What question did this study set out to answer?

This research aims to analyze the effect of random diffusion in 3D stochastic incompressible Euler equations and establish conditions for well-posedness.

June 14, 2026Open Access

On the Stochastic Regularization of 3D Euler Equations Through Random Diffusion

Key Points

This research aims to analyze the effect of random diffusion in 3D stochastic incompressible Euler equations and establish conditions for well-posedness.
Applied a multiplicative noise term to the 3D stochastic incompressible Euler equations.
Utilized the Itô integral to introduce a random field in the diffusion coefficient.
Established local and global well-posedness through contraction arguments and a priori estimates.
Established local well-posedness for stochastic parabolic equations.
Demonstrated global well-posedness for small initial data with high probability.
Showed that the induced noise leads to a structure similar to Navier-Stokes equations.

Abstract

In this paper, we consider 3D stochastic incompressible Euler equations perturbed by a multiplicative noise term acting as a random diffusion. By introducing a random field to the diffusion coefficient and applying the Itô integral, we show that this noise induces a mathematical structure analogous to the Navier-Stokes equations. We establish local well-posedness through a contraction argument and a priori estimates for these stochastic parabolic equations. In addition, we show the global well-posedness results for small initial data belonging to Gevrey-type Fourier-Bessel potential spaces with high probability.

On the Stochastic Regularization of 3D Euler Equations Through Random Diffusion

Key Points

Abstract

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