2024/04/23 by James Coe, Leonardo Tolomeo, Coe, James +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Waves and Solitons #Geometry and complex manifolds
paper · pdf · doi:10.2140/apde.2026.19.1107
We consider the 1-dimensional cubic Szegő equation with data distributed according to the Gaussian measure with inverse covariance operator (1 -∂ 2x ) s , where s > 1 2 .We show that, for s > 1, this measure is quasi-invariant under the flow of the equation, while for s < 1, s ̸ = 3 4 , the transported measure and the initial Gaussian measure are mutually singular for almost every time.This is the first observation of a transition from quasi-invariance to singularity in the context of the transport of Gaussian measures under the flow of Hamiltonian PDEs.