论文标题
圆圈上的benjamin-ono方程的多项式结合和非线性平滑
Polynomial Bound and Nonlinear Smoothing for the Benjamin-Ono Equation on the Circle
论文作者
论文摘要
For initial data in Sobolev spaces $H^s(\mathbb T)$, $\frac 12 < s \leqslant 1$, the solution to the Cauchy problem for the Benjamin-Ono equation on the circle is shown to grow at most polynomially in time at a rate $(1+t)^{3(s-\frac 12) + ε}$, $0<ε\ll 1$.确定该结果的关键是发现本杰明·荷兰方程的非线性平滑效果，根据该方程，通过某个量规变换满足方程的解决方案，该方程在凯奇问题的良好态度理论中广泛使用，一旦取出了自由解决方案，就变得更加轻松。
For initial data in Sobolev spaces $H^s(\mathbb T)$, $\frac 12 < s \leqslant 1$, the solution to the Cauchy problem for the Benjamin-Ono equation on the circle is shown to grow at most polynomially in time at a rate $(1+t)^{3(s-\frac 12) + ε}$, $0<ε\ll 1$. Key to establishing this result is the discovery of a nonlinear smoothing effect for the Benjamin-Ono equation, according to which the solution to the equation satisfied by a certain gauge transform, which is widely used in the well-posedness theory of the Cauchy problem, becomes smoother once its free solution is removed.
