin layer of fluid coats a horizontal rotating cylinder. The main result reduces the computation of the instability index to a finite-dimensional space of trigonometric polynomials. The proof uses Lyapunov's method to associate the differential operator with a quadratic form, whose maximal positive subspace has dimension equal to the instability index. The quadratic form is given by a solution of Lyapunov's equation, which here takes the form of a fourth order linear PDE in two variables. Elliptic estimates for the solution of this PDE play a key role. We include some numerical examples.
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