Closed Estimates of Leray Projected Transport Noise and Strong Solutions of the Stochastic Euler Equations

2025/07/01 by Goodair, Daniel
#Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2507.00787

Abstract

We consider the incompressible Euler and Navier-Stokes equations on the three dimensional torus, in velocity form, perturbed by a transport or transport-stretching Stratonovich noise. Closed control of the noise contributions in energy estimates are demonstrated, for any positive integer ordered Sobolev Space and the equivalent Stokes Space; difficulty arises due to the presence of the Leray Projector disrupting cancellation of the top order derivative. This is particularly pertinent in the case of a transport noise without stretching, where the vorticity form cannot be used. As a consequence we obtain, for the first time, the existence of a local strong solution to the corresponding stochastic Euler equation. Furthermore, smooth solutions are shown to exist until blow-up in L¹([0,T];W^1,∞).

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Related

• High Order Smoothness for Stochastic Navier-Stokes Equations with Transport and Stretching Noise on Bounded Domains
• Global smooth solutions by transport noise of 3D Navier-Stokes equations with small hyperviscosity
• A note on stochastic Navier-Stokes equations with not regular\n multiplicative noise
• Well-posedness for a stochastic 2D Euler equation with transport noise
• Navier-Stokes Equations with Navier Boundary Conditions and Stochastic Lie Transport: Well-Posedness and Inviscid Limit