OxPDE Lunchtime Seminar

Sard’s theorem asserts that the set of critical values of a smooth map from one Euclidean space to another one has measure zero. A version of this result for infinite-dimensional Banach manifolds was proven by Smale for maps with Fredholm differential. However, when the domain is infinite dimensional and the range is finite dimensional, the result is not true – even under the assumption that the map is “polynomial” – and a general theory is still lacking. In this seminar, I will provide sharp quantitative criteria for the validity of Sard’s theorem in this setting, obtained combining a functional analysis approach with new tools in semialgebraic geometry. As an application, I will present new results on the Sard conjecture in sub-Riemannian geometry. Based on a joint work with A. Lerario and L. Rizzi.

The Alfvén waves are fundamental wave phenomena in magnetized plasmas. Mathematically, the dynamics of Alfvén waves are governed by a system of nonlinear partial differential equations called the magnetohydrodynamics (MHD) equations. Let us introduce some recent results about inverse scattering of Alfvén waves in ideal MHD, which are intended to establish the relationship between Alfvén waves emanating from the plasma and their scattering fields at infinities.The proof is mainly based on the weighted energy estimates. Moreover, the null structure inherent in MHD equations is thoroughly examined, especially when we estimate the pressure term.

Several physical models are available to understand the dynamics of fluid mixtures, including the so-called Baer-Nunziato models. The partial differential equations associated with these models look like those of Navier-Stokes, with the addition of new relaxation terms. One strategy to obtain these models is homogenisation: starting from a mesoscopic mixture, where two pure fluids satisfying the compressible Navier-Stokes equations share the space between them, a change of scale is performed to obtain a macroscopic mixture, where the two fluids can coexist at any point in space.

This problem concerns the study of the Navier-Stokes equations with strongly oscillating initial data. We'll start by explaining some results in this framework, in one dimension of space and on the torus, for barotropic fluids. We will then detail the various steps involved in demonstrating homogenisation. Finally, we'll explain how to adapt this reasoning to homogenisation for perfect gases, with and without heat conduction.