Ill-posedness of Leray solutions for the ipodissipative Navier-Stokes equations

We prove the ill-posedness of Leray solutions to the Cauchy problem for the ipodissipative Navier--Stokes equations, when the dissipative term is a fractional Laplacian $(-\Delta)^\alpha$ with exponent $\alpha < \frac{1}{5}$. The proof follows the ''convex integration methods'' introduced by the second author and L\'aszl\'o Sz\'ekelyhidi Jr. for the incomprresible Euler equations. The methods yield indeed some conclusions even for exponents in the range $[\frac{1}{5}, \frac{1}{2}[$.Comment: arXiv admin note: text overlap with arXiv:1302.281

On the lower semicontinuous envelope of functionals defined on polyhedral chains

In this note we prove an explicit formula for the lower semicontinuous envelope of some functionals defined on real polyhedral chains. More precisely, denoting by $H \colon \mathbb{R} \to \left[ 0,\infty \right)$ an even, subadditive, and lower semicontinuous function with $H(0)=0$, and by $\Phi_H$ the functional induced by $H$ on polyhedral $m$-chains, namely \Phi_{H}(P) := \sum_{i=1}^{N} H(\theta_{i}) \mathcal{H}^{m}(\sigma_{i}), \quad\mbox{for every }P=\sum_{i=1}^{N} \theta_{i} [[ \sigma_{i} ]] \in\mathbf{P}_m(\mathbb{R}^n), we prove that the lower semicontinuous envelope of $\Phi_H$ coincides on rectifiable $m$-currents with the $H$-mass \mathbb{M}_{H}(R) := \int_E H(\theta(x)) \, d\mathcal{H}^m(x) \quad \mbox{ for every } R= [[ E,\tau,\theta ]] \in \mathbf{R}_{m}(\mathbb{R}^{n}). Comment: 14 page

From ancient Assyria to European stages and screens

UID/HIS/04666/2019 Copyright Year 2020“Adieu, Assyria! / I loved thee well”. These were the last words of king Sardanapalus, the last king of Assyria, according to Lord Byron. Throughout the centuries, Europe was confronted with the tragic story of Mesopotamia’s last monarch, a king more effeminate than a woman, a lascivious and idle man, a governor who loathed all expressions of militarism and war. But this story was no more than it proposed to be: a story, not history. Sardanapalus was not even real! The Greeks conceived him; artists, play writers, and cineastes preserved him. Through the imaginative minds of early Modern and Modern historians, artists and dramaturgs, Sardanapalus’ legend endured well into the 20th-century in several different media. Even after the first excavations in Assyria, and the exhumation of its historical archives, where no king by the name of Sardanapalus was recorded, fantasy continued to surpass history.authorsversionpublishe