Volume 56, Number 6, November-December 2022
|Page(s)||1993 - 2019|
|Published online||14 September 2022|
Convergence of a spectral method for the stochastic incompressible Euler equations
Centre for Applicable Mathematics, Tata Institute of Fundamental Research, P.O. Box 6503, GKVK Post Office, Bangalore 560065, India
* Corresponding author: firstname.lastname@example.org
Accepted: 6 July 2022
We propose a spectral viscosity method (SVM) to approximate the incompressible Euler equations driven by a multiplicative noise. We show that the SVM solution converges to a dissipative measure-valued martingale solution of the underlying problem. These solutions are weak in the probabilistic sense i.e. the probability space and the driving Wiener process are an integral part of the solution. We also exhibit a weak (measure-valued)-strong uniqueness principle. Moreover, we establish strong convergence of approximate solutions to the regular solution of the limit system at least on the lifespan of the latter, thanks to the weak (measure-valued)–strong uniqueness principle for the underlying system.
Mathematics Subject Classification: 35R60 / 60H15 / 65M12 / 65M70 / 76B03 / 76D03
Key words: Euler system / incompressible fluids / stochastic forcing / multiplicative noise / spectral method / dissipative measure-valued martingale solution / weak-strong uniqueness
© The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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