Injections de Sobolev probabilistes et applications

Burq, Nicolas; Lebeau, Gilles

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 6, pp. 917-962.

On démontre dans cet article des versions probabilistes des injections de Sobolev sur une variété riemannienne compacte, $(M,g)$. Plus précisément on démontre que pour des mesures de probabilité naturelles sur l’espace $L^2\left(M\right)$, presque toute fonction appartient à tous les espaces $L^p\left(M\right)$, $p<+\infty$. On donne ensuite des applications à l’étude des harmoniques sphériques sur la sphère ${𝕊}^d$ : on démontre (encore pour des mesures de probabilité naturelles) que presque toute base hilbertienne de $L^2\left({𝕊}^d\right)$ formée d’harmoniques sphériques a tous ses éléments uniformément bornés dans tous les espaces $L^p\left({𝕊}^d\right),p<+\infty$. On démontre aussi des résultats similaires sur les tores ${𝕋}^d$. On donne aussi une application à l’étude du taux de décroissance de l’équation des ondes amortie dans un cadre où la condition de contrôle géométrique de Bardos, Lebeau et Rauch n’est pas vérifiée. En supposant le flot ergodique, on démontre qu’il existe sur des ensembles de mesure arbitrairement proche de $1$ (dans l’espace des données initiales d’énergie finie), un taux de décroissance uniforme. Finalement, on conclut avec une application à l’étude de l’équation des ondes semi-linéaire $H^1$-surcritique, pour laquelle on démontre que pour presque toute donnée initiale, les solutions faibles sont fortes et uniques (localement en temps).

In this article, we give probabilistic versions of Sobolev embeddings on any Riemannian manifold $(M,g)$. More precisely, we prove that for natural probability measures on $L^2\left(M\right)$, almost every function belongs to all spaces $L^p\left(M\right)$, $p<+\infty$. We then give applications to the study of the growth of the $L^p$ norms of spherical harmonics on spheres ${𝕊}^d$: we prove (again for natural probability measures) that almost every Hilbert base of $L^2\left({𝕊}^d\right)$ made of spherical harmonics has all its elements uniformly bounded in all $L^p\left({𝕊}^d\right),p<+\infty$ spaces. We also prove similar results on tori ${𝕋}^d$. We give then an application to the study of the decay rate of damped wave equations in a framework where the geometric control property of Bardos-Lebeau-Rauch is not satisfied. Assuming that it is violated for a measure $0$ set of trajectories, we prove that there exists almost surely a rate. Finally, we conclude with an application to the study of the $H^1$-supercritical wave equation, for which we prove that for almost all initial data, the weak solutions are strong and unique, locally in time.

[1] N. Anantharaman, Spectral deviations for the damped wave equation, Geom. Funct. Anal. 20 (2010), 593-626. | MR

[2] M. Asch & G. Lebeau, The spectrum of the damped wave operator for a bounded domain in ${𝐑}^2$, Experiment. Math. 12 (2003), 227-241. | MR

[3] A. Ayache & N. Tzvetkov, $L^p$ properties for Gaussian random series, Trans. Amer. Math. Soc. 360 (2008), 4425-4439. | MR

[4] C. Bardos, G. Lebeau & J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992), 1024-1065. | MR

[5] A. L. Besse, Manifolds all of whose geodesics are closed, Ergebnisse Math. Grenzg. 93, Springer, 1978. | MR

[6] M. D. Blair, H. F. Smith & C. D. Sogge, Strichartz estimates for the wave equation on manifolds with boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 1817-1829. | Numdam | MR

[7] D. D. Bleecker, Nonperturbative conformal quantum gravity, Classical Quantum Gravity 4 (1987), 827-849. | MR

[8] J. Bourgain, Eigenfunction bounds for the Laplacian on the $n$-torus, Int. Math. Res. Not. 1993 (1993), 61-66. | MR

[9] J. Bourgain, Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces, Israel J. Math. 193 (2013), 441-458. | MR

[10] J. Bourgain & Z. Rudnick, Restriction of toral eigenfunctions to hypersurfaces and nodal sets, Geom. Funct. Anal. 22 (2012), 878-937. | MR

[11] N. Burq, G. Lebeau & F. Planchon, Global existence for energy critical waves in 3-D domains, J. Amer. Math. Soc. 21 (2008), 831-845. | MR

