Boundary value problems for two dimensional steady incompressible fluids

On the Grad-Rubin boundary value problem for the two-dimensional magneto-hydrostatic equations

“…The reason to choose this manifold is the following: for Ω as in (1.4) we can choose the values of B ⋅ n in such a way that ∂Ω + ∩ ∂Ω − = ∅ and in particular we can guarantee that B ⋅n ≠ 0 at all points x ∈ ∂Ω. As it has been discussed in [2] at the points of the set ∂Ω + ∩ ∂Ω − some singular behaviour for B arise for generic domains Ω. In order to avoid the technical difficulties that should be considered in that situation, we will just work on the particular manifold (1.4).…”

“…It is worth to notice that several boundary value problems for the steady Euler or MHS equations have been considered in the literature [1,2,7,20,25,26]. We refer the interested reader to [2] for a thorough description of the currently available results considering the wellposedness of the different boundary value problems for the steady Euler or MHS equations.…”

“…It is worth to notice that there are several important differences regarding the problem treated here and previous works [1,2,7,20,25,26]. For a more detailed description of the different boundary value problems mentioned previously, we refer the reader to [2]. In the case treated by Alber [1] for the steady incompressible Euler equation, the vorticity ω 0 (or current j 0 in our case) on ∂Ω − can be readily obtained from the boundary values given in the problem, so roughly speaking ω 0 (or j 0 ) on ∂Ω − is already prescribed.…”

“…A fixed point argument for the operator Γ(⋅) solves the problem (1.8), (1.9), (1.10), (1.11) and (1.13). Using now the fact that ∇ × (j × B) = B ⋅ ∇j = 0 one can show, arguing as in [2], that there exists a pressure function p such that (B, p) satisfies (1.1) and (1.7).…”

“…It is worth to notice that there are several important differences regarding the problem treated here and previous works [1,2,7,20,25,26]. For a more detailed description of the different boundary value problems mentioned previously, we refer the reader to [2].…”