### 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 C^{1,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 language | English |
---|---|

Pages (from-to) | 781-811 |

Number of pages | 31 |

Journal | Journal de l'Ecole Polytechnique - Mathematiques |

Volume | 4 |

DOIs | |

Publication status | Published - 2017 Jan 1 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**On the persistence of H�lder regular patches of density for the inhomogeneous navier-stokes equations.** / Danchin, Raphaöl L.; Zhang, Xin.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - On the persistence of H�lder regular patches of density for the inhomogeneous navier-stokes equations

AU - Danchin, Raphaöl L.

AU - Zhang, Xin

PY - 2017/1/1

Y1 - 2017/1/1

N2 - 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.

AB - 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.

KW - C1

KW - Inhomogeneous Navier-Stokes equations

KW - Striated regularity

KW - Ε density patch

UR - http://www.scopus.com/inward/record.url?scp=85034805696&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85034805696&partnerID=8YFLogxK

U2 - 10.5802/jep.56

DO - 10.5802/jep.56

M3 - Article

AN - SCOPUS:85034805696

VL - 4

SP - 781

EP - 811

JO - Journal de l'Ecole Polytechnique - Mathematiques

JF - Journal de l'Ecole Polytechnique - Mathematiques

SN - 2429-7100

ER -