Abstract: I will discuss two recent results on the weak–strong uniqueness principle for two-phase compressible viscous fluid systems with two different closure laws: algebraic and differential. For the algebraic closure, starting from the already available weak solutions and local-in-time strong solutions, we show that a suitable convexification of a parametrised, piecewise-defined relative entropy functional allows us to identify the two solutions on the lifespan of the strong solution. For the differential closure, corresponding to the Baer–Nunziato system, we first construct dissipative measure-valued solutions and compare them with the corresponding local-in-time strong solutions. We show that augmenting the relative entropy by a pressure-gap dissipation term arising from the closure relation allows us to establish the measure-valued–strong uniqueness principle. The result holds for arbitrary finite-energy initial data and adiabatic exponents \gamma^{\pm}>1. The main difficulties arise from the singular and discontinuous behaviour of the pressure-relaxation source as the volume fractions vanish, the associated concentration effects, and the lack of direct coercivity of the standard thermodynamic relative energy with respect to the volume fraction. These difficulties are overcome by an endpoint cutoff argument combined with a reversal of the usual order of approximation. As a byproduct, the so-called “no-pure-phase” restriction on the initial data can be removed. 

Abstract: In two papers, 20 years apart, Beckner proved two sharp inequalities for the Fourier transform each with a corresponding uncertainty principle. The inequalities in question were Hausdorff-Young and Pitt’s inequality, with the uncertainty principles being entropic and logarithmic.

In my talk, I will discuss the same story for a class of operators called metaplectic operators, of which examples include the Fourier transform, the quantum harmonic oscillator and the partial Fourier transform. The Hausdorff-Young inequality for such operators was previously considered by Giacchi, and the remaining story is recent work with Giacchi and Oliveira. Here, the goal is to obtain dynamical uncertainty principles for Schrödinger-type evolution problems. Such a discussion is dependent on the symplectic geometry of the metaplectic operators which separates the results into natural cases. 

Abstract: The study of high-dimensional maximal functions began with Stein’s work in the 1980s. Subsequently, dimension-free estimates for maximal operators associated with convex bodies were studied by Bourgain, Carbery, and Müller. In this talk, we will discuss the asymptotic behavior of the Lp(Rd) norms of these operators and the corresponding extremizers as d → ∞. In particular, we will see that, for each 1 < p < ∞, the operator norms of maximal operators associated with radial log concave densities converge uniformly to a universal limit λ(p). Moreover, the Euclidean ball minimizes the asymptotic operator norm among all symmetric convex bodies. Our proof combines a multiplier-to-maximal transference principle with thin-shell estimates from high dimensional convex geometry. The talk is based on joint work with Błażej Wróbel.

Abstract: TBA.