The defocusing energy-supercritical NLS in four space dimensions

Abstract

We consider a class of defocusing energy-supercritical nonlinear Schrödinger equations in four space dimensions. Following a concentrationcompactness approach, we show that for 1 < sc < 3/2, any solution that remains bounded in the critical Sobolev spaceḢ sc x (R 4 ) must be global and scatter. Key ingredients in the proof include a long-time Strichartz estimate and a frequency-localized interaction Morawetz inequality.

The defocusing quintic NLS in four space dimensions

“…We prove that any solution to (1.1) that remains bounded in the critical Sobolev space, namelyḢ 3 2 x (R 4 ), must be global and scatter. In [29], we proved the analogous statement for (1.1) with the nonlinearity |u| p u for 2 < p < 4. In this paper, we treat the endpoint p = 4, where the techniques in [29] break down.…”

“…For the case of NLS at 'non-conserved critical regularity', one has no such a priori control; however, the success of the techniques developed to treat the massand energy-critical problems suggests that this should be the only missing ingredient for a proof of global well-posedness and scattering. Indeed, previous works have shown that criticalḢ s x -bounds imply scattering for NLS for a range of dimensions and nonlinearities [19,23,29,30,31,32,42]. In [29], the authors treated the nonlinearity |u| p u for 2 < p < 4 in four space dimensions.…”

“…Indeed, previous works have shown that criticalḢ s x -bounds imply scattering for NLS for a range of dimensions and nonlinearities [19,23,29,30,31,32,42]. In [29], the authors treated the nonlinearity |u| p u for 2 < p < 4 in four space dimensions. In this paper, we address the endpoint p = 4.…”

Scattering theory for the radialH˙12-critical wave equation with a cubic convolution

“…Furthermore, by exploiting the double Duhamel trick, Dodson and Lawrie [8] extend the result in [11] to dimension five. We also refer reader to [24,29,34] for the defocusing energy-supercritical Schrödinger equation.…”

The Intercritical Defocusing Nonlinear Schrödinger Equations with Radial Initial Data in Dimensions Four and Higher

“…For further discussion about Conjecture 1.3, we refer to [18,23,24,11]. Now we are in a position to state our main results.…”