论文标题
2D中陡峭的水波的四分之一集成性和长期存在
The quartic integrability and long time existence of steep water waves in 2D
论文作者
论文摘要
由于Dyachenko \＆Zakharov \ Cite {ZD}的工作是为了众所周知，对于弱非线性2D无限深度水波，没有3波相互作用，所有4波相互作用系数都消失在非整合谐振词段上。在本文中，我们从不同的角度研究了这种部分整合性。 我们直接在物理空间中构造了一系列能量函数$ \ Mathfrak e_j（t）$，它们在Riemann映射变量中显式，并涉及2D水浪方程的订单$ j $的材料衍生物，因此$ \ frac D {dt} dt} \ mathfrak e_j（t）$ quick is quintic n is quintic quintic。我们表明，如果某种规范的规范以及一个涉及一个空间导数的规范在初始数据的缩放范围之上的规模不超过$ \ varepsilon $，那么2D水浪方程的解决方案的寿命至少是$ o（\ varepsilon^{ - 3}）$的订单$ O（\ varepsilon^} { - 3}）$，以及该解决方案作为常规数据的定期数据。如果数据的缩放不变标准是大小$ \ varepsilon $，那么解决方案的寿命至少是$ o（\ varepsilon^{ - 5/2}）$的订单。我们的长时间存在结果不会对初始界面的斜率和初始速度的大小施加尺寸限制，它们允许界面具有任意的较大陡度和初始速度的任意大小。
It is known since the work of Dyachenko \& Zakharov \cite{zd} that for the weakly nonlinear 2d infinite depth water waves, there are no 3-wave interactions and all of the 4-wave interaction coefficients vanish on the non-trivial resonant manifold. In this paper we study this partial integrability from a different point of view. We construct, directly in the physical space, a sequence of energy functionals $\mathfrak E_j(t)$ which are explicit in the Riemann mapping variable and involve material derivatives of order $j$ of the solutions for the 2d water wave equation, so that $\frac d{dt} \mathfrak E_j(t)$ is quintic or higher order. We show that if some scaling invariant norm, and a norm involving one spatial derivative above the scaling of the initial data are of size no more than $\varepsilon$, then the lifespan of the solution for the 2d water wave equation is at least of order $O(\varepsilon^{-3})$, and the solution remains as regular as the initial data during this time. If only the scaling invariant norm of the data is of size $\varepsilon$, then the lifespan of the solution is at least of order $O(\varepsilon^{-5/2})$. Our long time existence results do not impose size restrictions on the slope of the initial interface and the magnitude of the initial velocity, they allow the interface to have arbitrary large steepnesses and initial velocities to have arbitrary large magnitudes.
