Convex integration solutions to the transport equation with full dimensional concentration

Modena, Stefano; Sattig, Gabriel

doi:10.1016/j.anihpc.2020.03.002

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

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

Cet article a éte moissonné depuis la source Numdam

Voir la notice de l'article

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

@article{AIHPC_2020__37_5_1075_0,
     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 = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.03.002/}
}

TY  - JOUR
AU  - Modena, Stefano
AU  - Sattig, Gabriel
TI  - Convex integration solutions to the transport equation with full dimensional concentration
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2020
SP  - 1075
EP  - 1108
VL  - 37
IS  - 5
PB  - Elsevier
UR  - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.03.002/
DO  - 10.1016/j.anihpc.2020.03.002
LA  - en
ID  - AIHPC_2020__37_5_1075_0
ER  -

%0 Journal Article
%A Modena, Stefano
%A Sattig, Gabriel
%T Convex integration solutions to the transport equation with full dimensional concentration
%J Annales de l'I.H.P. Analyse non linéaire
%D 2020
%P 1075-1108
%V 37
%N 5
%I Elsevier
%U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.03.002/
%R 10.1016/j.anihpc.2020.03.002
%G en
%F AIHPC_2020__37_5_1075_0

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

Cité par Sources :

Parcourir par

Geodesic

Parcourir par