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Standing Ring Blow up Solutions to the N-Dimensional Quintic Nonlinear Schrödinger Equation

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Abstract

We consider the quintic nonlinear Schrödinger equation \({i\partial_tu=-\Delta u-|u|^{4}u}\) in dimension N ≥ 3. This problem is energy critical in dimension N = 3 and energy super critical for N ≥ 4. We prove the existence of a radially symmetric blow up mechanism with L 2 concentration along the unit sphere of \({\mathbb{R}^{N}}\). This singularity formation is moreover stable by smooth and radially symmetric perturbation of the initial data. This result extends the result obtained for N = 2 in [29] and is the first result of description of a singularity formation in the energy supercritical class for (NLS) type problems. Our main tool is the proof of the propagation of regularity outside the blow up sphere in the presence a so-called log-log type singularity.

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Authors and Affiliations

  1. Institut de Mathématiques, Université Paul Sabatier, Toulouse, France

    Pierre Raphaël

  2. Institut de Mathématiques, Université Bordeaux 1, Bordeaux, France

    Jérémie Szeftel

  3. Department of Mathematics, Fine Hall, Princeton University, Princeton, NJ, 08544-1000, USA

    Jérémie Szeftel

Authors
  1. Pierre Raphaël
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  2. Jérémie Szeftel
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Correspondence to Jérémie Szeftel.

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Communicated by P. Constantin

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Raphaël, P., Szeftel, J. Standing Ring Blow up Solutions to the N-Dimensional Quintic Nonlinear Schrödinger Equation. Commun. Math. Phys. 290, 973–996 (2009). https://doi.org/10.1007/s00220-009-0796-2

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  • DOI: https://doi.org/10.1007/s00220-009-0796-2

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