### Abstract

We address the persistence of Hölder continuity for weak solutions of the linear drift-diffusion equation with nonlocal pressureu _{t}+b·∇ u-Δu=∇p, Δu=0 on [0,∞) × ℝ^{n}, with n≥2. The drift velocity b is assumed to be at the critical regularity level, with respect to the natural scaling of the equations. The proof draws on Campanato's characterization of Hölder spaces, and uses a maximum-principle-type argument by which we control the growth in time of certain local averages of u. We provide an estimate that does not depend on any local smallness condition on the vector field b, but only on scale invariant quantities.

Original language | English (US) |
---|---|

Pages (from-to) | 637-652 |

Number of pages | 16 |

Journal | Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire |

Volume | 29 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2012 |

### Fingerprint

### ASJC Scopus subject areas

- Analysis
- Mathematical Physics

### Cite this

*Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire*,

*29*(4), 637-652. https://doi.org/10.1016/j.anihpc.2012.02.003

**Hölder continuity for a drift-diffusion equation with pressure.** / Silvestre, Luis; Vicol, Vlad.

Research output: Contribution to journal › Article

*Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire*, vol. 29, no. 4, pp. 637-652. https://doi.org/10.1016/j.anihpc.2012.02.003

}

TY - JOUR

T1 - Hölder continuity for a drift-diffusion equation with pressure

AU - Silvestre, Luis

AU - Vicol, Vlad

PY - 2012/1/1

Y1 - 2012/1/1

N2 - We address the persistence of Hölder continuity for weak solutions of the linear drift-diffusion equation with nonlocal pressureu t+b·∇ u-Δu=∇p, Δu=0 on [0,∞) × ℝn, with n≥2. The drift velocity b is assumed to be at the critical regularity level, with respect to the natural scaling of the equations. The proof draws on Campanato's characterization of Hölder spaces, and uses a maximum-principle-type argument by which we control the growth in time of certain local averages of u. We provide an estimate that does not depend on any local smallness condition on the vector field b, but only on scale invariant quantities.

AB - We address the persistence of Hölder continuity for weak solutions of the linear drift-diffusion equation with nonlocal pressureu t+b·∇ u-Δu=∇p, Δu=0 on [0,∞) × ℝn, with n≥2. The drift velocity b is assumed to be at the critical regularity level, with respect to the natural scaling of the equations. The proof draws on Campanato's characterization of Hölder spaces, and uses a maximum-principle-type argument by which we control the growth in time of certain local averages of u. We provide an estimate that does not depend on any local smallness condition on the vector field b, but only on scale invariant quantities.

UR - http://www.scopus.com/inward/record.url?scp=84864130981&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84864130981&partnerID=8YFLogxK

U2 - 10.1016/j.anihpc.2012.02.003

DO - 10.1016/j.anihpc.2012.02.003

M3 - Article

AN - SCOPUS:84864130981

VL - 29

SP - 637

EP - 652

JO - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

JF - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

SN - 0294-1449

IS - 4

ER -