We address the local well-posedness for the active scalar equation where . This equation reduces to the Euler equation if and to the quasi-geostrophic equation for . In this note, we prove the local existence for the equation in the space , where , for . An earlier result by Chae, Constantin, Córdoba, Gancedo, and Wu shows the local existence in . 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.
Dans cette Note, nous étudions l'existence locale des solutions pour l'équation du scalaire actif où . Cette équation correspond à l'équation d'Euler quand et à l'équation quasi-géostrophique quand . Nous montrons l' existence locale dans l'espace , où , pour . Un résultat récent de Chae, Constantin, Córdoba, Gancedo et Wu montre l'existence locale dans . L'amélioration est due à des nouvelles estimations plus précises du commutateur, et l'exposant fractionnaire est obtenu par un traitement different de la non-linéarité en utilisant une inégalité double du commutateur.
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Weiwei Hu 1; Igor Kukavica 1; Mohammed Ziane 1
@article{CRMATH_2015__353_3_241_0, author = {Weiwei Hu and Igor Kukavica and Mohammed Ziane}, title = {Sur l'existence locale pour une \'equation de scalaires actifs}, journal = {Comptes Rendus. Math\'ematique}, pages = {241--245}, publisher = {Elsevier}, volume = {353}, number = {3}, year = {2015}, doi = {10.1016/j.crma.2014.12.008}, language = {fr}, }
Weiwei Hu; Igor Kukavica; Mohammed Ziane. Sur l'existence locale pour une équation de scalaires actifs. Comptes Rendus. Mathématique, Volume 353 (2015) no. 3, pp. 241-245. doi : 10.1016/j.crma.2014.12.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.12.008/
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