Theses
THESIS DEFENSE -- Analysis of weak solutions of the Euler equations and magnetohydrodynamics
Speaker: Javier Peñafiel Tomás ()Advisor/s: Alberto Enciso Carrasco (ICMAT), Daniel Peralta Salas (ICMAT)Date: Thursday, 02 July 2026 - 11:00Place: Aula Naranja, ICMAT
Abstract:
The goal of this thesis is to study the properties of weak solutions of the Euler equations and ideal magnetohydrodynamics (MHD), with special emphasis on their qualitative behavior. The main technique used is convex integration, a method that has been highly successful in fluid mechanics over the past fifteen years. The first problem we address is the extension of smooth solutions of the three-dimensional Euler equations on bounded domains to weak solutions on R^3, without altering the solution inside the original domain. We develop techniques that provide fine local control of the solution and that can also be used to construct examples exhibiting pathological behavior. We then introduce a novel convex integration scheme that preserves the topological properties of the fields throughout the iteration process. Using this scheme, we construct weak solutions of the steady Euler equations with any prescribed streamline topology. As an application, we produce low-regularity counterexamples to a conjecture posed by Grad. We further extend these methods to ideal MHD, which allows us to construct Holder-continuous solutions that dissipate the Elsässer energies while maintaining nonzero (constant) magnetic helicity.
