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

Mathematics > Analysis of PDEs [Submitted on 18 Jun 2026] Title:Solutions of the 3D inhomogeneous incompressible Navier-Stokes system with initial velocity in $VMO^{-1}$ View PDFAbstract: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. References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.