Global well-posedness and Nonsqueezing property for the higher-order KdV-type flow

Abstract

In this paper, we prove that the periodic higher-order KdV-type equation \[\left\{\begin{array}{ll} \partial_t u + (-1)^{j+1} \partial_x^{2j+1}u + \frac12 \partial_x(u^2)=0, \hspace{1em} &(t,x) \in \mathbb{R} \times \mathbb{T}, \\ u(0,x) = u_0(x), &u_0 \in H^s(\mathbb{T}). \end{array} \right.\] is globally well-posed in $H^s$ for $s \ge -\frac{j}{2}$, $j \ge 3$. The proof of the global well-posedness is based on "I-method" introduced by Colliander et al. \cite{CKSTT1}. To apply "I-method", we factorize the resonant functions by using the different ways from Hirayama \cite{Hirayama}. Furthermore, we prove the nonsqueezing property of the periodic higher-order KdV-type equation as well. The proof relies on Gromov's nonsqueezing theorem for the finite dimensional Hamiltonian system and approximation for the solution flow. More precisely, after taking the frequency truncation to the solution flow, we applied the nonsqueezing theorem and then the result is transferred to the infinite dimensional original flow. This argument was introduced by Kuksin \cite{Kuksin:1995ue} and made concretely by Bourgain \cite{Bourgain:1994tr} for 1D cubic NLS flow, and Colliander et. al. \cite{CKSTT3} for the KdV flow. One of our observation is that the higher-order KdV-type equation has the better modulation effect from the non-resonant interaction than KdV equation. Hence, unlike the work of Colliander et. al. \cite{CKSTT3}, we can get the nonsqueezing property for the solution flow without the Miura transform.