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

Bahouri, Hajer; Majdoub, Mohamed; Masmoudi, Nader

Comptes Rendus. Mathématique, Tome 350 (2012) no. 3-4, pp. 177-181.

Cette Note est consacrée à lʼétude du défaut de compacité de lʼinjection de Sobolev de $H^1({\mathbb{R}}^2)$ dans lʼespace dʼOrlicz critique $\mathcal{L}({\mathbb{R}}^2)$. Nous démontrons que la déscription donnée dans Bahouri et al. (2011) [5] concernant le cas radial reste valable dans le cas général (à des translations près par des coeurs de concentration). La preuve utilise des arguments de capacité ainsi quʼun processus dʼextraction de concentrations.

This Note is devoted to the description of the lack of compactness of the Sobolev embedding of $H^1({\mathbb{R}}^2)$ in the critical Orlicz space $\mathcal{L}({\mathbb{R}}^2)$. It turns out that up to cores our result is expressed in terms of the concentration-type examples derived by J. Moser (1971) in [16] as in the radial setting investigated in Bahouri et al. (2011) [5]. 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the one conducted in the radial case which is based on an $L^{\infty }$ estimate far away from the origin and which is no longer valid in the general frame work. 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.

[1] Bahouri, H.; Chemin, J.-Y.; Danchin, R. Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, Springer, 2011

[2] Bahouri, H.; Cohen, A.; Koch, G. A general wavelet-based profile decomposition in the critical embedding of function spaces, Confluentes Mathematici, Volume 3 (2011) no. 3, pp. 1-25

[3] Bahouri, H.; Gallagher, I. Weak stability of the set of global solutions to the Navier–Stokes equations, 2011 | arXiv

[4] Bahouri, H.; Gérard, P. High frequency approximation of solutions to critical nonlinear wave equations, American Journal of Mathematics, Volume 121 (1999), pp. 131-175

[5] Bahouri, H.; Majdoub, M.; Masmoudi, N. On the lack of compactness in the 2D critical Sobolev embedding, Journal of Functional Analysis, Volume 260 (2011), pp. 208-252

[6] Bahouri, H.; Majdoub, M.; Masmoudi, N. Lack of compactness in the 2D critical Sobolev embedding, the general case, 2011 | arXiv

[7] Brezis, H.; Wainger, S. A note on limiting cases of Sobolev embeddings and convolution inequalities, Communications in Partial Differential Equations, Volume 5 (1980), pp. 773-789

[8] Gérard, P. Description du défaut de compacité de lʼinjection de Sobolev, ESAIM: Control, Optimisation and Calculus of Variations, Volume 3 (1998), pp. 213-233 http://www.emath.fr/cocv/ (electronic)

[9] Henrot, A.; Pierre, M. Variations et optimisation de formes, Mathématiques et Applications, vol. 48, Springer, 2005

[10] Ibrahim, S.; Majdoub, M. Comparaison des ondes linéaires et non linéaires à coefficients variables, Bulletin de la Société Mathématique de Belgique, Volume 10 (2003), pp. 299-312

[11] Jaffard, S. Analysis of the lack of compactness in the critical Sobolev embeddings, Journal of Functional Analysis, Volume 161 (1999), pp. 384-396

[12] Kenig, C.E.; Merle, F. Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation, Acta Mathematica, Volume 201 (2008), pp. 147-212

[13] Keraani, S. On the defect of compactness for the Strichartz estimates of the Shrödinger equation, Journal of Differential Equations, Volume 175 (2001), pp. 353-392

[14] Lions, P.-L. The concentration–compactness principle in the calculus of variations. The limit case. I, Revista Matemática Iberoamericana, Volume 1 (1985), pp. 145-201

[15] Majdoub, M. Qualitative study of the critical wave equation with a subcritical perturbation, Journal of Mathematics Analysis and Applications, Volume 301 (2005), pp. 354-365

[16] Moser, J. A sharp form of an inequality of N. Trudinger, Indiana University Mathematics Journal, Volume 20 (1971), pp. 1077-1092

[17] Rao, M.-M.; Ren, Z.-D. Applications of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 250, Marcel Dekker Inc., 2002

[18] Ruf, B. A sharp Trudinger–Moser type inequality for unbounded domains in ${\mathbb{R}}^2$, Journal of Functional Analysis, Volume 219 (2005), pp. 340-367

[19] Struwe, M. Critical points of embeddings of $H_0^{1,n}$ into Orlicz spaces, Annales de lʼInstitut Henri Poincaré Analyse Non Linéaire, Volume 5 (1988), pp. 425-464