Description of the lack of compactness for the Sobolev imbedding

P. Gérard

doi:10.1051/cocv:1998107

Abstract

We prove that any bounded sequence in a Hilbert homogeneous Sobolev space has a subsequence which can be decomposed as analmost-orthogonal sum of a sequence going strongly to zero in the corresponding Lebesgue space, and of a superposition of termsobtained from fixed profiles by applying sequences of translations and dilations. This decomposition contains in particular the variousversions of the concentration-compactness principle.

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Jaffard, Stéphane 1999. Analysis of the Lack of Compactness in the Critical Sobolev Embeddings. Journal of Functional Analysis, Vol. 161, Issue. 2, p. 384.

Keraani, Sahbi 2001. On the Defect of Compactness for the Strichartz Estimates of the Schrödinger Equations. Journal of Differential Equations, Vol. 175, Issue. 2, p. 353.

Gallagher, Isabelle and Gérard, Patrick 2002. Nonlinear Partial Differential Equations and their Applications - Collège de France Seminar Volume XIV. Vol. 31, Issue. , p. 367.

Carles, Rémi Fermanian Kammerer, Clotilde and Gallagher, Isabelle 2003. On the role of quadratic oscillations in nonlinear Schrödinger equations. Journal of Functional Analysis, Vol. 203, Issue. 2, p. 453.

IBRAHIM, SLIM 2004. GEOMETRIC-OPTICS FOR NONLINEAR CONCENTRATING WAVES IN FOCUSING AND NON-FOCUSING TWO GEOMETRIES. Communications in Contemporary Mathematics, Vol. 06, Issue. 01, p. 1.

Kenig, Carlos E. and Merle, Frank 2006. Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case. Inventiones mathematicae, Vol. 166, Issue. 3, p. 645.

Hmidi, Taoufik and Keraani, Sahbi 2006. Remarks on the Blowup for the $L^2$‐Critical Nonlinear Schrödinger Equations. SIAM Journal on Mathematical Analysis, Vol. 38, Issue. 4, p. 1035.

Visan, Monica and Zhang, Xiaoyi 2007. On the Blowup for the $L^2$‐Critical Focusing Nonlinear Schrödinger Equation in Higher Dimensions below the Energy Class. SIAM Journal on Mathematical Analysis, Vol. 39, Issue. 1, p. 34.

Kenig, Carlos E. and Merle, Frank 2008. Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. Acta Mathematica, Vol. 201, Issue. 2, p. 147.

Côte, Raphaël Kenig, Carlos E. and Merle, Frank 2008. Scattering Below Critical Energy for the Radial 4D Yang-Mills Equation and for the 2D Corotational Wave Map System. Communications in Mathematical Physics, Vol. 284, Issue. 1, p. 203.

Chemin, Jean-Yves and Gallagher, Isabelle 2009. Wellposedness and stability results for the Navier–Stokes equations in $ R^{3} $. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Vol. 26, Issue. 2, p. 599.

PAUSADER, BENOIT and SHAO, SHUANGLIN 2010. THE MASS-CRITICAL FOURTH-ORDER SCHRÖDINGER EQUATION IN HIGH DIMENSIONS. Journal of Hyperbolic Differential Equations, Vol. 07, Issue. 04, p. 651.

Koch, Gabriel S. 2010. Profile decompositions and applications to Navier-Stokes. Journées équations aux dérivées partielles, p. 1.

BAHOURI, HAJER COHEN, ALBERT and KOCH, GABRIEL 2011. A GENERAL WAVELET-BASED PROFILE DECOMPOSITION IN THE CRITICAL EMBEDDING OF FUNCTION SPACES. Confluentes Mathematici, Vol. 03, Issue. 03, p. 387.

Pocovnicu, Oana 2011. Traveling waves for the cubic Szegő equation on the real line. Analysis & PDE, Vol. 4, Issue. 3, p. 379.

Lenzmann, Enno and Lewin, Mathieu 2011. On singularity formation for theL2-critical Boson star equation. Nonlinearity, Vol. 24, Issue. 12, p. 3515.

Bahouri, Hajer Majdoub, Mohamed and Masmoudi, Nader 2011. On the lack of compactness in the 2D critical Sobolev embedding. Journal of Functional Analysis, Vol. 260, Issue. 1, p. 208.

Bahouri, Hajer and Cohen, Albert 2011. Refined Sobolev Inequalities in Lorentz Spaces. Journal of Fourier Analysis and Applications, Vol. 17, Issue. 4, p. 662.

Fang, DaoYuan Xie, Jian and Cazenave, Thierry 2011. Scattering for the focusing energy-subcritical nonlinear Schrödinger equation. Science China Mathematics, Vol. 54, Issue. 10, p. 2037.

Laurent, Camille 2011. On stabilization and control for the critical Klein–Gordon equation on a 3-D compact manifold. Journal of Functional Analysis, Vol. 260, Issue. 5, p. 1304.

Download full list

Article contents

Description of the lack of compactness for the Sobolev imbedding

Abstract

Keywords

Access options

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Article contents

Description of the lack of compactness for the Sobolev imbedding

Abstract

Keywords

Access options

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests