Statistical solutions of the incompressible Euler equations

TL;DR

This paper introduces a framework for dissipative statistical solutions to the incompressible Euler equations, proving their well-posedness and proposing a Monte Carlo algorithm for their approximation with convergence guarantees.

Contribution

It develops a new statistical solution framework for Euler equations, establishes partial well-posedness, and introduces a convergent spectral viscosity-based Monte Carlo method.

Findings

Proved partial well-posedness of dissipative statistical solutions.

Developed a Monte Carlo algorithm with proven convergence.

Numerical experiments illustrate the theoretical results.

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.