Almost sure global well-posedness for the energy supercritical NLS on the unit ball of $\mathbb{R}^3$

Abstract

In this paper, we present two almost sure global well-posedness (GWP) results for the energy supercritical nonlinear Schrödinger equations (NLS) on the unit ball of R 3 using two different approaches. First, for the NLS with algebraic nonlinearities with the subcritical initial data, we show the almost sure global well-posedness and the invariance of the underlying measures, and establish controls on the growth of Sobolev norms of the solutions. This global result is based on a deterministic local theory and a probabilistic globalization. Second, for the NLS with generic power nonlinearities with critical and supercritical initial conditions, we prove the almost sure global well-posedness, and the invariance of the measure under the solution flows. This global result is built on a compactness argument and the Skorokhod representation theorem.

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

“…The other way around, the analysis is more challenging. In [50] a similar identity was established with the use of an auxiliary estimate on a quantity of type E(u)E (u). But in our context, such an estimate is not available.…”

“…Now, without a control on E(u)E (u), we can only handle the weaker quantity M(u)M (u) from (4.12). So our proof here will be more tricky than that in [50]. Coming back to the proof of the remaining inequality, we write for some fixed frequency F, the frequency decomposition…”

“…However for energy supercritical equations, both approaches come across serious obstructions. The first author initiated a new approach that combines the two methods and applied it to the energy-supercritical NLS [48], this approach was developed further by the authors in [50]. This approach utilizes a fluctuation-dissipation argument on the Galerkin approximations of the equation, we then have a double approximation: a finite-dimensional one and a viscous one.…”

Invariant measures and global well-posedness for a fractional Schrödinger equation with Moser-Trudinger type nonlinearity

“…Sy [22] and Sy and Yu [23] combined these two approaches into what they call Inviscid-Infinite-dimensional limits, and constructed invariant measures for a class of energy-supercritical NLS equations with pure-power nonlinearities. This method consists in using a fluctuation-dissipation argument on N -dimensional Galerkin approximations.…”

Global well-posedness for the cubic fractional NLS on the unit disk

“…In this local theory, as one did in the NLS case, we need a Strichartz-type estimate to run the contraction mapping argument. To this end, we adapt the proof of bilinear estimates for NLS on the unit ball in [1] (see also [45] for the multilinear estimates for NLS on the unit ball) in Section 3. However, it is worth pointing out that due to the fractionality of the dispersion operator, it is impossible to periodize the time in the bilinear estimates and count the integer points on its Fourier characteristic surface.…”