We characterize the lack of compactness in the critical embedding of functions spaces X ⊂ Y having similar scaling properties in the following terms: a sequence (un)n≥0 bounded in X has a subsequence that can be expressed as a finite sum of translations and dilations of functions (ϕl)l>0 such that the remainder converges to zero in Y as the number of functions in the sum and n tend to +∞. Such a decomposition was established by Gérard in [13] for the embedding of the homogeneous Sobolev space X = Ḣs into the Y = Lp in d dimensions with 0 < s = d/2 - d/p, and then generalized by Jaffard in [15] to the case where X 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 X and Y that are of key use in building the profile decomposition. These properties may then easily be checked for typical choices of X and Y satisfying critical embedding properties. These includes Sobolev, Besov, Triebel-Lizorkin, Lorentz, Hölder and BMO spaces.
@article{CML_2011__3_3_387_0,
author = {Bahouri, Hajer and Cohen, Albert and Koch, Gabriel},
title = {A general wavelet-based profile decomposition in the critical embedding of function spaces},
journal = {Confluentes Mathematici},
pages = {387--411},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {3},
number = {3},
year = {2011},
doi = {10.1142/S1793744211000370},
language = {en},
url = {http://www.numdam.org/articles/10.1142/S1793744211000370/}
}
TY - JOUR AU - Bahouri, Hajer AU - Cohen, Albert AU - Koch, Gabriel TI - A general wavelet-based profile decomposition in the critical embedding of function spaces JO - Confluentes Mathematici PY - 2011 SP - 387 EP - 411 VL - 3 IS - 3 PB - World Scientific Publishing Co Pte Ltd UR - http://www.numdam.org/articles/10.1142/S1793744211000370/ DO - 10.1142/S1793744211000370 LA - en ID - CML_2011__3_3_387_0 ER -
%0 Journal Article %A Bahouri, Hajer %A Cohen, Albert %A Koch, Gabriel %T A general wavelet-based profile decomposition in the critical embedding of function spaces %J Confluentes Mathematici %D 2011 %P 387-411 %V 3 %N 3 %I World Scientific Publishing Co Pte Ltd %U http://www.numdam.org/articles/10.1142/S1793744211000370/ %R 10.1142/S1793744211000370 %G en %F CML_2011__3_3_387_0
Bahouri, Hajer; Cohen, Albert; Koch, Gabriel. A general wavelet-based profile decomposition in the critical embedding of function spaces. Confluentes Mathematici, Tome 3 (2011) no. 3, pp. 387-411. doi : 10.1142/S1793744211000370. http://www.numdam.org/articles/10.1142/S1793744211000370/
[1] R. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65 (Academic Press, 1975).
[2] H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1999) 131–175.
[3] H. Bahouri, M. Majdoub and N. Masmoudi, On the lack of compactness in the 2D critical Sobolev embedding, J. Funct. Anal. 260 (2011) 208–252.
[4] J. Ben Ameur, Description du défaut de compacité de l’injection de Sobolev sur le groupe de Heisenberg, Bull. Soc. Math. Belgique 15 (2008) 599–624.
[5] H. Brezis and J. M. Coron, Convergence of solutions of H-Systems or how to blow bubbles, Arch. Rational Mech. Anal. 89 (1985) 21–86.
[6] A. Cohen, Numerical Analysis of Wavelet Methods (Elsevier, 2003).
[7] I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992).
[8] R. DeVore, Nonlinear Approximation, Acta Nume. 7 (1998) 51–150.
[9] R. DeVore, B. Jawerth and V. Popov, Compression of wavelet decompositions, Amer. J. Math. 114 (1992) 737–785.
[10] I. Gallagher and P. Gérard, Profile decomposition for the wave equation outside convex obstacles, J. Math. Pures Appl. 80 (2001) 1–49.
[11] I. Gallagher, G. S. Koch and F. Planchon, A profile decomposition approach to the Navier–Stokes regularity criterion, arXiv:1012.0145.
[12] I. Gallagher, Profile decomposition for solutions of the Navier–Stokes equations, Bull. Soc. Math. France 129 (2001) 285–316.
[13] P. Gérard, Description du défaut de compacité de l’injection de Sobolev, ESAIM Control Optim. Calc. Var. 3 (1998) 213–233.
[14] S. Ibrahim, Comparaison des ondes linéaires et non linéaires à coefficients variables, Bull. Soc. Math. Belgique 10 (2003) 299–312.
[15] S. Jaffard, Analysis of the lack of compactness in the critical Sobolev embeddings, J. Funct. Anal. 161 (1999) 384–396.
[16] C. E. Kenig and G. S. Koch, An alternative approach to the Navier–Stokes equations in critical spaces, Ann. l’Inst. Henri Poincaré, Anal. Non Linéaire, DOI:10.1016/j.anihpc.2010.10.004.
[17] C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy critical focusing nonlinear wave equation, Acta Math. 201 (2008) 147–212.
[18] S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equation, J. Differential Equations 175 (2001) 353–392.
[19] G. Kyriasis, Nonlinear approximation and interpolation spaces, J. Approx. Theory 113 (2001) 110–126.
[20] C. Laurent, On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold, to appear in J. Funct. Anal.
[21] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iber. 1 (1985) 145–201.
[22] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iber. 1 (1985) 45–121.
[23] M. Majdoub, Qualitative study of the critical wave equation with a subcritical per- turbation, J. Math. Anal. Appl. 301 (2005) 354–365.
[24] Y. Meyer, Ondelettes et Opérateurs (Hermann, 1990).
[25] I. Schindler and K. Tintarev, An abstract version of the concentration compactness principle, Rev. Math. Complut. 15 (2002) 417–436.
[26] S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subset of a Sobolev space, Ann. l’IHP Anal. Non linéaire 12 (1995) 319–337.
[27] M. Struwe, A global compactness result for boundary value problems involving lim- iting nonlinearities, Math. Z. 187 (1984) 511–517.
[28] T. Tao, An inverse theorem for the bilinear L2 Strichartz estimate for the wave equa- tion, arXiv:0904.2880.
[29] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, 2nd edn. (Johann Ambrosius Barth, 1995).
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