BBM方程在Besov空间$B_{2,r}^s\left(ℝ\right)$中的全局适定性

Global Well-Posedness for the BBM Equation in Besov Spaces $B_{2,r}^s\left(ℝ\right)$

摘要: 本文研究了Benjamin-Bona-Mahony (BBM)方程在非齐次Besov空间$B_{2,r}^s\left(ℝ\right)$ 中的全局适定性。首先用了压缩映射原理证明了当$1\le p\le \infty ,1<r\le \infty$ 及$s>\frac{1}{p}$ (或$1\le p\le \infty$ ，$r=1$ 及$s\ge \frac{1}{p}$ )时，BBM方程在$B_{p,r}^s\left(ℝ\right)$ 中局部适定的。接着，用高低频分解技巧及算子半群理论证明了当$1/2<s\le 1$ ，$2\le r<\infty$ 时，BBM方程在$B_{2,r}^s\left(ℝ\right)$ 中全局适定。

Abstract: In this study, we devoted to the global well-posedness for the Benjamin-Bona-Mahony (BBM) equation in the Nonhomogeneous Besov spaces $B_{2,r}^s\left(ℝ\right)$ First, using the contraction mapping principle, it is proved that when $1\le p\le \infty ,1<r\le \infty$ and $s>\frac{1}{p}$ (or $1\le p\le \infty$ , $r=1$ and $s\ge \frac{1}{p}$ ), the BBM is locally well-posed in $B_{p,r}^s\left(ℝ\right)$ (or in $B_{p,1}^s\left(ℝ\right)$ ). Then using Bourgain’s low-high frequency decomposition technique, it is proved that when $\frac{1}{2}<s\le 1$ and $2\le r<\infty$ , BBM is globally well-posed in Besov spaces $B_{2,r}^s\left(ℝ\right)$ .