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Sharp quasi-invariance threshold for the cubic Szeg\H{o} equation

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arXiv:2404.14950v1 Announce Type: new
Abstract: We consider the 1-dimensional cubic Szeg\H{o} equation with data distributed according to the Gaussian measure with inverse covariance operator $(1-\partial_x^2)^\frac s2$, where $s>\frac12$. We show that, for $s>1$, this measure is quasi-invariant under the flow of the equation, while for $s<1$, $s\neq \frac34$, 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.

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