RESEARCH TALKS
Gianni Arioli (Polytechnic University of Milan, Italy)
Epochs of irregularity of Leray-Hopf solutions to 3D Navier-Stokes equations
Abstract
We study global Leray-Hopf solutions to Cauchy problems for the 3D Navier-Stokes equations in a cube under Navier boundary conditions. Since the pioneering work by Jean Leray it is known that finite energy solutions exist for any initial data, but it is not known yet whether their enstrophy may blow up in finite time in the so-called epochs of irregularity. We present computer assisted results which bring strong evidence to the fact that the enstrophy blow-up may indeed occur in finite time.
Nicolas Brisebarre (Ecole Normale Superieure de Lyon, France)
Series Expansions in Classical Orthogonal Polynomials for Solutions of Linear Differential Equations
Abstract
We consider series expansions in bases of classical orthogonal polynomials. When such a series solves a linear differential equation with polynomial coefficients, its coefficients satisfy a linear recurrence equation. Thanks to a framework due to Ore, we interpret this equation as the numerator of a fraction of linear recurrence operators. This interpretation lets us give a simple and unified view of previous algorithms for computing a suitable recurrence relation. (This is joint work with Alexandre Benoit and Bruno Salvy).
Florent Brehard (Ecole Polytechnique Lille, France)
Validated Numerics for Multiple Root-Finding of Univariate and Bivariate Polynomials
Abstract
A posteriori validation methods based on fixed-point theorems are used extensively in the field of validated numerics. In most cases, the problem is locally invertible around the solution and tight error bounds can be obtained by using a Newton(-like) operator. For singular (and thus degenerate) problems, however, one typically needs to solve overdetermined systems rigorously, which admit no solution under generic, arbitrarily small perturbations. Therefore, even prope