Hölder continuity for a drift-diffusion equation with pressure

Silvestre, Luis; Vicol, Vlad

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 4, pp. 637-652.

Nous abordons la question de la persistance de la continuité Hölder pour les solutions faibles de lʼéquation linéaire de dérive-diffusion avec une pression non-locale

$$u_t+b·\nabla u-△u=\nabla p,\nabla ·u=0$$

sur $[0,\infty )\times {ℝ}^n$, avec $n⩾2$. On suppose que la vitesse de dérive b est au niveau critique de régularité par rapport au changement dʼéchelle de lʼéquation. La démonstration sʼappuie sur la définition des espaces Hölder de Campanato, et elle utilise un argument de principe du maximum par lequel nous contrôlons la croissance en temps de certaines moyennes locales de u. Nous fournissons une estimation qui ne dépend dʼaucune condition de petitesse locale sur le champ de vecteur b, mais seulement sur des quantités invariantes par changement dʼéchelle.

We address the persistence of Hölder continuity for weak solutions of the linear drift-diffusion equation with nonlocal pressure

$$u_t+b·\nabla u-△u=\nabla p,\nabla ·u=0$$

on $[0,\infty )\times {ℝ}^n$, 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 Campanatoʼs 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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