On the energy of inviscid singular flows

2008/03/13 by Roman Shvydkoy, Shvydkoy, Roman · 2 citations
Engineering · Mathematics · #76B47 (Secondary) #76F02 (Primary) #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions #math.AP #msc:76B47 #msc:76F02

paper · pdf · doi:10.48550/arxiv.0803.2056

19 pages

arxiv created 2008/03/13 · openalex publication_date 2008/03/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is known that the energy of a weak solution to the Euler equation is conserved if it is slightly more regular than the Besov space B^1/3_3,∞. When the singular set of the solution is (or belongs to) a smooth manifold, we derive various L^p-space regularity criteria dimensionally equivalent to the critical one. In particular, if the singular set is a hypersurface the energy of u is conserved provided the one sided non-tangential limits to the surface exist and the non-tangential maximal function is L³ integrable, while the maximal function of the pressure is L^3/2 integrable. The results directly apply to prove energy conservation of the classical vortex sheets in both 2D and 3D at least in those cases where the energy is finite.

Cited by

• Fine dissipative properties of Euler solutions with measure first derivatives
• A note on helicity conservation for compressible Euler equations in a bounded domain with vacuum

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