Research reports

Years: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

On the vanishing viscosity limit of statistical solutions of the incompressible Navier-Stokes equations

by U. Fjordholm and S. Mishra and F. Weber

(Report number 2021-34)

Abstract
We study statistical solutions of the incompressible Navier--Stokes equation and their vanishing viscosity limit. We show that a formulation using correlation measures, which are probability measures accounting for spatial correlations, and moment equations is equivalent to statistical solutions in the Foiac{s}--Prodi sense. Under the assumption of weak scaling, a weaker version of Kolmogorov's self-similarity at small scales hypothesis that allows for intermittency corrections, we show that the limit is a statistical solution of the incompressible Euler equations. To pass to the limit, we derive a K'arm'an--Howarth--Monin relation for statistical solutions and combine it with the weak scaling assumption and a compactness theorem for correlation measures.

Keywords: Statistical Solutions, Vanishing Viscosity, Euler equations, Navier-Stokes equations

@Techreport{FMW21_976,
  author = {U. Fjordholm and S. Mishra and F. Weber},
  title = {On the vanishing viscosity limit of statistical solutions of the incompressible Navier-Stokes equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2021-34},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2021/2021-34.pdf },
  year = {2021}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).