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

2020/11/24 by Camillo De Lellis, De Lellis, Camillo, Vikram Giri +1 · 1 citation
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2011.12093

openalex publication_date 2020/11/24 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We give an example of a bounded divergence free autonomous vector field in\n mathbb R³ (and of a nonautonomous bounded divergence free vector field in\n mathbb R²) and of a bounded initial data for which the Cauchy problem for\nthe corresponding transport equation has 2 distinct solutions. We then show\nthat both solutions are limits of classical solutions of transport equations\nfor appropriate smoothings of the vector fields and of the initial data.\n

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