On the ill-posedness result for the BBM equation

We prove that the initial value problem (IVP) for the BBM equation is ill-posed for data in Hs(R), s < 0 in the sense that the ow-map u0 7! u(t) that associates to initial data u0 the solution u cannot be continuous at the origin from Hs(R) to even D0(R) at any _xed t > 0 small enough. This result is sharp.Fundação para a Ciência e a Tecnologia (FCT

A para-differential renormalization technique for nonlinear dispersive equations

For \alpha \in (1,2) we prove that the initial-value problem \partial_t u+D^\alpha\partial_x u+\partial_x(u^2/2)=0 on \mathbb{R}_x\times\mathbb{R}_t; u(0)=\phi, is globally well-posed in the space of real-valued L^2-functions. We use a frequency dependent renormalization method to control the strong low-high frequency interactions.Comment: 42 pages, no figure

Global well-posedness for the KP-I equation on the background of a non localized solution

We prove that the Cauchy problem for the KP-I equation is globally well-posed for initial data which are localized perturbations (of arbitrary size) of a non-localized (i.e. not decaying in all directions) traveling wave solution (e.g. the KdV line solitary wave or the Zaitsev solitary waves which are localized in $x$ and $y$ periodic or conversely)

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

Integral representation of the linear Boltzmann operator for granular gas dynamics with applications

We investigate the properties of the collision operator associated to the linear Boltzmann equation for dissipative hard-spheres arising in granular gas dynamics. We establish that, as in the case of non-dissipative interactions, the gain collision operator is an integral operator whose kernel is made explicit. One deduces from this result a complete picture of the spectrum of the collision operator in an Hilbert space setting, generalizing results from T. Carleman to granular gases. In the same way, we obtain from this integral representation of the gain operator that the semigroup in L^1(\R \times \R,\d \x \otimes \d\v) associated to the linear Boltzmann equation for dissipative hard spheres is honest generalizing known results from the first author.Comment: 19 pages, to appear in Journal of Statistical Physic

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

Are Alexandrium catenella Blooms Spreading Offshore in Southern Chile? An In-Depth Analysis of the First PSP Outbreak in the Oceanic Coast

The blooms of Alexandrium catenella, the main producer of paralytic shellfish toxins worldwide, have become the main threat to coastal activities in Southern Chile, such as artisanal fisheries, aquaculture and public health. Here, we explore retrospective data from an intense Paralytic Shellfish Poisoning outbreak in Southern Chile in Summer&ndash;Autumn 2016, identifying environmental drivers, spatiotemporal dynamics, and detoxification rates of the main filter-feeder shellfish resources during an intense A. catenella bloom, which led to the greatest socio-economic impacts in that area. Exponential detoxification models evidenced large differences in detoxification dynamics between the three filter-feeder species surf clam (Ensis macha), giant barnacle (Austromegabalanus psittacus), and red sea squirt (Pyura chilensis). Surf clam showed an initial toxicity (9054 &micro;g STX-eq&middot;100 g&minus;1) around 10-fold higher than the other two species. It exhibited a relatively fast detoxification rate and approached the human safety limit of 80 &micro;g STX-eq&middot;100 g&minus;1 towards the end of the 150 days. Ecological implications and future trends are also discussed. Based on the cell density evolution, data previously gathered on the area, and the biology of this species, we propose that the bloom originated in the coastal area, spreading offshore thanks to the resting cysts formed and transported in the water column

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

Prenatal diagnosis of Kagami-Ogata Syndrome

Kagami-Ogata syndrome (KOS14) is a rare congenital disorder associated with defective genomic imprinting of the chromosome 14q32 domain. Typical features include polyhydramnios, small and bell-shaped thorax, coat-hanger ribs, dysmorphic facial features, abdominal wall defects, placentomegaly, severe postnatal respiratory distress and intellectual disability. To the best of our knowledge, this may be the first case where ultrasound findings such as: severe polyhydramnios, a small bell- shaped thorax, a protuberant abdomen and characteristic dysmorphic face prompted directed family interrogation finally leading to the prenatal diagnosis of KOS14

On the probabilistic Cauchy theory for nonlinear dispersive PDEs

In this note, we review some of the recent developments in the well-posedness theory of nonlinear dispersive partial differential equations with random initial data.Comment: 26 pages. To appear in Landscapes of Time-Frequency Analysis, Appl. Numer. Harmon. Ana