论文标题
$ h^s（\ mathbb r）$的立方NLS和MKDV的尖锐适应性
Sharp well-posedness for the cubic NLS and mKdV in $H^s(\mathbb R)$
论文作者
论文摘要
我们证明，对于任何规律性$ s> - \ frac12 $，我们在$ h^s（\ mathbb r）$中全球范围均得到了良好的范围。长期以来以$ s \ geq 0 $而闻名，请参见[55]，但以前不以任何$ s <0 $。关键的缩放值$ s = - \ frac12 $在这里必然排除，因为已知瞬时规范通货膨胀会发生[11，40，48]。 我们还（以平行方式）证明了对任何$ s> - \ frac12 $ h^s（\ mathbb r）$ in $ h^s（\ mathbb r）$中的实现和复杂的修改后的korteweg-de vries方程。以前实现的最佳规律性是$ s \ geq \ frac14 $;参见[15，24，33，39]。 为了克服数据到解决方案图的均匀连续性的失败，我们采用了[37]中引入的通勤流的方法。与[37]中的论点形成鲜明对比的是，本文的一种基本要素是证明了两个方程式的局部平滑效果。尽管拟态性具有非扰动性质，但衍生物的增益与潜在线性方程的增益相匹配。为了弥补平滑估计值的局部性质，我们还证明了轨道的紧密度。当地平滑和紧密度的证据都取决于我们发现了一个新的一参参数的强制微观保护定律，该法律在这种较低的规律性下仍然有意义。
We prove that the cubic nonlinear Schrödinger equation (both focusing and defocusing) is globally well-posed in $H^s(\mathbb R)$ for any regularity $s>-\frac12$. Well-posedness has long been known for $s\geq 0$, see [55], but not previously for any $s<0$. The scaling-critical value $s=-\frac12$ is necessarily excluded here, since instantaneous norm inflation is known to occur [11, 40, 48]. We also prove (in a parallel fashion) well-posedness of the real- and complex-valued modified Korteweg-de Vries equations in $H^s(\mathbb R)$ for any $s>-\frac12$. The best regularity achieved previously was $s\geq \frac14$; see [15, 24, 33, 39]. To overcome the failure of uniform continuity of the data-to-solution map, we employ the method of commuting flows introduced in [37]. In stark contrast with our arguments in [37], an essential ingredient in this paper is the demonstration of a local smoothing effect for both equations. Despite the non-perturbative nature of the well-posedness, the gain of derivatives matches that of the underlying linear equation. To compensate for the local nature of the smoothing estimates, we also demonstrate tightness of orbits. The proofs of both local smoothing and tightness rely on our discovery of a new one-parameter family of coercive microscopic conservation laws that remain meaningful at this low regularity.
