Haroune Houamed, NYU Abu Dhabi

Abstract: The analysis of the Euler and Navier-Stokes equations constitutes a highly active area of research. Beyond its significance as an independent problem, a comprehensive understanding of its aspects is pivotal for refining our comprehension of other models based on these equations.
The primary objective of my talk is to revisit a general stability analysis of the two-dimensional Euler equations within Yudovich’s class of solutions. As a motivational preamble, I will commence with a succinct introduction to certain plasma models, specifically the Navier-Stokes-Maxwell and Euler-Maxwell systems. I will quickly review recent advances in the analysis of these systems, explain their relevance to the classical MHD model, and shed light on some open questions. In a specific configuration to be precisely clarified, these models can be reformulated as perturbations of the (simple-decoupled) MHD model where the leading part is the Euler (or Navier-Stokes) equations.
This will lead us to the main second part of my talk -- analysis of perturbations of Euler’s equations. Thus, I will step over the key elements of our new approach to studying the stability of solutions in an endpoint setting, based on a simple extrapolation-compactness argument. To underscore its significance, I will delve into the details of this method and, furthermore, show how it can be employed to obtain an improvement on the rate of convergence in the inviscid limit of the Navier-Stokes equations within a subclass of Yudovich.
The content of my talk is based on recent joint work with Diogo Arsénio from NYU Abu Dhabi.