Transport of gaussian measures by the flow of the nonlinear Schrödinger equation

Abstract

We prove a new smoothing type property for solutions of the 1d quintic Schrödinger equation. As a consequence, we prove that a family of natural gaussian measures are quasi-invariant under the flow of this equation. In the defocusing case, we prove global in time quasi-invariance while in the focusing case because of a blow-up obstruction we only get local in time quasi-invariance. Our results extend as well to generic odd power nonlinearities.

Presentation improved

References in corpus (3)

• On the growth of Sobolev norms for NLS on $2d$ and $3d$ manifolds
• Quasi-invariant gaussian measures for one dimensional Hamiltonian PDE's
• An Improvement on the Brézis-Gallouët technique for 2D NLS and 1D half-wave equation

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• On the transport of Gaussian measures under the one-dimensional fractional nonlinear Schrödinger equations
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• Quasi-Invariance of Gaussian Measures Transported by the Cubic NLS with Third-Order Dispersion on $\mathbf{T}$
• Quasi-invariance of fractional Gaussian fields nonlinear wave equation with polynomial nonlinearity
• New bounds on the high Sobolev norms of the 1d NLS solutions
• Modified energies for the periodic generalized KdV equation and applications
• Transport of low regularity Gaussian measures for the 1d quintic nonlinear Schrödinger equation
• Sharp quasi-invariance threshold for the cubic Szegő equation