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Finite-Time Blow-up of Solutions of an Aggregation Equation in R n

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Abstract

We consider the aggregation equation \(u_t + \nabla \cdot(u \nabla K\,*\,u) = 0\) in R n, n ≥ 2, where K is a rotationally symmetric, nonnegative decaying kernel with a Lipschitz point at the origin, e.g. K(x) =  e −|x|. We prove finite-time blow-up of solutions from specific smooth initial data, for which the problem is known to have short time existence of smooth solutions.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Los Angeles, 90095, USA

    Andrea L. Bertozzi

  2. Department of Mathematics, Duke University, Durham, NC, 27708, USA

    Andrea L. Bertozzi & Thomas Laurent

Authors
  1. Andrea L. Bertozzi
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  2. Thomas Laurent
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Correspondence to Andrea L. Bertozzi.

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Communicated by P. Constantin

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Bertozzi, A.L., Laurent, T. Finite-Time Blow-up of Solutions of an Aggregation Equation in R n . Commun. Math. Phys. 274, 717–735 (2007). https://doi.org/10.1007/s00220-007-0288-1

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  • DOI: https://doi.org/10.1007/s00220-007-0288-1

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