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arXiv · 2606.20207

Solutions of the 3D inhomogeneous incompressible Navier-Stokes system with initial velocity in $VMO^{-1}$

Ruilin Hu·Quoc-Hung Nguyen·Feng Shao·Dongyi Wei·Ping Zhang·Zhifei Zhang
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Abstract

In this paper, we establish local existence of strong solutions for the three-dimensional inhomogeneous incompressible Navier-Stokes equations with initial data $(\rho_0,u_0)$ lying in $C^1 \times (L^2 \cap VMO^{-1})$, where $\rho_0$ has a positive lower bound. Furthermore, if $\rho_0 \in C^2$ and $||\rho_0-1||_{L^\infty}+||u_0||_{BMO^{-1}}$ is sufficiently small, we prove global existence of the solution. To achieve this, we employ an estimate for the transport equation to obtain regularity for the density and apply a new freezing-coefficient method for the momentum equation.

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Ruilin Hu, Quoc-Hung Nguyen, Feng Shao, Dongyi Wei, Ping Zhang, Zhifei Zhang. 2026-06-18. Solutions of the 3D inhomogeneous incompressible Navier-Stokes system with initial velocity in $VMO^{-1}$. https://arxiv.org/abs/2606.20207

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