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 we only get local in time quasi-invariance because of a blow-up obstruction. Our results extend as well to generic odd power nonlinearities.Again by integration by parts we getwhere we have used the property ∂ 4k−5x J ∂ x J ∂ x J ∈ L 2k,k 0 for a suitable k 0 , whose proof follows again by looking at the proof of (64). Hence by iteration of the previous argument we getwhere at the last step we have used (92). Summarizing we get J 0 ≡ −J 0 which implies J 0 ≡ 0.

On the transport of Gaussian measures under the one-dimensional fractional nonlinear Schrödinger equations

“…This should be contrasted with the local-in-time quasi-invariance result in [38,Theorem 1.5] for the focusing quintic NLS on T. In that setting, a global flow does not exist in view of the presence of finite-time blow-up solutions (see for example [29]). Thus, it is impossible to remove the 'local-in-time' restriction.…”

“…• Method 3: Introduced by Planchon, Tzvetkov and Visciglia [38], where they studied the quasi-invariance of Gaussian measures under the flow of the (super-)quintic NLS on T, the third approach is similar in spirit to Method 2. The fundamental feature of this method is the use of deterministic growth bounds on the H s− 1 2 −ε -norm of solutions (see Proposition 4.6), so that the analysis can be restricted to a closed ball B R ⊂ H s− 1 2 −ε (T).…”

“…It can be viewed as a replacement of the explicit factorisations available for the phase function (1.16) of NLS φ(n) = n 2 1 − n 2 2 + n 2 3 − n 2 = −2(n − n 1 )(n − n 3 ) when n = n 1 − n 2 + n 3 , and 4NLS (see [33,Lemma 3.1]). Following the argument in [38], we obtain quasi-invariance of Gaussian measures µ s under the flow of ( Our next goal is to attempt to lower the regularity restriction from Method 3 by using Method 4. This requires us to construct a suitable weighted Gaussian measure (Subsection 5.2) and establish an effective energy estimate of the form (1.15).…”

“…Methodology and discussion. Theorem 1.2 is an addition to a recent program initiated by Tzvetkov [43] based on understanding the role dispersion has on the transport properties of Gaussian measures under the flow of nonlinear Hamiltonian PDEs; see also [33,35,31,32,38,19]. Within the context of abstract Wiener space, the classical work of Ramer [39] (see also [7]) studied the quasi-invariance of Gaussian measures under general nonlinear transformations.…”

“…Our proof of Theorem 1.2 and Theorem 1.3 is split into two parts. In the first, we employ the recent argument in [38] (see Method 3 below) to obtain quasi-invariance for all α > 1 2 , for some range of regularities s. We then apply the argument in [19] (see Method 4 below) to improve upon the previous regularity restriction for α > 5 6 . We now go over the recent developments in the study of quasi-invariance of Gaussian measures under the flow of nonlinear Hamiltonian PDEs.…”