Convex integration solutions to the transport equation with full dimensional concentration

Modena, Stefano; Sattig, Gabriel

doi:10.1016/j.anihpc.2020.03.002

Modena, Stefano ¹ ; Sattig, Gabriel ²

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

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Résumé

We construct infinitely many incompressible Sobolev vector fields $u \in C_{t} W_{x}^{1, \tilde{p}}$ on the periodic domain $T^{d}$ for which uniqueness of solutions to the transport equation fails in the class of densities $ρ \in C_{t} L_{x}^{p}$ , provided $1 / p + 1 / \tilde{p} > 1 + 1 / d$ . The same result applies to the transport-diffusion equation, if, in addition, $p^{'} < d$ .

MR Zbl

DOI : 10.1016/j.anihpc.2020.03.002

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

Affiliations des auteurs :

Modena, Stefano ¹ ; Sattig, Gabriel ²

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

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     author = {Modena, Stefano and Sattig, Gabriel},
     title = {Convex integration solutions to the transport equation with full dimensional concentration},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1075--1108},
     year = {2020},
     publisher = {Elsevier},
     volume = {37},
     number = {5},
     doi = {10.1016/j.anihpc.2020.03.002},
     mrnumber = {4138227},
     zbl = {1458.35363},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2020.03.002/}
}

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Modena, Stefano; Sattig, Gabriel. Convex integration solutions to the transport equation with full dimensional concentration. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 5, pp. 1075-1108. doi: 10.1016/j.anihpc.2020.03.002

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