Sur l'existence locale pour une équation de scalaires actifs

Weiwei Hu, Igor Kukavica, Mohammed Ziane

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We address the local well-posedness for the active scalar equation ∂tθ+u{dot operator}∇θ=0 where u=-∇(-δ)-1+β/2θ. This equation reduces to the Euler equation if β=0 and to the quasi-geostrophic equation for β=1. In this note, we prove the local existence for the equation in the space H1+β+ε, where ε>0, for 1<β<2. An earlier result by Chae, Constantin, Córdoba, Gancedo, and Wu shows the local existence in H4. The improvement is due to a sharper commutator estimate, while the fractional exponent is obtained through a different treatment of the nonlinearity using a double commutator inequality.

Original languageFrench
Pages (from-to)241-245
Number of pages5
JournalComptes Rendus Mathematique
Issue number3
StatePublished - Mar 1 2015
Externally publishedYes

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