Mathematical Analysis Seminars @ DISIM

Title: On the topology of the magnetic lines of large solutions to the Magnetohydrodynamic equations in R^3 

Abstract: The purpose of this talk is to present the article [1], in which we establish two results: first, we introduce a new class of global strong solutions to the magnetohydrodynamic system in R^3 with initial data (u_0,b_0) of arbitrarily large size in any critical space. To do so, we impose a smallness condition on the difference u_0-b_0. Then we use this result to prove magnetic reconnection for a suitable class of (large) solutions. With this, we mean a change of topology of the integral lines of the magnetic field b under the evolution. The proof relies on counting the number of hyperbolic critical points of the solutions, and this instance is structurally stable.

[1] R. Lucà, C. Peña: "On the topology of the magnetic lines of large solutions to the Magnetohydrodynamic equations in R^3". Preprint: arXiv:2505.09340.

Title: H-compactness for nonlocal linear operators in fractional divergence form

Abstract: In this seminar we address the problem of the H-convergence of nonlocal linear operators in fractional divergence form, where the oscillations of the matrices are prescribed outside the reference domain. Our compactness argument bypasses the failure of the classical localisation techniques that mismatch with the nonlocal nature of the operators involved. 

If symmetry is also assumed, we extend the equivalence between the H-convergence of the operators and the $\Gamma$-convergence of the associated energies. If time allows we will talk about further generalizations. The seminar is based on a joint work with M. Caponi (Univaq) and A. Maione (CRM).