Conserved quantities and regularity in fluid dynamics
Abstract
Conserved or dissipated quantities, like energy or entropy, are at the heart of the study of many classes of time-dependent PDEs in connection with fluid mechanics. This is the case, for instance, for the Euler and Navier-Stokes equations, for systems of conservation laws, and for transport equations. In all these cases, a formally conserved quantity may no longer be constant in time for a weak solution at low regularity. The delicate interplay between regularity and conservation of the respective quantity relates to renormalisation in the DiPerna-Lions theory of transport and continuity equations, and to Onsager's conjecture in the realm of ideal incompressible fluids. We will review the classical commutator methods of DiPerna-Lions and Constantin-E-Titi, and then proceed to more recent results.
Cite
@article{arxiv.2003.07807,
title = {Conserved quantities and regularity in fluid dynamics},
author = {Emil Wiedemann},
journal= {arXiv preprint arXiv:2003.07807},
year = {2020}
}
Comments
This is a set of lecture notes for the 2019 EMS School in Applied Mathematics held in K\'acov, Czech Republic. It partially contains previously uploaded material, in particular from arXiv:1412.2476