arXiv subjects
Publications and source records attributed to Daniel Sánchez-Simón del Pino.
We study steady solutions of the two dimensional incompressible Euler equations in the exterior of a compact domain. We assume that fluid enters the exterior domain through every point of the obstacle boundary. Under suitable assumptions on the direction and decay of the velocity at infinity, we prove that the flow is irrotational if and only if the velocity has no stagnation points.
In this article we study a one dimensional model for Magnetic Relaxation. This model was introduced by Moffatt and describes a low resistivity viscous plasma, in which the pressure and the inercia are much smaller than the magnetic pressure. In the limit of resistivity $\varepsilon\rightarrow 0$, we prove the existence of two time scales for the evolution of the magnetic field: a fast one for times of order $\log(\varepsilon^{-1})$ in which the resistivity plays no role and the energy is dissipated only via viscosity; and a slow one for times of order $\varepsilon^{-1}$ characterized by the influence of the resistivity. We show that in this second time scale, as $\varepsilon\rightarrow 0$, the modulus of magnetic field approaches a function that depends only on time. We also prove that, in this regime, the magnetic field $b_\varepsilon(t,x)$ can be approximated as $\varepsilon \rightarrow 0$ by the solution of a PDE whose solutions exhibit blow up for some choices of initial data.
In this paper we study the solvability of the magnetohydrostatic equations with Grad-Rubin boundary conditions in general domains. Earlier results for this problem were obtained in the recent years by D. Alonso-Orán and J. L. L. Velázquez, where particularly simple geometries were considered. In this article we develop a theory that allows to solve these boundary value problems for a larger class of domains. We will give precise applications to more physically relevant situations, like the case of the space between two circumferences or spheres and domains close to them.
In this work, we study the well-posedness of the three dimensional magneto-hydrostatic equation under Grad-Rubin boundary value conditions. The proof relies on a fixed point argument to construct solutions to an elliptic-hyperbolic problem in a perturbative regime by means of pseudo-differential operators with symbols with limited regularity in Hölder spaces. As a byproduct, the employed technique in this work is more flexible and simplifies the arguments of the proof for the previous two-dimensional setting.