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Statistical solutions of the incompressible Euler equations

by S. Lanthaler and S. Mishra and C. Parés-Pulido

(Report number 2019-51)

Abstract
We propose and study the framework of dissipative statistical solutions for the incompressible Euler equations. Statistical solutions are time-parameterized probability measures on the space of square-integrable functions, whose time-evolution is determined from the underlying Euler equations. We prove partial well-posedness results for dissipative statistical solutions and propose a Monte Carlo type algorithm, based on spectral viscosity spatial discretizations, to approximate them. Under verifiable hypotheses on the computations, we prove that the approximations converge to a statistical solution in a suitable topology. In particular, multi-point statistical quantities of interest converge on increasing resolution. We present several numerical experiments to illustrate the theory.

Keywords: incompressible-Euler spectral-viscosity statistical-solutions turbulence Monte-Carlo

@Techreport{LMP19_855,
  author = {S. Lanthaler and S. Mishra and C. Parés-Pulido},
  title = {Statistical solutions of the incompressible Euler equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-51},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-51.pdf },
  year = {2019}
}

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