Gibbs Measure Dynamics for the Fractional NLS

Abstract

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

“…In this subsection, we prove Proposition 1.11. The argument is the same as that of Corollary 1.3 in [51] and diverges only when we prove the stability in L p loc (dµ s ). We will include details for the benefit of the reader.…”

“…The application of quasi-invariance of Gaussian measures (more generally, of arbitrary probability measures) to the L 1 -stability statement, as in (1.13), was first noted by Sun and Tzvetkov in [51]. The result of Proposition 1.11 shows that the higher (local) integrability of the Radon-Nikodym derivative with respect to the reference Gaussian measure µ s implies stability locally in L p (dµ s ), for p > 1.…”

“…In the remaining range 1 2 < α ≤ 2 3 , one no longer has deterministic well-posedness in the support of µ α . In [52], Sun and Tzvetkov proved almost sure global well-posedness of FNLS (1.1) with respect to µ α for any α > 31− √ 233 28 ≈ 0.562 and hence the corresponding invariance of ρ α ; see also [51]. These results also imply the quasi-invariance of µ α .…”

“…This global well-posedness result is sharp for the cubic NLS (1.1) (α = 1) and 4NLS (α = 2) as they are ill-posed in negative Sobolev spaces in the sense of non-existence of solutions [25]; see also [5,32,10,45,41,34,46,28]. In the latter case (ii), the cubic FNLS (1.1) is locally well-posed in H σ (T) for σ ≥ 1−α 2 [9,51] and globally well-posed when σ > 10α+1 12 [17]. This latter result is not expected to be sharp.…”

Global well-posedness and long-time behavior of the fractional NLS

“…FNLS with Hartree type nonlinearity was studied in [13]. On the circle, Gibbs measures were constructed in [50] and the dynamics on full measure sets with respect to these measures was studied. The authors of the present paper proved the deterministic global well-posedness below the energy space for FNLS posed on the unit disc by extending the I-method (introduced by Colliander, Keel, Staffilani, Takaoka and Tao [15]) to the fractional context [55].…”