[Paper Review] Global solutions to 2-d inhomogeneous navier-stokes system with general velocity

Jingchi Huang, Marius Paicu+1 more|arXiv (Cornell University)|Dec 17, 2012

Navier-Stokes equation solutions13 references3 citations

TL;DR

This paper establishes global wellposedness for the 2D inhomogeneous Navier-Stokes equations with variable viscosity in a critical Besov space framework, under a doubly exponential smallness condition on initial density fluctuations relative to initial velocity. It proves global existence for piecewise constant initial density with small jumps, removing the smallness requirement on initial velocity from prior results.

ABSTRACT

In this paper, we are concerned with the global wellposedness of 2-D density-dependent incompressible Navier-Stokes equations with variable viscosity, in a critical functional frame- work which is invariant by the scaling of the equations and under a non-linear smallness condition on fluctuation of the initial density which has to be doubly exponential small compared with the size of the initial velocity. In the second part of the paper, we apply our methods combined with the techniques of R. Danchin and P. B. Mucha to prove the global existence of solutions to inhomogeneous Navier-Stokes system with piecewise constant initial density which has small jump at the interface and is away from vacuum. In particular, this latter result removes the smallness condition for the initial velocity in a corresponding theorem of R. Danchin and P. B. Mucha.

Motivation & Objective

• Address the global wellposedness of 2D inhomogeneous Navier-Stokes equations with variable viscosity in a scaling-invariant critical functional framework.
• Overcome the limitation of prior results requiring small initial velocity by removing this smallness condition for piecewise constant initial density with small jumps.
• Provide a global existence result under a non-linear smallness condition on the fluctuation of initial density, which must be doubly exponential small relative to the initial velocity size.
• Extend the applicability of the theory to physical scenarios involving immiscible fluids with distinct densities and small interfacial jumps.

Proposed method

• Employ a critical functional framework based on Besov spaces invariant under the scaling of the equations.
• Use Littlewood-Paley theory and paraproduct estimates to control nonlinear terms in the Navier-Stokes system.
• Introduce a modified formulation involving the density fluctuation $ a = ho - 1 $ and the viscosity deviation $ ilde{ u}(a) - u $, transforming the system into a perturbed form.
• Apply energy estimates and Gronwall-type inequalities in anisotropic Besov spaces to control the growth of solutions over time.
• Use a fixed-point argument in a carefully chosen function space to establish local existence, then extend it globally via a contradiction argument assuming finite maximal time of existence.
• Combine techniques from [10] with new estimates on the density and viscosity to handle the piecewise constant initial density case.

Experimental results

Research questions

• RQ1 Can global solutions be constructed for the 2D inhomogeneous Navier-Stokes system with variable viscosity without assuming small initial velocity?
• RQ2 What is the optimal smallness condition on the initial density fluctuation that ensures global wellposedness in critical spaces?
• RQ3 Can the smallness assumption on the initial velocity be removed in the case of piecewise constant initial density with small interfacial jumps?
• RQ4 How does the regularity of the initial density interface propagate in time under the inhomogeneous Navier-Stokes dynamics?
• RQ5 What is the role of the viscosity function $ ilde{ u}(a) $ in stabilizing the system under large initial velocity?

Key findings

• Global wellposedness is established for the 2D inhomogeneous Navier-Stokes system with variable viscosity in a critical Besov framework, under a doubly exponential smallness condition on the initial density fluctuation relative to the initial velocity size.
• The solution exists globally in time for initial data with piecewise constant density and small jump at the interface, even without smallness on the initial velocity, thus removing a key restriction from earlier results.
• The proof relies on a refined energy estimate in Besov spaces that controls the growth of the solution via a Gronwall-type inequality with a double exponential factor in the initial data size.
• The maximal time of existence $ au^* $ is shown to be infinite by contradiction, assuming a finite $ au^* $ leads to a violation of the smallness condition on the solution norm.
• The method successfully handles the nonlinearity and coupling between density and velocity fields through a perturbation approach in the $ ilde{ u}(a) - u $ and $ a $ variables.
• The result provides a partial answer to Lions' open question on the propagation of boundary regularity in density-patch solutions.