 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We consider the periodic defocusing cubic nonlinear KleinGordon 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 wellposedness nor any uniform control on the local time of existence for arbitrary initial data. We will present several nonsqueezing 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 globalintime nonsqueezing. As a consequence of the conditional result, we will see that we can conclude nonsqueezing for certain subsets of the phase space. In particular, we obtain deterministic small data nonsqueezing 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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