164 research outputs found
A general wavelet-based profile decomposition in the critical embedding of function spaces
We characterize the lack of compactness in the critical embedding of
functions spaces having similar scaling properties in the
following terms : a sequence bounded in has a subsequence
that can be expressed as a finite sum of translations and dilations of
functions such that the remainder converges to zero in as
the number of functions in the sum and tend to . Such a
decomposition was established by G\'erard for the embedding of the homogeneous
Sobolev space into the in dimensions with
, and then generalized by Jaffard to the case where is a Riesz
potential space, using wavelet expansions. In this paper, we revisit the
wavelet-based profile decomposition, in order to treat a larger range of
examples of critical embedding in a hopefully simplified way. In particular we
identify two generic properties on the spaces and that are of key use
in building the profile decomposition. These properties may then easily be
checked for typical choices of and satisfying critical embedding
properties. These includes Sobolev, Besov, Triebel-Lizorkin, Lorentz, H\"older
and BMO spaces.Comment: 24 page
Concentration analysis and cocompactness
Loss of compactness that occurs in may significant PDE settings can be
expressed in a well-structured form of profile decomposition for sequences.
Profile decompositions are formulated in relation to a triplet , where
and are Banach spaces, , and is, typically, a
set of surjective isometries on both and . A profile decomposition is a
representation of a bounded sequence in as a sum of elementary
concentrations of the form , , , and a remainder that
vanishes in . A necessary requirement for is, therefore, that any
sequence in that develops no -concentrations has a subsequence
convergent in the norm of . An imbedding with this
property is called -cocompact, a property weaker than, but related to,
compactness. We survey known cocompact imbeddings and their role in profile
decompositions
Analyse fréquentielle du signal
International audienceLes signaux sont prĂ©sents partout autour de nous. L'analyse du signal est au coeur des mathĂ©matiques appliquĂ©es et permet de dĂ©coder l'information. Un des principaux outils est l'analyse des frĂ©quences. Nous prĂ©sentons ici l'analyse frĂ©quentielle d'abord dans le cas de signaux pĂ©riodiques avant d'ouvrir quelques perspectives sur des cas plus gĂ©nĂ©raux. L'article contient de nombreuses images et animations pour illustrer les concepts de la façon la plus concrĂšte possible. L'article est auto-contenu et devraitĂȘtre accessible aux lycĂ©ens. La version interactive permet de jouer directement avec les paramĂštres des animations. Sauf mention contraire, nous avons produit chaque video, spĂ©cialement pour cet article
Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system
We describe the asymptotic behavior as time goes to infinity of solutions of
the 2 dimensional corotational wave map system and of solutions to the 4
dimensional, radially symmetric Yang-Mills equation, in the critical energy
space, with data of energy smaller than or equal to a harmonic map of minimal
energy. An alternative holds: either the data is the harmonic map and the
soltuion is constant in time, or the solution scatters in infinite time
WKB analysis for nonlinear Schr\"{o}dinger equations with potential
We justify the WKB analysis for the semiclassical nonlinear Schr\"{o}dinger
equation with a subquadratic potential. This concerns subcritical, critical,
and supercritical cases as far as the geometrical optics method is concerned.
In the supercritical case, this extends a previous result by E. Grenier; we
also have to restrict to nonlinearities which are defocusing and cubic at the
origin, but besides subquadratic potentials, we consider initial phases which
may be unbounded. For this, we construct solutions for some compressible Euler
equations with unbounded source term and unbounded initial velocity.Comment: 25 pages, 11pt, a4. Appendix withdrawn, due to some inconsistencie
Concerning the Wave equation on Asymptotically Euclidean Manifolds
We obtain KSS, Strichartz and certain weighted Strichartz estimate for the
wave equation on , , when metric
is non-trapping and approaches the Euclidean metric like with
. Using the KSS estimate, we prove almost global existence for
quadratically semilinear wave equations with small initial data for
and . Also, we establish the Strauss conjecture when the metric is radial
with for .Comment: Final version. To appear in Journal d'Analyse Mathematiqu
Global well-posedness for a Smoluchowski equation coupled with Navier-Stokes equations in 2D
We prove global existence for a nonlinear Smoluchowski equation (a nonlinear
Fokker-Planck equation) coupled with Navier-Stokes equations in two dimensions.
The proof uses a deteriorating regularity estimate and the tensorial structure
of the main nonlinear terms
Generalised GagliardoâNirenberg inequalities using weak Lebesgue spaces and BMO
Using elementary arguments based on the Fourier transform we prove that for
, if then and there
exists a constant such that
where . In
particular, in we obtain the generalised Ladyzhenskaya inequality
. We also
show that for the norm in can be replaced by the
norm in BMO. As well as giving relatively simple proofs of these inequalities,
this paper provides a brief primer of some basic concepts in harmonic analysis,
including weak spaces, the Fourier transform, the Lebesgue Differentiation
Theorem, and Calderon-Zygmund decompositions
Existence of global strong solutions in critical spaces for barotropic viscous fluids
This paper is dedicated to the study of viscous compressible barotropic
fluids in dimension . We address the question of the global existence
of strong solutions for initial data close from a constant state having
critical Besov regularity. In a first time, this article show the recent
results of \cite{CD} and \cite{CMZ} with a new proof. Our result relies on a
new a priori estimate for the velocity, where we introduce a new structure to
\textit{kill} the coupling between the density and the velocity as in
\cite{H2}. We study so a new variable that we call effective velocity. In a
second time we improve the results of \cite{CD} and \cite{CMZ} by adding some
regularity on the initial data in particular is in . In this
case we obtain global strong solutions for a class of large initial data on the
density and the velocity which in particular improve the results of D. Hoff in
\cite{5H4}. We conclude by generalizing these results for general viscosity
coefficients
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