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Smooth approximation is not a selection principle for the transport equation with rough vector field

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Abstract

In this paper we analyse the selection problem for weak solutions of the transport equation with rough vector field. We answer in the negative the question whether solutions of the equation with a regularized vector field converge to a unique limit, which would be the selected solution of the limit problem. To this aim, we give a new example of a vector field which admits infinitely many flows. Then we construct a smooth approximating sequence of the vector field for which the corresponding solutions have subsequences converging to different solutions of the limit equation.

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Acknowledgements

This research has been supported by the ERC Starting Grant 676675 FLIRT. We thank the anonymous referee and Làszlò Szèkelyhidi for the careful reading of the paper and for several useful comments.

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Authors and Affiliations

  1. GSSI - Gran Sasso Science Institute, Viale Francesco Crispi 7, 67100, L’Aquila, Italy

    Gennaro Ciampa

  2. Department Mathematik und Informatik, Universität Basel, Spiegelgasse 1, 4051, Basel, Switzerland

    Gennaro Ciampa & Gianluca Crippa

  3. DISIM - Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila, Via Vetoio, 67100, L’Aquila, Italy

    Stefano Spirito

Authors
  1. Gennaro Ciampa
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  2. Gianluca Crippa
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  3. Stefano Spirito
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Corresponding author

Correspondence to Gianluca Crippa.

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Communicated by C. De Lellis.

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Ciampa, G., Crippa, G. & Spirito, S. Smooth approximation is not a selection principle for the transport equation with rough vector field. Calc. Var. 59, 13 (2020). https://doi.org/10.1007/s00526-019-1659-0

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