On the persistence of H\"older regular patches of density for the inhomogeneous Navier-Stokes equations

TL;DR

This paper proves that in slightly inhomogeneous viscous incompressible fluids, patches of density with H"older regularity persist over time, extending previous results from Boussinesq equations to more complex Navier-Stokes systems.

Contribution

It demonstrates the persistence of H"older regular density patches for inhomogeneous Navier-Stokes equations, using advanced commutator estimates and handling the quasilinear coupling.

Findings

Density patches with $ ext{C}^{1, ext{ε}}$ regularity propagate for all time.

The analysis extends previous work on Boussinesq equations to more complex Navier-Stokes systems.

The proof involves conservation of H"older regularity along flow-moving vector fields.

Abstract

In our recent work dedicated to the Boussinesq equations [Danchin and Zhang 2016], we established the persistence of solutions with piecewise constant temperature along interfaces with H\"older regularity. We here address the same problem for the inhomogeneous Navier-Stokes equations satisfied by a viscous incompressibleand inhomogeneous fluid. We establish that, indeed, in the slightly inhomogeneous case, patches of densities with $\mathcal{C}^{1, \varepsilon}$ regularity propagate for all time. As in [Danchin and Zhang 2016], our result follows from the conservation of H\"older regularityalong vector fields moving with the flow. The proof of that latter result is based on commutator estimates involving para-vector fields, and multiplier spaces. The overall analysis is more complicated than in [Danchin and Zhang 2016] however, since the coupling between the mass and velocity equations in the inhomogeneous Navier-Stokes equations is \emph{quasilinear} while it is linear for the Boussinesq equations.