The authors construct invariant probability measures supported on level sets of the renormalized mass for the defocusing cubic nonlinear Schrodinger equation on the one- and two-dimensional torus.
Quasi-invariance of Gaussian measures for the $3d$ energy critical nonlinear Schr\" odinger equation
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abstract
We consider the $3d$ energy critical nonlinear Schr\" odinger equation with data distributed according to the Gaussian measure with covariance operator $(1-\Delta)^{-s}$, where $\Delta$ is the Laplace operator and $s$ is sufficiently large. We prove that the flow sends full measure sets to full measure sets. We also discuss some simple applications. This extends a previous result by Planchon-Visciglia and the second author from $1d$ to higher dimensions.
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New invariant surface measures for the cubic Schr\"odinger equation
The authors construct invariant probability measures supported on level sets of the renormalized mass for the defocusing cubic nonlinear Schrodinger equation on the one- and two-dimensional torus.