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首页> 外文期刊>Journal of Differential Equations >Numerical proof of stability of roll waves in thesmall-amplitude limit for inclined thin film flow
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Numerical proof of stability of roll waves in thesmall-amplitude limit for inclined thin film flow

机译：倾斜薄膜流小振幅极限下滚动波稳定性的数值证明

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摘要

We present a rigorous numerical proof based on interval arithmetic computations categorizing the lin-earized and nonlinear stability of periodic viscous roll waves of the KdV-KS equation modeling weakly unstable flow of a thin fluid film on an incline in the small-amplitude KdV limit. The argument proceeds by verification of a stability condition derived by Bar-Nepomnyashchy and Johnson-Noble-Rodrigues-Zumbrun involving inner products of various elliptic functions arising through the KdV equation. One key point in the analysis is a bootstrap argument balancing the extremely poor sup norm bounds for these func-tions against the extremely good convergence properties for analytic interpolation in order to obtain a fea-sible computation time. Another is the way of handling analytic interpolation in several variables by a two-step process carving up the parameter space into manageable pieces for rigorous evaluation. These and other general aspects of the analysis should serve as blueprints for more general analyses of spectral stability.
机译：我们提供基于间隔算术计算的严格数字证明，该函数对KdV-KS方程的周期性粘性滚动波的线性化和非线性稳定性进行分类，该方程模拟了在小振幅KdV极限内倾斜的薄流体膜的弱不稳定流动。通过验证由Bar-Nepomnyashchy和Johnson-Noble-Rodrigues-Zumbrun推导的涉及KdV方程产生的各种椭圆函数的内积的稳定性条件来进行论证。分析中的一个关键点是一个自举参数，用于平衡这些函数的极差上界与解析插值的极佳收敛性，从而获得可行的计算时间。另一个方法是通过两步过程将参数空间划分为易于管理的部分以进行严格评估，从而处理多个变量中的解析插值。分析的这些和其他一般方面应作为对光谱稳定性进行更一般分析的蓝图。

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