论文标题
通用Korteweg-De Vries和Benjamin-Ono方程的长时间动力学
Long time dynamics for generalized Korteweg-de Vries and Benjamin-Ono equations
论文作者
论文摘要
我们提供了对一个维圆环（没有外部参数的无需外部参数）的广义korteweg-de vries（GKDV）和本杰明·诺（Benjamin-ono（GBO）方程）的长时间动力学的准确描述，并且几乎任何（概率和密度和密度）发行了。我们强调，这两个方程式具有无限的非线性。特别是，我们证明了Sobolev Norm的长期稳定性结果：给定较大的常数r和足够小的参数$ε$，对于通用的初始基准u（0）尺寸$ $ε$的u（0），我们控制解决方案u（t）的sobolev norm of sobolev u（t）的订单$ε$^{ - r}。这些结果是通过将系统以合理的正常形式放置而获得的：我们共轭，最多可以在一定的高阶剩余条件下，这些方程的向量字段置于可在高索博莱夫规律性的围绕原点的大开放式上的可集成的矢量字段。
We provide an accurate description of the long time dynamics of the solutions of the generalized Korteweg-De Vries (gKdV) and Benjamin-Ono (gBO) equations on the one dimension torus, without external parameters, and that are issued from almost any (in probability and in density) small and smooth initial data. We stress out that these two equations have unbounded nonlinearities. In particular, we prove a long-time stability result in Sobolev norm: given a large constant r and a sufficiently small parameter $ε$, for generic initial datum u(0) of size $ε$, we control the Sobolev norm of the solution u(t) for times of order $ε$^{--r}. These results are obtained by putting the system in rational normal form : we conjugate, up to some high order remainder terms, the vector fields of these equations to integrable ones on large open sets surrounding the origin in high Sobolev regularity.
