The Navier-Stokes equations with transport noise in critical $H^{1/2}$ space

TL;DR

This work analyzes the 3D Navier–Stokes equations with transport-type stochastic forcing in the scaling-critical space $H^{1/2}$. A truncated-advecting-term strategy with a cutoff and nested fixed-point iterations is employed, together with Itô and Burkholder–Davis–Gundy techniques, to obtain a local probabilistically strong solution for small initial data and small noise. Energy-type inequalities ensure the solution remains small in $H^{1/2}$ and $H^{3/2}$, and the probability of global existence can be made arbitrarily close to 1 by taking the initial data norm small, yielding almost global well-posedness on both the torus and the whole space. The results extend the stochastic regularization program by demonstrating robustness of transport-noise-induced well-posedness in critical spaces beyond compact domains.

Abstract

We study the Navier-Stokes equations with transport noise in critical function spaces. Assuming the initial data belongs to $H^{1/2}$ almost surely, we establish the existence and uniqueness of a local-in-time probabilistically strong solution. Moreover, we show that the probability of global existence can be made arbitrarily close to $1$ by choosing the initial data norm sufficiently small, and that the solution norm remains small for all time. Our analysis is independent of the compactness of the spatial domain, and consequently, the results apply both to the three-dimensional torus and to the whole space.

Theorems & Definitions (14)

• Theorem 2.1
• Remark 2.2
• Lemma 3.1
• proof : Proof of Lemma \ref{['L03']}
• Lemma 3.2
• proof : Proof of Lemma \ref{['L04']}
• Lemma 3.3
• proof : Proof of Lemma \ref{['L08']}
• Lemma 3.4
• proof : Proof of Lemma \ref{['L06']}