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Finite-Time Singularities of an Aggregation Equation in \({\mathbb {R}^n}\) with Fractional Dissipation

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Abstract

We consider an aggregation equation in \({\mathbb {R}^n}\), n ≥ 2 with fractional dissipation, namely, \({u_t + \nabla\cdot(u \nabla K*u)=-\nu (-\Delta)^{\gamma/2} u}\), where 0 ≤ γ < 1 and K is a nonnegative decreasing radial kernel with a Lipschitz point at the origin, e.g. K(x) = e −|x|. We prove that for a class of smooth initial data, the solutions develop blow-up in finite time.

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Authors and Affiliations

  1. School of Mathematics, Institute For Advanced Studies, Einstein Drive, Princeton, NJ, 08540, USA

    Dong Li

  2. Warwick University, Coventry, CV4 7AL, UK

    Jose Rodrigo

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  1. Dong Li
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  2. Jose Rodrigo
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Correspondence to Jose Rodrigo.

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Communicated by P. Constantin

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Li, D., Rodrigo, J. Finite-Time Singularities of an Aggregation Equation in \({\mathbb {R}^n}\) with Fractional Dissipation. Commun. Math. Phys. 287, 687–703 (2009). https://doi.org/10.1007/s00220-008-0669-0

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  • DOI: https://doi.org/10.1007/s00220-008-0669-0

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