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lagrangian flows

Proceedings of the Royal Society of Edinburgh Section A: Mathematics (1)

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Optimal stability estimates for continuity equations
Part of
Christian Seis
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 148 / Issue 6 / December 2018

Published online by Cambridge University Press:

16 July 2018, pp. 1279-1296

Print publication:

December 2018
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This review paper is concerned with the stability analysis of the continuity equation in the DiPerna–Lions setting in which the advecting velocity field is Sobolev regular. Quantitative estimates for the equation were derived only recently, but optimality was not discussed. We revisit the results from our 2017 paper, compare the new estimates with previously known estimates for Lagrangian flows and demonstrate how these can be applied to produce optimal bounds in applications from physics, engineering and numerical analysis.