TY - JOUR

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

AU - Silvestre, Luis

AU - Vicol, Vlad

N1 - Funding Information:
Luis Silvestre was partially supported by NSF grant DMS-1001629 and the Sloan Foundation. Vlad Vicol was partially supported by an AMS-Simmons travel grant.

PY - 2012

Y1 - 2012

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.

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U2 - 10.1016/j.anihpc.2012.02.003

DO - 10.1016/j.anihpc.2012.02.003

M3 - Article

AN - SCOPUS:84864130981

SN - 0294-1449

VL - 29

SP - 637

EP - 652

JO - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

JF - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

IS - 4

ER -