A symplectic non-squeezing theorem for BBM equation

Abstract

We study the initial value problem for the BBM equation: $$\left\{\begin{array}{l} u_t+u_x+uu_x-u_{txx}=0 \qquad x\in \T, t \in \R u(0,x)=u_0(x) \end{array} \right. .$$ We prove that the BBM equation is globaly well-posed on $H^s(\T)$ for $s\geq0$ and a symplectic non-squeezing theorem on $H^{1/2}(\T)$. That is to say the flow-map $u_0 \mapsto u(t)$ that associates to initial data $u_0 \in H^{1/2}(\T)$ the solution $u$ cannot send a ball into a symplectic cylinder of smaller width.