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{∂tu+(−1)j+1∂x2j+1u+12∂x(u2)=0,(t,x)∈R×T,u(0,x)=u0(x),u0∈Hs(T), is globally well-posed in Hs for s≥−j2, j≥3. The proof is based on “I-method” introduced by Colliander et al. [4]. We also prove the nonsqueezing property of the periodic higher-order KdV-type equation. The proof relies on Gromov's nonsqueezing theorem for the finite dimensional Hamiltonian system and an approximation argument for the solution flow. More precisely, after taking the frequency truncation to the solution flow, we apply the nonsqueezing theorem. By using the approximation argument, we extend this result to the infinite dimensional system. This argument was introduced by Kuksin [14] and made concretely by Bourgain [2] for the 1D cubic NLS flow, and Colliander et al. [5] for the KdV flow. One of our observations is that the higher-order KdV-type equation has the better modulation effect from the non-resonant interaction than that the KdV equation has. Hence, unlike the work of Colliander et al. [5], we can get the nonsqueezing property for the solution flow without the Miura transform.