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 concentration-compactness approach, we show that for $1<s_c<3/2$, any solution that remains bounded in the critical Sobolev space ${\dot{H}}_x^{s_c}({\mathbb{R}}^4)$ must be global and scatter. Key ingredients in the proof include a long-time Strichartz estimate and a frequency-localized interaction Morawetz inequality.

52 pages

References in corpus (5)

• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation
• Global well-posedness and scattering for the defocusing, $L^2$-critical, nonlinear Schr{ö}dinger equation when $d=2$
• The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions
• Global Well-posedness and Scattering of Defocusing Energy subcritical Nonlinear Wave Equation in dimension 3 with radial data
• Inter-critical NLS: critical ${\dot{H}}^s$-bounds imply scattering

Cited by in corpus (5)

• The radial defocusing energy-supercritical NLS in dimension four
• The radial defocusing nonlinear Schrödinger equation in three space dimensions
• Global well-posedness and scattering for the defocusing ${\dot{H}}^{\frac{1}{2}}$-critical nonlinear Schrödinger equation in ${\mathbb{R}}^2$
• The interctitical defocusing nonlinear Schrödinger equations with radial initial data in dimensions four and higher
• Scattering theory for energy-supercritical Klein-Gordon equation