Dissipative continuous Euler flows

We show the existence of continuous periodic solutions of the 3D incompressible Euler equations which dissipate the total kinetic energy

Asymptotic stability of the Cauchy and Jensen functional equations

The aim of this note is to investigate the asymptotic stability behaviour of the Cauchy and Jensen functional equations. Our main results show that if these equations hold for large arguments with small error, then they are also valid everywhere with a new error term which is a constant multiple of the original error term. As consequences, we also obtain results of hyperstability character for these two functional equations

Weak Solutions to the Stationary Incompressible Euler Equations

We consider weak stationary solutions to the incompressible Euler equations and show that the analogue of the h-principle obtained in [5, 7] for time-dependent weak solutions continues to hold. The key difference arises in dimension d = 2, where it turns out that the relaxation is strictly smaller than what one obtains in the time-dependent case.Comment: 16 pages, 2 figures. Corrected a mistake in the proof of Theorem 17. Results unchanged. Corrected a typographical erro

h-Principles for the Incompressible Euler Equations

Recently, De Lellis and Sz\'ekelyhidi constructed H\"older continuous, dissipative (weak) solutions to the incompressible Euler equations in the torus $\mathbb T^3$. The construction consists in adding fast oscillations to the trivial solution. We extend this result by establishing optimal h-principles in two and three space dimensions. Specifically, we identify all subsolutions (defined in a suitable sense) which can be approximated in the $H^{-1}$-norm by exact solutions. Furthermore, we prove that the flows thus constructed on $\mathbb T^3$ are genuinely three-dimensional and are not trivially obtained from solutions on $\mathbb T^2$.Comment: 29 pages, no figure

Some functional equations related to the characterizations of information measures and their stability

The main purpose of this paper is to investigate the stability problem of some functional equations that appear in the characterization problem of information measures.Comment: 36 pages. arXiv admin note: text overlap with arXiv:1307.0657, arXiv:1307.0631, arXiv:1307.0664, arXiv:1307.065