Conserved quantities and regularity in fluid dynamics

2020/03/17 by Emil Wiedemann, Wiedemann, Emil
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #math.AP #physics.flu-dyn

paper · pdf · doi:10.48550/arxiv.2003.07807

This is a set of lecture notes for the 2019 EMS School in Applied Mathematics held in Kácov, Czech Republic. It partially contains previously uploaded material, in particular from arXiv:1412.2476

arxiv created 2020/03/17 · arxiv updated 2020/03/18

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.

Citations

• Failure of the chain rule for the divergence of bounded vector fields

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