The stochastic Navier-Stokes equations with general $L^{3}$ data

TL;DR

This work studies local-in-time probabilistically strong solutions to the 3D stochastic Navier–Stokes equations with multiplicative noise for critical initial data in $L^{3}$ and $H^{1/2}$. The authors introduce a data-decomposition strategy that splits the initial data into a small subcritical part and a regular part, uses existing subcritical theories to handle the large component, and constructs the remainder via a truncated-difference scheme across a sequence of subcritical components. The main contributions are a local well-posedness result for general $L^{3}$ data and an analogous result for $H^{1/2}$ data, both accompanied by energy-type inequalities and pathwise uniqueness on a short time interval. The techniques extend the deterministic critical-space analyses to the stochastic setting, leveraging a combination of decompositions, cutoff controls, and stopping-time arguments to manage the impact of rough data and noise.

Abstract

We consider the stochastic Navier-Stokes equations with multiplicative noise with critical initial data. Assuming that the initial data $u_0$ belongs to the critical space $L^{3}$ almost surely, we construct a unique local-in-time probabilistically strong solution. We also prove an analogous result for data in the critical space~$H^\frac{1}{2}$.

Theorems & Definitions (15)

• Theorem 2.1: A local-in-time solution with $L^{3}$ initial data
• Lemma 2.2: Decomposition of initial data
• Lemma 2.3
• Corollary 2.4: $L^3$ estimate for $\bar{w}$
• Proposition 3.1
• Lemma 3.2: An $L^{6}$ solution
• Lemma 3.3: An $L^{p}$-energy control
• Lemma 3.4: KX2
• Remark 3.5
• proof : Proof of Lemma \ref{['L11']}