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arXiv · 2606.19060

Delayed blow-up by transport noise for the 3D Navier-Stokes equation with Navier-slip boundary conditions

Meng Zhao
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Abstract

We study the vorticity formulation of the 3D Navier-Stokes equation driven by transport noise in a periodic channel with Navier-slip boundary conditions. We consider both non-degenerate transport noise and degenerate tangential transport noise. For any prescribed $T>0$ and $ε>0$, we prove that, by choosing the noise intensity sufficiently large and concentrating the noise on sufficiently high modes, the solution exists up to $T$ with probability at least $1-ε$. A main contribution of this work is to identify and analyze the interaction between enhanced dissipation induced by transport noise and physical boundary effects. The no-flux condition breaks the isotropy of the noise and changes the scaling limit of the Itô-Stratonovich corrector. In the non-degenerate case, a boundary feedback term appears in the limiting effective operator; in the degenerate case, the limiting operator is a nonlocal anisotropic tangential dissipation. The proof is based on a combination of a boundary correction operator, a Meyers-type estimate, a scaling-limit analysis of the Itô-Stratonovich corrector, and resolvent estimates for the deterministic limiting equations.

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Meng Zhao. 2026-06-17. Delayed blow-up by transport noise for the 3D Navier-Stokes equation with Navier-slip boundary conditions. https://arxiv.org/abs/2606.19060

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