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Regularity results for a model in magnetohydrodynamics with imposed pressure

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 $W^{1,p}(Ω)$ for $p\geq 2$ and in $W^{2,p}(Ω)$ for $p\geq 6/5$ 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 $W^{1,p}(Ω)$ for $3/2<p<2$ by using a fixed-point technique over a linearized (MHD) problem.