Regularity results for a model in magnetohydrodynamics with imposed pressure

Poirier, Julien; Seloula, Nour

Comptes Rendus. Mathématique, Tome 358 (2020) no. 9-10, pp. 1033-1043.

La plupart des travaux sur le système de la magnétohydrodynamique (MHD) considèrent une condition aux limites de type Dirichlet pour le champ de vitesses. Dans cette Note, nous étudions le système (MHD) avec une pression donnée au bord, ainsi qu’une trace tangentielle nulle pour la vitesse du fluide et le champ magnétique. Dans un ouvert borné tridimensionnel, éventuellement multiplement connexe, on commence par prouver l’existence de solutions faibles dans le cas Hilbertien, et ensuite, nous montrons la régularité $W^{1,p}\left(Ø\right)$ pour $p\ge 2$ et $W^{2,p}\left(Ø\right)$ pour $p\ge 6/5$ en utilisant les résultats de régularité pour certains problèmes de Stokes avec ce type de conditions aux limites. De plus, pour des données petites, nous démontrons l’existence et l’unicité des solutions dans $W^{1,p}\left(Ø\right)$ pour $3/2<p<2$ en utilisant un théorème de point fixe appliqué au problème linéarisé de (MHD).

The magnetohydrodynamics (MHD) problem is most often studied in a framework where Dirichlet type boundary conditions on the velocity field is imposed. In this Note, we study the (MHD) system with pressure boundary condition, together with zero tangential trace for the velocity and the magnetic field. In a three-dimensional bounded possibly multiply connected domain, we first prove the existence of weak solutions in the Hilbert case, and later, the regularity in $W^{1,p}\left(Ø\right)$ for $p\ge 2$ and in $W^{2,p}\left(Ø\right)$ for $p\ge 6/5$ using the regularity results for some Stokes and elliptic problems with this type of boundary conditions. Furthermore, under the condition of small data, we obtain the existence and uniqueness of solutions in $W^{1,p}\left(Ø\right)$ for $3/2<p<2$ by using a fixed-point technique over a linearized (MHD) problem.

Classification : 35J60, 35Q35, 35Q60

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