Inviscid incompressible limits of the full Navier-Stokes-Fourier system

We consider the full Navier-Stokes-Fourier system in the singular limit for the small Mach and large Reynolds and Peclet numbers, with ill prepared initial data on the three dimensional Euclidean space. The Euler-Boussinesq approximation is identified as the limit system

Birkhoff Normal form for Gravity Water Waves

We consider the gravity water waves system with a one-dimensional periodic interface in infinite depth, and present the proof of the rigorous reduction of these equations to their cubic Birkhoff normal form (Berti et al. in Birkhoff normal form and long-time existence for periodic gravity Water Waves. arXiv:1810.11549, 2018). This confirms a conjecture of Zakharov\u2013Dyachenko (Phys Lett A 190:144\u2013148, 1994) based on the formal Birkhoff integrability of the water waves Hamiltonian truncated at degree four. As a consequence, we also obtain a long-time stability result: periodic perturbations of a flat interface that are of size \u3b5 in a sufficiently smooth Sobolev space lead to solutions that remain regular and small up to times of order \u3b5 123

Time quasi-periodic gravity water waves in finite depth

We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions\u2014namely periodic and even in the space variable x\u2014of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the fully nonlinear nature of the gravity water waves equations\u2014the highest order x-derivative appears in the nonlinear term but not in the linearization at the origin\u2014and the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these problems by first reducing the linearized operators, obtained at each approximate quasi-periodic solution along a Nash\u2013Moser iterative scheme, to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme which requires very weak Melnikov non-resonance conditions which lose derivatives both in time and space. Despite the fact that the depth parameter moves the linear frequencies by just exponentially small quantities, we are able to verify such non-resonance conditions for most values of the depth, extending degenerate KAM theory

Stability with respect to domain of the low Mach number limit of compressible viscous fluids

We study the asymptotic limit of solutions to the barotropic Navier-Stokes system, when the Mach number is proportional to a small parameter \ep \to 0 and the fluid is confined to an exterior spatial domain \Omega_\ep that may vary with \ep. As $\epsilon \rightarrow 0$, it is shown that the fluid density becomes constant while the velocity converges to a solenoidal vector field satisfying the incompressible Navier-Stokes equations on a limit domain. The velocities approach the limit strongly (a.a.) on any compact set, uniformly with respect to a certain class of domains. The proof is based on spectral analysis of the associated wave propagator (Neumann Laplacian) governing the motion of acoustic waves.Comment: 32 page

Cross standard form : a solution to improve a given controller with H2 or Hoo specifications

This paper introduces in cross standard form (CSF) as a solution to the inverse optimal control problem. That is, the CSF is a canonical standard problem whose unique H1 or H2 optimal controller is a given controller. From the control design point of view, the general idea is to apply the CSF to a given controller in order to set up a standard problem which can be completed to handle frequency domain H2 or H1 specification. The analytical formulation of the CSF proposed in this paper can be applied to reduced-, full- or augmented-order compensators or two-degree of freedom compensations. Numerical and academic examples are given

An algorithmic reduction theory for binary codes: LLL and more

In this article, we propose an adaptation of the algorithmic reduction theory of lattices to binary codes. This includes the celebrated LLL algorithm (Lenstra, Lenstra, Lovasz, 1982), as well as adaptations of associated algorithms such as the Nearest Plane Algorithm of Babai (1986). Interestingly, the adaptation of LLL to binary codes can be interpreted as an algorithmic version of the bound of Griesmer (1960) on the minimal distance of a code. Using these algorithms, we demonstrate —both with a heuristic analysis and in practice— a small polynomial speed-up over the Information-Set Decoding algorithm of Lee and Brickell (1988) for random binary codes. This appears to be the first such speed-up that is not based on a time-memory trade-off. The above speed-up should be read as a very preliminary example of the potential of a reduction theory for codes, for example in cryptanalysis

Wave: A New Family of Trapdoor One-Way Preimage Sampleable Functions Based on Codes

We present here a new family of trapdoor one-way Preimage Sampleable Functions (PSF) based on codes, the Wave-PSF family. The trapdoor function is one-way under two computational assumptions: the hardness of generic decoding for high weights and the indistinguishability of generalized $(U,U+V)$-codes. Our proof follows the GPV strategy [GPV08]. By including rejection sampling, we ensure the proper distribution for the trapdoor inverse output. The domain sampling property of our family is ensured by using and proving a variant of the left-over hash lemma. We instantiate the new Wave-PSF family with ternary generalized $(U,U+V)$-codes to design a "hash-and-sign" signature scheme which achieves existential unforgeability under adaptive chosen message attacks (EUF-CMA) in the random oracle model. For 128 bits of classical security, signature sizes are in the order of 15 thousand bits, the public key size in the order of 4 megabytes, and the rejection rate is limited to one rejection every 10 to 12 signatures.Comment: arXiv admin note: text overlap with arXiv:1706.0806