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arXiv · 1909.06615

Statistical solutions of the incompressible Euler equations

Samuel Lanthaler·Siddhartha Mishra·Carlos Parés-Pulido
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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.

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Samuel Lanthaler, Siddhartha Mishra, Carlos Parés-Pulido. 2019-09-14. Statistical solutions of the incompressible Euler equations. https://doi.org/10.1142/s0218202521500068

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