2025/02/24 by Knezevitch, Alexis · 3 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2502.17094
We consider the 1d quintic nonlinear Schrödinger equation (NLS) on the torus with initial data distributed according to the Gaussian measures with covariance operator (1-Δ)-s, and denoted μs. For the full range s>(9)/(10), we prove that these Gaussian measures are quasi-invariant along the flow of (NLS), meaning that the law of the solution at any time is absolutely continuous with respect to the initial Gaussian measure. Moreover, the condition s>(9)/(10) corresponds to the threshold where the Sobolev space H(2)/(5)+(\mathbbT) is of μs-full measure (it is of zero μs-measure otherwise). This is the lower regularity Sobolev space where we currently know that (NLS) is globally well-posed, thanks to a work by LI-WU-XU. The present work extends the known threshold s>(3)/(2) for the quasi-invariance down to s>(9)/(10), but we do not obtain here quantitative results on the Radon-Nikodym derivatives. Our approach is based on a work of Sun-Tzvetkov, combining a Poincaré-Dulac normal form reduction with energy estimates. However, our main tool to obtain these energy estimates differs: we use the Boué-Dupuis variational formula instead of Wiener Chaos.