A symplectic non-squeezing theorem for BBM equation

David Roumegoux

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arXiv · 1007.1359

A symplectic non-squeezing theorem for BBM equation

David Roumegoux

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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.

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David Roumegoux. 2010-12-10. A symplectic non-squeezing theorem for BBM equation. https://arxiv.org/abs/1007.1359

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