[Paper Review] On the global well-posedness of 2-D density-dependent Navier-Stokes system with variable viscosity

Hammadi Abidi, Ping Zhang|arXiv (Cornell University)|Jan 11, 2013

Navier-Stokes equation solutions13 references4 citations

TL;DR

This paper establishes the global well-posedness of the 2D inhomogeneous Navier-Stokes equations with variable viscosity by proving existence and uniqueness of global solutions under smallness conditions on the initial density fluctuation and viscous coefficient. Using Littlewood-Paley theory and critical Besov space estimates, it extends global well-posedness to almost critical regularity regimes and scaling-invariant spaces.

ABSTRACT

Given solenoidal vector $u_0\in H^{-2\d}\cap H^1(\R^2),$ $ _0-1\in L^2(\R^2),$ and $ _0 \in L^\infty\cap\dot{W}^{1,r}(\R^2)$ with a positive lower bound for $\d\in (0,\f12)$ and $2

Motivation & Objective

• To establish global existence and uniqueness of solutions for the 2D incompressible inhomogeneous Navier-Stokes system with variable viscosity.
• To extend previous results by allowing less regular initial data and removing smallness assumptions on density fluctuations in certain regimes.
• To prove global well-posedness in scaling-invariant and almost critical Besov spaces under smallness conditions on the initial density perturbation.
• To analyze the propagation of regularity for solutions under improved initial data regularity.

Proposed method

• Employing Littlewood-Paley theory and frequency localization techniques to decompose solutions and control nonlinear terms.
• Using Chemin-Lerner type spaces to capture time-integrability and frequency localization of solution norms.
• Applying commutator estimates to control the interaction between velocity and density fluctuations in frequency space.
• Establishing energy estimates in Besov spaces with critical regularity indices to handle scaling invariance.
• Introducing a smallness condition on the $ L^rown $ norm of $ ho_0 - 1 $ relative to the initial energy and Besov norm of velocity.
• Using Bony's decomposition to handle the product of low- and high-frequency components in nonlinear terms.

Experimental results

Research questions

• RQ1 Under what conditions on initial data does the 2D inhomogeneous Navier-Stokes system with variable viscosity admit a unique global solution?
• RQ2 Can global well-posedness be established in scaling-invariant Besov spaces without requiring smallness of the initial velocity in $ L^2 $?
• RQ3 What is the minimal regularity required for initial data to ensure global existence and propagation of regularity?
• RQ4 How does the size of the viscous coefficient $ ho o ho $ affect the global well-posedness when $ ho_0 $ is close to 1 in $ L^rown $?
• RQ5 Can global well-posedness be extended to almost critical Besov spaces under a smallness condition on $ ho_0 - 1 $?

Key findings

• For $ ho_0 - 1 o L^2 imes ho_0 o L^rown imes ho_0 o ext{dot}{W}^{1,r} $ with $ r o (2, 2/(1-2 heta)) $, global well-posedness holds if $ ho_0 $ is close to 1 in $ L^rown $ norm relative to $ heta $ and initial data norms.
• If $ ( ho_0 - 1, u_0) o ext{dot}{B}^{2/p}_{p,1} imes ( ext{dot}{B}^{-1+2/p}_{p,1} imes L^2) $ with $ 1 < p < 4 $, global well-posedness holds when $ ho_0 - 1 $ is small compared to $ ext{exp}(C(rown{u}_0^2 + rown{u}_0_{ ext{dot}{B}^{-1+2/p}_{p,1}})) $.
• In the almost critical Besov space regime, global well-posedness is achieved if $ ho_0 - 1 $ is small in $ L^rown $, independent of the size of $ u_0 $.
• The solution propagates regularity: if initial data are smoother, the solution remains smooth for all time.
• The result improves upon prior work by removing smallness assumptions on density fluctuations and reducing regularity requirements on initial velocity.
• The proof relies on a refined use of Littlewood-Paley theory and commutator estimates to control the variable viscosity term in frequency space.