论文标题
水波系统的准线性的几何证明
A geometric proof of the Quasi-linearity of the water-waves system
论文作者
论文摘要
在本文的第一部分中，我们证明了与汉堡方程相关的流量与$ \ partial_x | d | d |^{α-1} u $的非本地术语相关的术语，从$ h^s（{\ m mathbb d}）$ c^0（$ c^0（$ c^0）$ h^s（$ h^0（$ c^0）$ h^s（$ c^0（$ c^0）， $ s> \ frac {1} {2}+2 $，$ 0 \leqα<2 $，$ {\ mathbb d} = {\ mathbb r} \ \ \ \ \ \ \ \ \ text {or} \ {\ mathbb t} $。此外，我们表明，从$ h^s（{\ mathbb d}）$的有限集中，流量不能为$ c^1 $。我们将此结果推广到任何维度上的一大批非线性传输局限性方程，特别是包含有或没有表面张力的水波系统的whitham方程和寄生力。目前的结果是最佳的，因为对于$α= 2 $和$ {\ mathbb d} = {\ mathbb t} $，与本杰明·莫诺方程相关的流量是LipsChitz在功能上，$ 0 $均值均值$均值$ h^s_0 $。 在本文的第二部分中，我们应用这种方法来推断水波系统的准线性，这是本文的主要结果。
In the first part of this paper we prove that the flow associated to the Burgers equation with a non local term of the form $\partial_x |D|^{α-1} u$ fails to be uniformly continuous from bounded sets of $H^s({\mathbb D})$ to $C^0([0,T],H^s({\mathbb D}))$ for $T>0$, $s>\frac{1}{2}+2$, $0\leq α<2$, ${\mathbb D}={\mathbb R} \ \text{or} \ {\mathbb T} $. Furthermore we show that the flow cannot be $C^1$ from bounded sets of $H^s({\mathbb D})$ to $C^0([0,T],H^{s-1+(α-1)^+ +ε}({\mathbb D}))$ for $ε>0$. We generalize this result to a large class of nonlinear transport-dispersive equations in any dimension, that in particular contains the Whitham equation and the paralinearization of the water waves system with and without surface tension. The current result is optimal in the sense that for $α=2$ and ${\mathbb D}={\mathbb T}$ the flow associated to the Benjamin-Ono equation is Lipschitz on function with $0$ mean value $H^s_0$. In the second part of this paper we apply this method to deduce the quasi-linearity of the water waves system, which is the main result of this paper.