[12] N. Burq & N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math. 173 (2008), 449-475. | MR

[13] N. Burq & N. Tzvetkov, Probabilistic well-posedness for the cubic wave equation, preprint arXiv:1103.2222. | MR

[14] M. Dimassi & J. Sjöstrand, Spectral asymptotics in the semi-classical limit, Lecture Note Series 268, Cambridge University Press, 1999. | MR

[15] I. Gallagher & P. Gérard, Profile decomposition for the wave equation outside a convex obstacle, J. Math. Pures Appl. 80 (2001), 1-49. | MR

[16] E. Grosswald, Representations of integers as sums of squares, Springer, 1985. | MR

[17] V. Guillemin, Some classical theorems in spectral theory revisited, in Seminar on singularities of solutions of linear partial differential equations (Inst. Adv. Study, Princeton, N.J., 1977/78), Ann. of Math. Stud. 91, Princeton Univ. Press, 1979, 219-259. | MR

[18] L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193-218. | MR

[19] L. Hörmander, The analysis of linear partial differential operators. IV, Grund. Math. Wiss. 275, Springer, 1985.

[20] O. Ivanovici, Counterexamples to Strichartz estimates for the wave equation in domains, Math. Ann. 347 (2010), 627-673. | MR

[21] V. Ivrii, Microlocal analysis and precise spectral asymptotics, Springer Monographs in Math., Springer, 1998. | MR

[22] S. Kakutani, On equivalence of infinite product measures, Ann. of Math. 49 (1948), 214-224. | MR

[23] Y. Kifer, Large deviations in dynamical systems and stochastic processes, Trans. Amer. Math. Soc. 321 (1990), 505-524. | MR

[24] G. Lebeau, Équation des ondes amorties, in Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), Math. Phys. Stud. 19, Kluwer Acad. Publ., 1996, 73-109. | MR

[25] M. Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs 89, Amer. Math. Soc., 2001. | MR

[26] D. Li & H. Queffélec, Introduction à l'étude des espaces de Banach, Cours Spécialisés 12, Soc. Math. France, 2004.

[27] A. Martinez, An introduction to semiclassical and microlocal analysis, Universitext, Springer, 2002. | MR

[28] F. Morgan, Measures on spaces of surfaces, Arch. Rational Mech. Anal. 78 (1982), 335-359. | MR

[29] B. Shiffman & S. Zelditch, Random polynomials of high degree and Levy concentration of measure, Asian J. Math. 7 (2003), 627-646. | MR

[30] J. Sjöstrand, Asymptotic distribution of eigenfrequencies for damped wave equations, Publ. Res. Inst. Math. Sci. 36 (2000), 573-611. | MR

[31] H. F. Smith & C. D. Sogge, On the $L^p$ norm of spectral clusters for compact manifolds with boundary, to appear in Acta Math. | MR

[32] C. D. Sogge, Concerning the $L^p$ norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988), 123-138. | MR

[33] C. D. Sogge, Eigenfunction and Bochner Riesz estimates on manifolds with boundary, Math. Res. Lett. 9 (2002), 205-216. | MR

[34] M. Struwe, On uniqueness and stability for supercritical nonlinear wave and Schrödinger equations, Int. Math. Res. Not. 2006 (2006), Art. ID 76737, 14. | MR

[35] N. Tzvetkov, Invariant measures for the defocusing nonlinear Schrödinger equation, Ann. Inst. Fourier (Grenoble) 58 (2008), 2543-2604. | Numdam | MR

[36] N. Tzvetkov, Riemannian analogue of a Paley Sygmund theorem, Séminaire É.D.P., exposé no XV (2008-2009), 1-14. | MR

[37] J. M. Vanderkam, $L^{\infty }$ norms and quantum ergodicity on the sphere, Int. Math. Res. Not. 1997 (1997), 329-347. | MR

[38] S. Zelditch, A random matrix model for quantum mixing, Int. Math. Res. Not. 1996 (1996), 115-137. | MR

[39] S. Zelditch, Real and complex zeros of Riemannian random waves, in Spectral analysis in geometry and number theory, Contemp. Math. 484, Amer. Math. Soc., 2009, 321-342. | MR

[40] M. Zworski, Semiclassical analysis, Graduate Studies in Math. 138, Amer. Math. Soc., 2012. | MR