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Optimal stability estimates for continuity equations

Part of: Smooth dynamical systems: general theory Partial differential equations General first-order equations and systems

Published online by Cambridge University Press:  16 July 2018

Christian Seis
Christian Seis*
Affiliation:
Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
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Abstract

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.

Keywords

continuity equationSobolev vector fieldsstability estimatesKantorovich–Rubinstein distanceLagrangian flows

MSC classification

Primary: 35F10: Initial value problems for linear first-order equations
Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters 37C10: Vector fields, flows, ordinary differential equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 6 , December 2018 , pp. 1279 - 1296
Copyright
Copyright © Royal Society of Edinburgh 2018 

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Footnotes

*

Present address: Institut für Analysis und Numerik, Universität Münster, Orléans-Ring 10, 48149 Münster, Germany (seis@wwu.de).