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

We consider the periodic defocusing cubic nonlinear Klein-Gordon equation in three dimensions in the symplectic phase space $H^{\frac{1}{2}}(\bT^3) \times H^{-\frac{1}{2}}(\bT^3)$. 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 will present several non-squeezing results for this equation: a local in time result and a conditional result which states that uniform bounds on the Strichartz norms of solutions for initial data in bounded subsets of the phase space implies global-in-time non-squeezing. As a consequence of the conditional result, we will see that we can conclude non-squeezing for certain subsets of the phase space. In particular, we obtain deterministic 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.

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