On the blow up phenomenon of the critical nonlinear Schrödinger equation

By Sahbi Keraani. 2006

150 citations recorded by the source corpus.

Abstract In this paper we consider the blow up phenomenon of critical nonlinear Schrodinger equations in dimension 1D and 2D. We define the minimal mass as the L 2 norm necessary to ignite a wave collapse and we stress its role in the blow up mechanism. Asymptotic compactness properties and L 2 -concentration are proved. The proof relies on linear and nonlinear profile decompositions.

Subfields: Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c30ec2a9718b01086631e6ba2a600db1c14d0d42

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In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions (2007, background): “, [33], [7], [6], [2], [20], [21], [5], [18], [1], [29], [32]).”
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The mass-critical nonlinear Schr\"odinger equation with radial data in dimensions three and higher (2007, methodology): “Building on earlier developments in [1, 4, 24, 25, 30], Kenig and Merle [23] used this method to treat the energycritical focusing problem with radial data in dimensions three, four, and five.”
The focusing energy-critical nonlinear Schr\"odinger equation in dimensions five and higher (2008, methodology): “[21] S. Keraani, On the defect of compactness for the Strichartz estimates for the Schrödinger equations, J. Diff.”
An alternative approach to regularity for the Navier–Stokes equations in critical spaces (2009, background): “Introduction In recent studies, the idea of establishing the existence of so-called “critical elements” (or the earlier “minimal blow-up solutions”) has led to significant progress in the theory of “critical” dispersive and hyperbolic equations such as the energy-critical nonlinear Schrödinger equat”
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Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1 (2010, background): “The argument in [55] is almost identical when d = 1.”
Quintic NLS in the exterior of a strictly convex obstacle (2012, background): “MR2257393 [42] S. Keraani, On the defect of compactness for the Strichartz estimates for the Schrödinger equations.”