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Self-similar solutions for the Schrödinger map equation

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Abstract

We study in this article the equivariant Schrödinger map equation in dimension 2, from the Euclidean plane to the sphere. A family of self-similar solutions is constructed; this provides an example of regularity breakdown for the Schrödinger map. These solutions do not have finite energy, and hence do not fit into the usual framework for solutions. For data of infinite energy but small in some norm, we build up associated global solutions.

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012-1185, USA

    Pierre Germain & Jalal Shatah

  2. School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA, 30332-0160, USA

    Chongchun Zeng

Authors
  1. Pierre Germain
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  2. Jalal Shatah
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  3. Chongchun Zeng
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Correspondence to Pierre Germain.

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Germain, P., Shatah, J. & Zeng, C. Self-similar solutions for the Schrödinger map equation. Math. Z. 264, 697–707 (2010). https://doi.org/10.1007/s00209-009-0492-0

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  • DOI: https://doi.org/10.1007/s00209-009-0492-0

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