Description of the lack of compactness of some critical Sobolev embedding

Bahouri, Hajer

Journées équations aux dérivées partielles (2011), article no. 1, 13 p.

In this text, we present two recent results on the characterization of the lack of compactness of some critical Sobolev embedding. The first one derived in [5] deals with an abstract framework including Sobolev, Besov, Triebel-Lizorkin, Lorentz, Hölder and BMO spaces. The second one established in [3] concerns the lack of compactness of $H^1\left({ℝ}^2\right)$ into the Orlicz space. Although the two results are expressed in the same manner (by means of defect measures) and rely on the defect of compactness due to concentration as in [17] and [18], they are actually of different nature. In fact, both in [5] and [3] it is proved that the lack of compactness can be described in terms of an asymptotic decomposition, but the elements involved in the decomposition are of completely different kinds in the two frameworks. We also highlight that contrary to semilinear cases like the wave equation studied in [2] and [9], the linearizability of the non linear wave equation with exponential growth is not directly related to the lack of compactness of $H^1\left({ℝ}^2\right)$ into the Orlicz space.

[1] S. Adachi and K. Tanaka, Trudinger type inequalities in ${ℝ}^N$ and their best exponents, Proc. Amer. Math. Soc. 128 (2000), no. 7, 2051–2057. | MR | Zbl

[2] H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1999), 131–175. | MR | Zbl

[3] H. Bahouri, M. Majdoub and N. Masmoudi, On the lack of compactness in the 2D critical Sobolev embedding, Journal of Functional Analysis, 260, 2011, pages 208-252. | MR | Zbl

[4] H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, Springer. | MR

[5] H. Bahouri, A. Cohen and G. Koch, A general wavelet-based profile decomposition in the critical embedding of function spaces, arXiv:1103.2468v1, 2011. | MR

[6] J. Ben Ameur, Description du défaut de compacité de l’injection de Sobolev sur le groupe de Heisenberg, Bulletin de la Société Mathématique de Belgique, 15-4, 2008, pages 599-624. | MR | Zbl

[7] H. Brezis and J. M. Coron, Convergence of solutions of H-Systems or how to blow bubbles, Archiv for Rational Mechanics and Analysis, 89, 1985, pages 21-86. | MR | Zbl

[8] A. Cohen, Numerical analysis of wavelet methods, Elsevier, 2003. | MR | Zbl

[9] P. Gérard, Oscillations and concentration effects in semilinear dispersive wave equations, J. Funct. Anal. 133 (1996), 50–68. | MR | Zbl

[10] P. Gérard, Description du défaut de compacité de l’injection de Sobolev, ESAIM Control Optim. Calc. Var. 3 (1998), 213–233 (electronic, URL: http://www.emath.fr/cocv/). | Numdam | MR | Zbl

[11] S. Ibrahim, M. Majdoub and N. Masmoudi, Global solutions for a semilinear, two-dimensional Klein-Gordon equation with exponential-type nonlinearity, Comm. Pure Appl. Math. 59 (2006), no. 11, 1639–1658. | MR | Zbl

[12] S. Jaffard, Analysis of the lack of compactness in the critical Sobolev embeddings, J. Funct. Anal., 161 (1999), 384–396. | MR | Zbl

[13] C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation, Acta Math., 201 (2008), 147–212. | MR | Zbl

[14] H. Kozono and H. Wadade, Remarks on Gagliardo-Nirenberg type inequality with critical Sobolev space and BMO, Math. Z., 259 (2008), 935–950. | MR | Zbl

[15] S. Keraani, On the defect of compactness for the Strichartz estimates of the Shrödinger equation, Journal of Differential equations, 175-2, 2001, pages 353-392. | MR | Zbl

[16] C. Laurent, On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold, to appear in Journal of Functional Analysis. | MR

[17] P.L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I., Rev. Mat. Iberoamericana 1 (1985), no. 1, 145–201. | MR | Zbl

[18] P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, I, Ann. Inst. H. Poincare Anal. Non Linéaire 1 (1984), 109–145. | Numdam | MR | Zbl

[19] Y. Meyer, Ondelettes et opérateurs, Hermann, 1990. | MR | Zbl

[20] J. Moser, A sharp form of an inequality of N. Trudinger, Ind. Univ. Math. J. 20(1971), pp. 1077-1092. | MR | Zbl

[21] M. Nakamura and T. Ozawa, The Cauchy problem for nonlinear Klein-Gordon equation in the Sobolev spaces, Publ. RIMS, Kyoto University 37 (2001), 255–293. | MR | Zbl

[22] I. Schindler and K. Tintarev, An abstract version of the concentration compactness principle, Revista Math Complutense, 15-2, 2002, pages 417-436. | MR | Zbl

[23] M. Struwe, A global compactness result for boundary value problems involving limiting nonlinearities, Mathematische Zeitschrift, 187, 511-517, 1984. | MR | Zbl

[24] M. Struwe, Critical points of embeddings of $H_0^{1,n}$ into Orlicz spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 425–464. | Numdam | MR | Zbl

[25] S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subset of a Sobolev space, Annales de l’IHP analyse non linéaire, 12-3, 1995, pages 319-337. | Numdam | MR | Zbl

[26] T. Tao, An inverse theorem for the bilinear $L^2$ Strichartz estimate for the wave equation, arXiv: 0904-2880, 2009.

[27] N.S. Trudinger, On imbedding into Orlicz spaces and some applications, J. Math. Mech. 17(1967), pp. 473-484. | MR | Zbl