Failure of the chain rule for the divergence of bounded vector fields

• Gianluca Crippa ^[1] ; Nikolay Gusev ^[2] ; Stefano Spirito ^[3] ; Emil Wiedemann ^[4]
  1. [1] University of Basel

     University of Basel

     Basilea, Suiza

     
  2. [2] Steklov Mathematical Institute

     Steklov Mathematical Institute

     Rusia

     
  3. [3] University of L'Aquila

     University of L'Aquila

     L'Aquila, Italia

     
  4. [4] Universität Hannover, Germany

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• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 17, Nº 1, 2017, págs. 1-18
• Idioma: inglés
• DOI: 10.2422/2036-2145.201506_006
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• Resumen

  • We provide a vast class of counterexamples to the chain rule for the divergence of bounded vector fields in three space dimensions. Our convex integration approach allows us to produce renormalization defects of various kinds, which in a sense quantify the breakdown of the chain rule. For instance, we can construct defects which are absolutely continuous with respect to the Lebesgue measure, or defects which are not even measures.