Symplectic nonsqueezing for the cubic nonlinear KleinGordon equation on $\mathbb{T}^3$.
Connections for Women: Dispersive and Stochastic PDE August 19, 2015  August 21, 2015
Location: MSRI: Simons Auditorium
defocusing cubic nonlinear KleinGordon equation
symplectic space
critical exponent case
local uniform existence
global wellposedness
negative results
nonsqueezing
53Dxx  Symplectic geometry, contact geometry [See also 37Jxx, 70Gxx, 70Hxx]
37J05  General theory, relations with symplectic geometry and topology
37Kxx  Infinitedimensional Hamiltonian systems [See also 35Axx, 35Qxx]
37K05  Hamiltonian structures, symmetries, variational principles, conservation laws
14329
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.
14329
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