On the persistence of Hölder regular patches of density for the inhomogeneous Navier-Stokes equations

Raphaöl L. Danchin, Xin Zhang

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

In our recent work dedicated to the Boussinesq equations [15], we established the persistence of solutions with piecewise constant temperature along interfaces with Hölder regularity. We here address the same question for the inhomogeneous Navier-Stokes equations satisfied by a viscous incompressible and inhomogeneous fluid. We prove that, indeed, in the slightly inhomogeneous case, patches of densities with C1,ε regularity propagate for all time. Our result follows from the conservation of Hölder 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 [15], since the coupling between the mass and velocity equations in the inhomogeneous Navier-Stokes equations is quasilinear while it is linear for the Boussinesq equations.

Original languageEnglish
Pages (from-to)781-811
Number of pages31
JournalJournal de l'Ecole Polytechnique - Mathematiques
Volume4
DOIs
Publication statusPublished - 2017 Jan 1

Keywords

  • C1
  • Inhomogeneous Navier-Stokes equations
  • Striated regularity
  • Ε density patch

ASJC Scopus subject areas

  • Mathematics(all)

Fingerprint Dive into the research topics of 'On the persistence of Hölder regular patches of density for the inhomogeneous Navier-Stokes equations'. Together they form a unique fingerprint.

  • Cite this