78 research outputs found
On the stability in weak topology of the set of global solutions to the Navier-Stokes equations
Let 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 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
On the lack of compactness in the 2D critical Sobolev embedding
This paper is devoted to the description of the lack of compactness of
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
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 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based on an estimate far away from the origin and which is no
longer valid in the general framework. Within the general framework of
, 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
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
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
Dispersive estimates for the Schr\"odinger operator on step 2 stratified Lie groups
The present paper is dedicated to the proof of dispersive estimates on
stratified Lie groups of step 2, for the linear Schr\"odinger equation
involving a sublaplacian. It turns out that the propagator behaves like a wave
operator on a space of the same dimension p as the center of the group, and
like a Schr\"odinger operator on a space of the same dimension k as the radical
of the canonical skew-symmetric form, which suggests a decay with exponant
-(k+p-1)/2. In this article, we identify a property of the canonical
skew-symmetric form under which we establish optimal dispersive estimates with
this rate. The relevance of this property is discussed through several
examples
- …