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.

References in corpus (5)

• Deep learning observables in computational fluid dynamics
• Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws
• Statistical solutions of hyperbolic conservation laws I: Foundations
• Computation of measure-valued solutions for the incompressible Euler equations
• On the convergence of the spectral viscosity method for the two-dimensional incompressible Euler equations with rough initial data

Cited by in corpus (5)

• Statistical solutions of the incompressible Euler equations
• On Bayesian data assimilation for PDEs with ill-posed forward problems
• Probabilistic Descriptions of Fluid Flow: A Survey
• Numerical approximation of statistical solutions of the incompressible Navier-Stokes Equations
• Efficient Computation of Large-Scale Statistical Solutions to Incompressible Fluid Flows