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On the conservation of energy in two-dimensional incompressible flows

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

(Report number 2020-06)

Abstract
We prove the conservation of energy for weak and statistical solutions of the two-dimensional Euler equations, generated as strong (in an appropriate topology) limits of the underlying Navier-Stokes equations and a Monte Carlo-Spectral Viscosity numerical approximation, respectively. We characterize this conservation of energy in terms of a uniform decay of the so-called structure function, allowing us to extend existing results on energy conservation. Moreover, we present numerical experiments with a wide variety of initial data to validate our theory and to observe energy conservation in a large class of two-dimensional incompressible flows.

Keywords: incompressible Euler, energy conservation, anomalous dissipation, turbulence, statistical solution, vorticity, structure function

@Techreport{LMP20_879,
  author = {S. Lanthaler and S. Mishra and C. Parés-Pulido},
  title = {On the conservation of energy in two-dimensional incompressible flows },
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-06},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-06.pdf },
  year = {2020}
}

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