Global well-posedness for an advection–diffusion equation arising in magneto-geostrophic dynamics

Friedlander, Susan; Vicol, Vlad

Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 2, pp. 283-301.

Nous utilisons des techniques de De Giorgi pour démontrer la continuité Hölder de solutions faibles pour une classe dʼéquations de dérive-diffusion, avec données initiales $L^2$ et champ de vitesse incompressible appartenant à $L_t^{\infty }{\mathrm{𝐵𝑀𝑂}}_x^{-1}$. Nous appliquons ce résultat pour démontrer la régularité globale pour une famille dʼéquations du scalaire actif qui comprend lʼéquation dʼadvection–diffusion qui a été proposée par Moffatt dans le contexte de la turbulence magnétostrophique dans le noyau fluide de la Terre.

We use De Giorgi techniques to prove Hölder continuity of weak solutions to a class of drift-diffusion equations, with $L^2$ initial data and divergence free drift velocity that lies in $L_t^{\infty }{\mathrm{𝐵𝑀𝑂}}_x^{-1}$. We apply this result to prove global regularity for a family of active scalar equations which includes the advection–diffusion equation that has been proposed by Moffatt in the context of magnetostrophic turbulence in the Earthʼs fluid core.

Classification : 76D03, 35Q35, 76W05
Mots clés : Global regularity, Weak solutions, De Giorgi, Parabolic equations, Magneto-geostrophic equations

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