论文标题
$ h^{s}（\ mathbb {t}，\ mathbb {r}）$ and and的定性特性
Sharp well-posedness results of the Benjamin-Ono equation in $H^{s}(\mathbb{T},\mathbb{R})$ and qualitative properties of its solution
论文作者
论文摘要
我们证明，在sobolev space $ h^{s}（\ mathbb {t}，\ mathbb {r}）$中，benjamin-圆环上的方程式在全球范围内均得到良好的序列。因此，Benjamin的关键Sobolev指数$ S_C = -1/2 $ - 方程是圆环上适当的阈值。获得的解决方案几乎是周期性的。此外，我们证明，圆环上的本杰明·诺克方程的波动解决方案在$ h^{s}（\ mathbb {t}，\ mathbb {r}）$中呈轨道稳定。新颖的保护定律和$ h^{s}上的非线性傅立叶变换（\ mathbb {t}，\ mathbb {r}）$，$ s> -1/2 $是这些结果证明的关键成分。
We prove that the Benjamin--Ono equation on the torus is globally in time well-posed in the Sobolev space $H^{s}(\mathbb{T},\mathbb{R})$ for any $s > - 1/2$ and ill-posed for $s \le - 1/2$. Hence the critical Sobolev exponent $s_c=-1/2$ of the Benjamin--Ono equation is the threshold for well-posedness on the torus. The obtained solutions are almost periodic in time. Furthermore, we prove that the traveling wave solutions of the Benjamin-Ono equation on the torus are orbitally stable in $H^{s}(\mathbb{T},\mathbb{R})$ for any $ s > - 1/2$. Novel conservation laws and a nonlinear Fourier transform on $H^{s}(\mathbb{T},\mathbb{R})$ with $s > - 1/2$ are key ingredients into the proofs of these results.
