We consider the Modified Kuramoto–Sivashinky Equation (MKSE) in one and two space dimensions and we obtain explicit and accurate estimates of various Sobolev norms of the solutions. In particular, by using the sharp constants which appear in the functional interpolation inequalities used in the analysis of partial differential equations, we evaluate explicitly the sup-norm of the solutions of the MKSE. Furthermore we introduce and then compute the so-called crest factor associated with the above solutions. The crest factor provides information on the distortion of the solution away from its space average and therefore, if it is large, gives evidence of strong turbulence. Here we find that the time average of the crest factor scales like \(\lambda ^{(2d-1)/8}\) for \(\lambda \) large, where \(\lambda \) is the bifurcation parameter of the source term and \(d=1,2\) is the space dimension. This shows that strong turbulence cannot be attained unless the bifurcation parameter is large enough.

中文翻译：

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一维和二维空间中修正的Kuramoto-Sivashinky方程解的Sup-norm和波峰因数的显式估计

我们考虑在一维和二维空间中的修正的Kuramoto-Sivashinky方程（MKSE），并且获得了溶液的各种Sobolev范数的明确而准确的估计。特别是，通过使用偏微分方程分析中使用的函数插值不等式中出现的尖锐常数，我们可以明确地评估MKSE解的超范数。此外，我们介绍并计算与上述解决方案相关的所谓波峰因数。波峰因数提供了关于溶液偏离其空间平均值的变形的信息，因此，如果溶液变形很大，则可以提供强烈湍流的证据。在这里，我们发现，时间平均波峰因数鳞片状\（\拉姆达^ {（2D-1）/ 8} \）为\（\拉姆达\）大，其中\（\ lambda \）是源项的分叉参数，\（d = 1,2 \）是空间尺寸。这表明除非分叉参数足够大，否则无法获得强湍流。