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Smoothing does not give a selection principle for transport equations with bounded autonomous fields

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Abstract

We give an example of a bounded divergence free autonomous vector field in \({\mathbb {R}}^3\) (and of a nonautonomous bounded divergence free vector field in \({\mathbb {R}}^2\)) and of a smooth initial data for which the Cauchy problem for the corresponding transport equation has 2 distinct solutions. We then show that both solutions are limits of classical solutions of transport equations for appropriate smoothings of the vector fields and of the initial data.

Résumé

Nous donnons un exemple de champ vectoriel autonome, borné et à divergence nulle dans \({\mathbb {R}}^3\) (et d’un champ vectoriel non autonome, borné et à divergence nulle dans \({\mathbb {R}}^2\)) et des données initiales lisses pour lesquels le problème de Cauchy pour l’équation de transport correspondante a deux solutions distinctes. Nous montrons ensuite que les deux solutions sont des limites des solutions classiques des équations de transport pour des lissages appropriés des champs vectoriels et des données initiales.

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Acknowledgements

The authors would like to thank the anonymous referee for their valuable comments that have much improved this paper.

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Correspondence to Camillo De Lellis.

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In honor of Sasha Shnirelman on occasion of his 75th birthday.

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The first author acknowledges the support of the NSF grants DMS-1946175 and DMS-1854147, while the second acknowledges the support of the NSF Grant DMS-FRG-1854344.

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De Lellis, C., Giri, V. Smoothing does not give a selection principle for transport equations with bounded autonomous fields. Ann. Math. Québec 46, 27–39 (2022). https://doi.org/10.1007/s40316-021-00160-y

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