Spiders (Araneae) of stony debris in North Bohemia

The arachnofauna was studied at five stony debris sites in northern Bohemia. In Central Europe, the northern and montane species inhabiting cold places live not only on mountain tops and peat bogs but also on the lower edges of boulder debris, where air streaming through the system of inner compartments gives rise to an exceedingly cold microclimate. At such cold sites, spiders can live either on bare stones (Bathyphantes simillimus, Wubanoides ura/ensis), or in the rich layers of moss and lichen (Dip/oeentria bidentata). Kratoehviliella bieapitata exhibits a diplostenoecious occurrence in stony debris and on tree bark. Latithorax faustus and Theonoe minutissima display diplostenoecious occurrence in stony debris and on peat bogs. The occurrence of the species Seotina eelans in the Czech Republic was documented for the first time

On the stability in weak topology of the set of global solutions to the Navier-Stokes equations

Let $X$ be a suitable function space and let \cG \subset X be the set of divergence free vector fields generating a global, smooth solution to the incompressible, homogeneous three dimensional Navier-Stokes equations. We prove that a sequence of divergence free vector fields converging in the sense of distributions to an element of \cG belongs to \cG if $n$ is large enough, provided the convergence holds "anisotropically" in frequency space. Typically that excludes self-similar type convergence. Anisotropy appears as an important qualitative feature in the analysis of the Navier-Stokes equations; it is also shown that initial data which does not belong to \cG (hence which produces a solution blowing up in finite time) cannot have a strong anisotropy in its frequency support.Comment: To appear in Archive for Rational and Mechanical Analysi

Lack of compactness in the 2D critical Sobolev embedding, the general case

This paper is devoted to the description of the lack of compactness of the Sobolev embedding of $H^1(\R^2)$ in the critical Orlicz space {\cL}(\R^2). It turns out that up to cores our result is expressed in terms of the concentration-type examples derived by J. Moser in \cite{M} as in the radial setting investigated in \cite{BMM}. However, the analysis we used in this work is strikingly different from the one conducted in the radial case which is based on an $L^ \infty$ estimate far away from the origin and which is no longer valid in the general framework. Within the general framework of $H^1(\R^2)$, the strategy we adopted to build the profile decomposition in terms of examples by Moser concentrated around cores is based on capacity arguments and relies on an extraction process of mass concentrations. The essential ingredient to extract cores consists in proving by contradiction that if the mass responsible for the lack of compactness of the Sobolev embedding in the Orlicz space is scattered, then the energy used would exceed that of the starting sequence.Comment: Submitte

On the lack of compactness in the 2D critical Sobolev embedding

This paper is devoted to the description of the lack of compactness of $H^1_{rad}(\R^2)$ in the Orlicz space. Our result is expressed in terms of the concentration-type examples derived by P. -L. Lions. The approach that we adopt to establish this characterization is completely different from the methods used in the study of the lack of compactness of Sobolev embedding in Lebesgue spaces and take into account the variational aspect of Orlicz spaces. We also investigate the feature of the solutions of non linear wave equation with exponential growth, where the Orlicz norm plays a decisive role.Comment: 38 page

Tempered distributions and Fourier transform on the Heisenberg group

The final goal of the present work is to extend the Fourier transform on the Heisenberg group \H^d, to tempered distributions. As in the Euclidean setting, the strategy is to first show that the Fourier transform is an isomorphism on the Schwartz space, then to define the extension by duality. The difficulty that is here encountered is that the Fourier transform of an integrable function on \H^dis no longer a function on \H^d : according to the standard definition, it is a family of bounded operators on $L^2(\R^d).$ Following our new approach in\ccite{bcdFHspace}, we here define the Fourier transform of an integrable functionto be a mapping on the set~\wt\H^d=\N^d\times\N^d\times\R\setminus\{0\}endowed with a suitable distance \wh d.This viewpoint turns out to provide a user friendly description of the range of the Schwartz space on \H^d by the Fourier transform, which makes the extension to the whole set of tempered distributions straightforward. As a first application, we give an explicit formula for the Fourier transform of smooth functions on \H^d that are independent of the vertical variable. We also provide other examples