We consider a long-time behavior of a stochastically forced nonlinear oscillator. In a long-time limit the force converges to fractional Brownian motion, a process that has memory. In contrast, we show that the  limit of the nonlinear oscillator driven by this force converges to diffusion driven by standard (not fractional) Brownian motion, and thus retains…

We will discuss some old and new results concerning the long-time behavior of solutions to the two-dimensional incompressible Euler equations. Specifically, we discuss whether steady states can be isolated, wandering for solutions starting nearby certain steady states, singularity formation at infinite time, and finally some results/conjectures on the infinite-time limit near and far from equilibrium.…

The so-called $l_0$ pseudonorm counts the number of nonzero components of a vector. It is standard in sparse optimization problems. However, as it is a discontinuous and nonconvex function, the l0 pseudonorm cannot be satisfactorily handled with the Fenchel conjugacy. In this talk, we review a series of recent results on a class of Capra…

During the preparation of a foundational chapter on manifolds of mappings for a book on geometric continuum mechanics I found out that the following object behaves surprisingly well as source of a manifold of mappings: — A Whitney manifold germ M˜ ⊃ M consists of an open manifold M˜ together with a closed subset M…

In this talk, I will first present a very simple quantitative form of the Young-Fenchel inequality.  I will then discuss some applications: a short proof of the Brøndsted-Rockafellar in Hilbert spaces and a primal-dual attainment for perturbed convex minimization problems. I will finally explain how this inequality (or some generalizations) can be used for quantitative…