Random data Cauchy theory for supercritical wave equations II : A global existence result

We prove that the subquartic wave equation on the three dimensional ball $\Theta$, with Dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in $\cap_{s<1/2} H^s(\Theta)$. We obtain this result as a consequence of a general random data Cauchy theory for supercritical wave equations developed in our previous work \cite{BT2} and invariant measure considerations which allow us to obtain also precise large time dynamical informations on our solutions

Global well-posedness of the KP-I initial-value problem in the energy space

We prove that the KP-I initial value problem is globally well-posed in the natural energy space of the equation

Quantum Zakharov Model in a Bounded Domain

We consider an initial boundary value problem for a quantum version of the Zakharov system arising in plasma physics. We prove the global well-posedness of this problem in some Sobolev type classes and study properties of solutions. This result confirms the conclusion recently made in physical literature concerning the absence of collapse in the quantum Langmuir waves. In the dissipative case the existence of a finite dimensional global attractor is established and regularity properties of this attractor are studied. For this we use the recently developed method of quasi-stability estimates. In the case when external loads are $C^\infty$ functions we show that every trajectory from the attractor is $C^\infty$ both in time and spatial variables. This can be interpret as the absence of sharp coherent structures in the limiting dynamics.Comment: 27 page

The universal Glivenko-Cantelli property

Let F be a separable uniformly bounded family of measurable functions on a standard measurable space, and let N_{[]}(F,\epsilon,\mu) be the smallest number of \epsilon-brackets in L^1(\mu) needed to cover F. The following are equivalent: 1. F is a universal Glivenko-Cantelli class. 2. N_{[]}(F,\epsilon,\mu)0 and every probability measure \mu. 3. F is totally bounded in L^1(\mu) for every probability measure \mu. 4. F does not contain a Boolean \sigma-independent sequence. It follows that universal Glivenko-Cantelli classes are uniformity classes for general sequences of almost surely convergent random measures.Comment: 26 page

Long-Time Behavior of Macroscopic Quantum Systems: Commentary Accompanying the English Translation of John von Neumann's 1929 Article on the Quantum Ergodic Theorem

The renewed interest in the foundations of quantum statistical mechanics in recent years has led us to study John von Neumann's 1929 article on the quantum ergodic theorem. We have found this almost forgotten article, which until now has been available only in German, to be a treasure chest, and to be much misunderstood. In it, von Neumann studied the long-time behavior of macroscopic quantum systems. While one of the two theorems announced in his title, the one he calls the "quantum H-theorem", is actually a much weaker statement than Boltzmann's classical H-theorem, the other theorem, which he calls the "quantum ergodic theorem", is a beautiful and very non-trivial result. It expresses a fact we call "normal typicality" and can be summarized as follows: For a "typical" finite family of commuting macroscopic observables, every initial wave function $\psi_0$ from a micro-canonical energy shell so evolves that for most times $t$ in the long run, the joint probability distribution of these observables obtained from $\psi_t$ is close to their micro-canonical distribution.Comment: 34 pages LaTeX, no figures; v2: minor improvements and additions. The English translation of von Neumann's article is available as arXiv:1003.213

Orbital stability: analysis meets geometry

We present an introduction to the orbital stability of relative equilibria of Hamiltonian dynamical systems on (finite and infinite dimensional) Banach spaces. A convenient formulation of the theory of Hamiltonian dynamics with symmetry and the corresponding momentum maps is proposed that allows us to highlight the interplay between (symplectic) geometry and (functional) analysis in the proofs of orbital stability of relative equilibria via the so-called energy-momentum method. The theory is illustrated with examples from finite dimensional systems, as well as from Hamiltonian PDE's, such as solitons, standing and plane waves for the nonlinear Schr{\"o}dinger equation, for the wave equation, and for the Manakov system

Difficult intubation and extubation in adult anaesthesia.

To provide an update to French guidelines about "Difficult intubation and extubation in adult anaesthesia 2006". A consensus committee of 13 experts was convened. A formal conflict-of-interest (COI) policy was developed at the onset of the process and enforced throughout. The entire guidelines process was conducted independent of any industry funding. The authors were advised to follow the principles of the Grading of Recommendations Assessment, Development and Evaluation (GRADE) system to guide assessment of quality of evidence. The potential drawbacks of making strong recommendations in the presence of low-quality evidence were emphasized. Few recommendations were ungraded. The panel focused on 6 questions: 1) Why must oxygen desaturation be avoided during intubation and what preoxygenation and oxygenation techniques should be used to prevent it? 2) Should videolaryngoscopes be used instead of standard laryngoscopy with or without a long stylet to achieve a better success rate of intubation after the first attempt during anticipated difficult intubation off fiberoptic intubation? 3) Should TCI or target controlled inhalation anaesthesia (TCIA) be used instead of bolus sedation for airway control in the event of suspected or proven difficulty in a patient spontaneously breathing? 4) What mode of anaesthesia should be performed in patients with difficult intubation criteria and potentially difficult mask ventilation? 5) In surgical patients, what criteria predict difficulties encountered during postoperative tracheal extubation? 6) Should decision trees and algorithms be employed to direct decision-making for the management of difficult intubation, whether foreseen or not? (based on the information from the preceding five issues). Population, intervention, comparison, and outcomes (PICO) questions were reviewed and updated as needed, and evidence profiles were generated. The analysis of the literature and the recommendations were then conducted according to the GRADE &lt;sup&gt;®&lt;/sup&gt; methodology. The SFAR Guideline panel provided 13 statements on difficult intubation and extubation in adult anaesthesia. After two rounds of discussion and various amendments, a strong agreement was reached for 99% of recommendations. Of these recommendations, five have a high level of evidence (Grade 1±), 8 have a low level of evidence (Grade 2±). No recommendation was provided for one question. Substantial agreement exists among experts regarding many strong recommendations for the best care of patients with difficult intubation and extubation in adult anaesthesia

Global well-posedness for a coupled modified kdv system

We prove the sharp global well-posedness result for the initial value problem (IVP) associated to the system of the modi ed Korteweg-de Vries (mKdV) equation. For the single mKdV equation such result has been obtained by using Mirura's Transform that takes the KdV equation to the mKdV equation [8]. We do not know the existence of Miura's Transform that takes a KdV system to the system we are considering. To overcome this di culty we developed a new proof of the sharp global well-posedness result for the single mKdV equation without using Miura's Transform. We could successfully apply this technique in the case of the mKdV system to obtain the desired result.Fundação para a Ciência e a Tecnologia (FCT

Liquid crystals and their defects

These lecture notes discuss classical models of liquid crystals, and the different ways in which defects are described according to the different models.Comment: CIME lecture course, Cetraro, 201