English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper, we first prove the local well-posedness of the 2-Dincompressible Navier-Stokes equations with variable viscosity in criticalBesov spaces with negative regularity indices, without smallness assumption onthe variation of the density. The key is to prove for $p\in(1,4)$ and$a\in\dot{B}_{p,1}^{\frac2p}(\mathbb{R}^2)$ that the solution mapping$\mathcal{H}_a:F\mapsto\nabla\Pi$ to the 2-D elliptic equation$\mathrm{div}\big((1+a)\nabla\Pi\big)=\mathrm{div} F$ is bounded on$\dot{B}_{p,1}^{\frac2p-1}(\mathbb{R}^2)$. More precisely, we prove that$$\|\nabla\Pi\|_{\dot{B}_{p,1}^{\frac2p-1}}\leqC\big(1+\|a\|_{\dot{B}_{p,1}^{\frac2p}}\big)^2\|F\|_{\dot{B}_{p,1}^{\frac2p-1}}.$$The proof of the uniqueness of solution to (1.2) relies on a Lagrangianapproach [15]-[17]. When the viscosity coefficient $\mu(\rho)$ is a positiveconstant, we prove that (1.2) is globally well-posed.

On the Well-posedness of 2-D Incompressible Navier-Stokes Equations with Variable Viscosity in Critical Spaces