Statistical solutions of the incompressible Euler equations
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.
Cite
@article{arxiv.1909.06615,
title = {Statistical solutions of the incompressible Euler equations},
author = {Samuel Lanthaler and Siddhartha Mishra and Carlos Parés-Pulido},
journal= {arXiv preprint arXiv:1909.06615},
year = {2021}
}