Convex integration solutions to the transport equation with full dimensional concentration

Modena, Stefano¹ ; Sattig, Gabriel²

¹ Technische Universität Darmstadt, Fachbereich Mathematik, D-64289 Darmstadt, Germany
² Universität Leipzig, Mathematisches Institut,D-04109 Leipzig, Germany

Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 5, pp. 1075-1108.

Résumé

We construct infinitely many incompressible Sobolev vector fields u∈CtWx1,p˜ on the periodic domain Td for which uniqueness of solutions to the transport equation fails in the class of densities ρ∈CtLxp, provided 1/p+1/p˜>1+1/d. The same result applies to the transport-diffusion equation, if, in addition, p′<d.

Détail

DOI : 10.1016/j.anihpc.2020.03.002

Keywords: Transport equation, Continuity equation, Convex integration, Non-uniqueness, Concentrated Mikado

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