Well-posedness and ill-posedness for the cubic fractional Schrödinger equations

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

“…(ii) For every R > 0, there exists T (R) > 0 and C(R) > 0 such that Proof. The uniform growth bounds (i) and (ii) follow from the local well-posedness argument in [9]. The approximation property (iii) follows from a modification of the contraction argument as in [41,Lemma 6.20] by using the nonlinear estimate (the trilinear estimate) in [9, Proposition 4.1] and the following uniform estimate which plays the same role as in [41,Lemma 6.19]: there exists N 0 = N 0 (ε, R) ∈ N such that we have…”

“…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.…”

A linearized conservative Galerkin–Legendre spectral method for the strongly coupled nonlinear fractional Schrödinger equations

“…For the well-posedness, global attractor, soliton dynamics and ground states related to the FSE, we refer to Refs. [5][6][7] and the references therein.…”

Global attractor for a one dimensional weakly damped half-wave equation

“…For the one dimensional case and for α ∈ (1, 2), the low regularity well-posedness for the evolution equation 4with cubic nonlinearity was studied in [11] in which the authors prove the local well-posedness in H s for s ≥ 2−α It should be emphasized that in most cases, the topic of local well-posedness for the problem defined by (4) has been considered in the light of [12] in which the authors derive dispersive estimates that generalize time decay and Strichartz estimates through an improvement result relating to the issue of oscillatory integrals.…”