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

The nonlinear Schrödinger equation ground states on product spaces

We study the nature of the nonlinear Schrödinger equation ground states on the product spaces Rn x Mk , where Mk is a compact Riemannian manifold. We prove that for small L2 masses the ground states coincide with the corresponding Rn ground states. We also prove that above a critical mass the ground states have nontrivial Mk dependence. Finally, we address the Cauchy problem issue, which transforms the variational analysis into dynamical stability results

RELATION BETWEEN THE REACTION TIME AND THE AMPLITUDE CHANGES OF THE H-REFLEX IN CASE OF SUCCESSIVE PLANTAR FLEXION OF THE FEET PERFORMED UNDER DIFFERENT EXPERIMENTAL CONDITIONS

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Endoscopic Dacryocystorhinostomy – Our Experience

Objectives: The aim of this study is to present our experience with endoscopic management of chronic obstruction of the nasolacrimal duct.Matherials and methods: Retrospective analysis of the patients with chronic stenosis and obstruction of nasolacrimal product managed with endoscopic DCR. Patients’ demographics, management techniques, complications and outcomes, patient satisfaction are reported.Results: A total of 11 patients with stenosis or obstruction of the nasolacrimal duct were treated in our clinic. All patients were managed with endoscopic DCR. 8 of the patients were female and 3 – male. 9 of the patients had unilateral complaints and 2 – bilateral. The results showed patency of the operated nasolacrimal ducts in the absence of recurrence to date. There was one case of postoperative epistaxis. Other postoperative complications were not observed.Conclusions: Our experience shows that endoscopic management of chronic obstruction of the nasolacrimal duct is safe and effective, and this is in line with data reported by other authors

Quasi-invariant Gaussian measures for the nonlinear wave equation in three dimensions

We prove quasi-invariance of Gaussian measures supported on Sobolev spaces under the dynamics of the three-dimensional defocusing cubic nonlinear wave equation. As in the previous work on the two-dimensional case, we employ a simultaneous renormalization on the energy functional and its time derivative. Two new ingredients in the three-dimensional case are (i) the construction of the weighted Gaussian measures, based on a variational formula for the partition function inspired by Barashkov and Gubinelli (2018), and (ii) an improved argument in controlling the growth of the truncated weighted Gaussian measures, where we combine a deterministic growth bound of solutions with stochastic estimates on random distributions.Comment: 35 pages. We now prove the full quasi-invariance resul

PHARMACOGENETIC STUDY OF THE ACETYLATION PHENOTYPE IN A BULGARIAN POPULATION

N-acetyltransferase, an enzyme involved in the metabolic inactivation of drugs like isoniazide, some sulfonamides and others is well-known to he under polymorphic genetic control. The acetylation phenotype of the patients may serve as an important guide in foretelling the therapeutic efficacy or tolerahility of a particular drug. In the present study we investigated the distribution of the acetylaiion phenotypes in a group of 100 healthy volunteers of both sexes using sulfadimidine as a substrate. The distribution was found to follow a bimodal pattern, as aspected, with a slight predominance of the "slow" acetylators - in 58 % of the cases, a finding similar to literature data from neighbouring and other European countries. In the men's group the distribution was approximately the same as that in the whole group whilst in the women's one the "rapid" inactivators prevailed. This work represents the first modest attempt in Bulgaria for phenotyping the population according to the individual acetylaiion status

Invariance of the white noise for KdV

We prove the invariance of the mean 0 white noise for the periodic KdV. First, we show that the Besov-type space \hat{b}^s_{p, \infty}, sp <-1, contains the support of the white noise. Then, we prove local well-posedness in \hat{b}^s_{p, \infty} for p= 2+, s = -{1/2}+ such that sp <-1. In establishing the local well-posedness, we use a variant of the Bourgain spaces with a weight. This provides an analytical proof of the invariance of the white noise under the flow of KdV obtained in Quastel-Valko.Comment: 18 pages. To appear in Comm. Math. Phy

Random data wave equations

Nowadays we have many methods allowing to exploit the regularising properties of the linear part of a nonlinear dispersive equation (such as the KdV equation, the nonlinear wave or the nonlinear Schroedinger equations) in order to prove well-posedness in low regularity Sobolev spaces. By well-posedness in low regularity Sobolev spaces we mean that less regularity than the one imposed by the energy methods is required (the energy methods do not exploit the dispersive properties of the linear part of the equation). In many cases these methods to prove well-posedness in low regularity Sobolev spaces lead to optimal results in terms of the regularity of the initial data. By optimal we mean that if one requires slightly less regularity then the corresponding Cauchy problem becomes ill-posed in the Hadamard sense. We call the Sobolev spaces in which these ill-posedness results hold spaces of supercritical regularity. More recently, methods to prove probabilistic well-posedness in Sobolev spaces of supercritical regularity were developed. More precisely, by probabilistic well-posedness we mean that one endows the corresponding Sobolev space of supercritical regularity with a non degenerate probability measure and then one shows that almost surely with respect to this measure one can define a (unique) global flow. However, in most of the cases when the methods to prove probabilistic well-posedness apply, there is no information about the measure transported by the flow. Very recently, a method to prove that the transported measure is absolutely continuous with respect to the initial measure was developed. In such a situation, we have a measure which is quasi-invariant under the corresponding flow. The aim of these lectures is to present all of the above described developments in the context of the nonlinear wave equation.Comment: Lecture notes based on a course given at a CIME summer school in August 201

The phase shift of line solitons for the KP-II equation

The KP-II equation was derived by [B. B. Kadomtsev and V. I. Petviashvili,Sov. Phys. Dokl. vol.15 (1970), 539-541] to explain stability of line solitary waves of shallow water. Stability of line solitons has been proved by [T. Mizumachi, Mem. of vol. 238 (2015), no.1125] and [T. Mizumachi, Proc. Roy. Soc. Edinburgh Sect. A. vol.148 (2018), 149--198]. It turns out the local phase shift of modulating line solitons are not uniform in the transverse direction. In this paper, we obtain the $L^\infty$-bound for the local phase shift of modulating line solitons for polynomially localized perturbations

Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation

The aim of this paper is the accurate numerical study of the KP equation. In particular we are concerned with the small dispersion limit of this model, where no comprehensive analytical description exists so far. To this end we first study a similar highly oscillatory regime for asymptotically small solutions, which can be described via the Davey-Stewartson system. In a second step we investigate numerically the small dispersion limit of the KP model in the case of large amplitudes. Similarities and differences to the much better studied Korteweg-de Vries situation are discussed as well as the dependence of the limit on the additional transverse coordinate.Comment: 39 pages, 36 figures (high resolution figures at http://www.mis.mpg.de/preprints/index.html