On the persistence of H\"older regular patches of density for the inhomogeneous Navier-Stokes equations
Résumé
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 incompressible
and 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 regularity
along 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.
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