Steady three-dimensional ideal flows with nonvanishing vorticity in domains with edges

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

“…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].…”

Boundary value problems for two dimensional steady incompressible fluids

“…Extensions of Alber's results to compressible flows with non-smooth domains have been obtained in [18]. Solutions to the three dimensional steady Euler equation with boundaries meeting at right angles have been constructed in [21]. An illustrative example of this situation are curved pipes domains.…”

“…The Grad-Shafranov method allows to solve the boundary value problems (A),(D) and (G). The case (A) has been already considered in [3,22]. In this paper we will discuss the application of the Grad-Shafranov approach in Section 2.…”

“…In the case of the boundary value problem (A), the construction of solutions to the Euler equation (1.1) reduces also to the study of an elliptic equation which can be treated with classical calculus of variations tools. This case has been studied before in the literature in [3,22] and we would not provide more details here.…”