Invariant measures for the periodic derivative nonlinear Schrödinger equation

Abstract

We construct invariant measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation (DNLS) for small data in L 2 and we show these measures to be absolutely continuous with respect to the Gaussian measure. The key ingredient of the proof is the analysis of the gauge group of transformations associated to DNLS. As an intermediate step for our main result, we prove quasi-invariance with respect to the gauge maps of the Gaussian measure on L 2 with covariance (I + (−∆) k ) −1 for any k 2.

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

“…In the next section we prove that Theorem 1.4 implies Theorem 1.3 by a generic argument. Let us observe that estimates (8) and (9) imply the bound…”

“…We believe that (8) can be improved to a tame estimate but this is not of importance for our purposes.…”

“…Estimates (8) and (9) show that the quintic NLS enjoys a new form of one derivative smoothing with respect to the nonlinearity. Such a one derivative smoothing is of course well-known for the nonlinear wave equation NLW and it may easily be seen when we write NLW as a first order system.…”

Quasi-invariance of low regularity Gaussian measures under the gauge map of the periodic derivative NLS

“…The restriction of the measure to a ball B(R) of L 2 is possible as G α leaves invariant the L 2 (T) norm for all α. It is worthy to remark that, unlike all the other works on the subject [31,23,11,12,3], we are not imposing any smallness assumption on R. This observation may be useful in the attempt of proving probabilistc global well-posedness for DNLS without imposing any smallness assumption on the L 2 norm. Remarkable results in this direction are [20,17] and [1] where the authors prove that DNLS is globally well-posed on the real line R in weighted and in translation invariant Sobolev spaces, respectively.…”

“…This gauge was introduced in the periodic setting in [16] in the context of the derivative nonlinear Schödinger equation (DNLS). It has been conveniently used in different contexts regarding the DNLS: just to mention few examples, the study of the local well-posedness at low regularity is based on the use of such a gauge transformation [16,15,8] and it revealed to be crucial also in the proof of the invariance of the Gibbs measures associated with the integrals of motions of DNLS [22,12].…”