Symplectic non-squeezing for the cubic nonlinear Klein-Gordon equation on ${\mathbb{T}}^3$

Abstract

We consider the periodic defocusing cubic nonlinear Klein-Gordon equation in three dimensions in the symplectic phase space $H^{\frac{1}{2}}({\mathbb{T}}^3)\times H^{-\frac{1}{2}}({\mathbb{T}}^3)$. This space is at the critical regularity for this equation, and in this setting there is no global well-posedness nor any uniform control on the local time of existence for arbitrary initial data. We prove a local-in-time non-squeezing result and a conditional global-in-time result which states that uniform bounds on the Strichartz norms of solutions imply global-in-time non-squeezing. As a consequence of the conditional result, we conclude non-squeezing for certain subsets of the phase space, and in particular, we prove small data non-squeezing for long times. The proofs rely on several approximation results for the flow, which we obtain using a combination of probabilistic and deterministic techniques.

57 pages

References in corpus (3)

• Ill-posedness for nonlinear Schrodinger and wave equations
• Global existence without decay for quadratic Klein-Gordon equations
• Invariant Gibbs Measure for 3D NLW in Infinite Volume

Cited by in corpus (4)

• Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ${\mathbb{R}}^2$
• Symplectic non-squeezing for the cubic NLS on the line
• Periodic fourth-order cubic NLS: Local well-posedness and Non-squeezing property
• Global well-posedness and Nonsqueezing property for the higher-order KdV-type flow