Almost Global Solutions of Capillary-Gravity Water Waves Equations on the Circle
Almost Global Solutions of Capillary-Gravity Water Waves Equations on the Circle
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DOI：
10.1007/978-3-319-99486-4
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发表时间：
2018-11
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影响因子：
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通讯作者：
M. Berti;Jean-Marc Delort
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文献类型：
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作者：
M. Berti;Jean-Marc Delort
本专著的目的是证明在一维空间中，具有周期的，甚至在空间中，初始数据为小尺寸的λ的毛细管重力水波方程的柯西问题的任何解，在Sobolev空间上几乎是在时间上全局定义的；也就是说，它存在于一个绝对值为ε−N的时间间隔上，对于任何N，只要初始数据足够光滑，重力毛细管参数被排除在零测量的特殊子集之外。与这些方程在实线上已知的许多结果相反，在柯西数据衰减的情况下，人们不能利用线性流的色散特性。相反，我们的方法是基于一个标准形式的程序，以消除那些贡献的索博列夫能量，是较低的均匀度的解决方案。由于水波方程是一个拟线性系统，通常的正规方法在无界变换中将面临众所周知的导数损失问题。在这篇专著中，为了克服这样的困难，在对毛细管重力水波方程进行平行化之后，必须获得能量估计，从而获得解的局部存在性，我们首先对方程进行了几次准微分约简，以获得具有常系数符号的对角线系统，直到平滑余数。然后我们可以从一个正规的过程开始，其中小因子由前面的准微分正则化补偿。水波方程的可逆结构，以及我们甚至在x中寻找解的事实，保证了一个键抵消，它阻止了解的Sobolev范数的增长。
The goal of this monograph is to prove that any solution of the Cauchy problem for the capillary-gravity water waves equations, in one space dimension, with periodic, even in space, initial data of small size ϵ, is almost globally defined in time on Sobolev spaces; ie it exists on a time interval of length of magnitude ϵ− N for any N, as soon as the initial data are smooth enough, and the gravity-capillary parameters are taken outside an exceptional subset of zero measure. In contrast to the many results known for these equations on the real line, with decaying Cauchy data, one cannot make use of dispersive properties of the linear flow. Instead, our method is based on a normal form procedure, in order to eliminate those contributions to the Sobolev energy that are of lower degree of homogeneity in the solution.Since the water waves equations are a quasi-linear system, usual normal form approaches would face the well-known problem of losses of derivatives in the unbounded transformations. In this monograph, to overcome such a difficulty, after a paralinearization of the capillary-gravity water waves equations, necessary to obtain energy estimates, and thus local existence of the solutions, we first perform several paradifferential reductions of the equations to obtain a diagonal system with constant coefficients symbols, up to smoothing remainders. Then we may start with a normal form procedure where the small divisors are compensated by the previous paradifferential regularization. The reversible structure of the water waves equations, and the fact that we look for solutions even in x, guarantees a key cancellation which prevents the growth of the Sobolev norms of the solutions.