Logarithmic estimates for mean-field models in dimension two and the Schrödinger–Poisson system

Dolbeault, Jean; Frank, Rupert L.; Jeanjean, Louis

doi:10.5802/crmath.272

Équations aux dérivées partielles

Logarithmic estimates for mean-field models in dimension two and the Schrödinger–Poisson system

Dolbeault, Jean ¹ ; Frank, Rupert L.² ; Jeanjean, Louis ³

¹ CEREMADE (CNRS UMR 7534), PSL university, Université Paris-Dauphine Place de Lattre de Tassigny, 75775 Paris 16, France
² R.L. Frank: Mathematisches Institut, Ludwig-Maximilans Universität München, Theresienstr. 39, 80333 München, Germany, and Munich Center for Quantum Science and Technology, Schellingstr. 4, 80799 München, Germany, and Mathematics 253-37, Caltech, Pasadena, CA 91125, USA
³ Laboratoire de Mathématiques (CNRS UMR 6623), Université of Bourgogne Franche-Comté, 25030 Besançon Cedex, France

Comptes Rendus. Mathématique, Tome 359 (2021) no. 10, pp. 1279-1293.

Résumé

In dimension two, we investigate a free energy and the ground state energy of the Schrödinger–Poisson system coupled with a logarithmic nonlinearity in terms of underlying functional inequalities which take into account the scaling invariances of the problem. Such a system can be considered as a nonlinear Schrödinger equation with a cubic but nonlocal Poisson nonlinearity, and a local logarithmic nonlinearity. Both cases of repulsive and attractive forces are considered. We also assume that there is an external potential with minimal growth at infinity, which turns out to have a logarithmic growth. Our estimates rely on new logarithmic interpolation inequalities which combine logarithmic Hardy–Littlewood–Sobolev and logarithmic Sobolev inequalities. The two-dimensional model appears as a limit case of more classical problems in higher dimensions.

Reçu le : 2021-07-01
Accepté le : 2021-09-17
Publié le : 2022-01-04

DOI : 10.5802/crmath.272

Classification : 35J50, 35Q55, 35J47

Affiliations des auteurs :

Dolbeault, Jean ¹ ; Frank, Rupert L. ² ; Jeanjean, Louis ³

@article{CRMATH_2021__359_10_1279_0,
     author = {Dolbeault, Jean and Frank, Rupert L. and Jeanjean, Louis},
     title = {Logarithmic estimates for mean-field models in dimension two and the {Schr\"odinger{\textendash}Poisson} system},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1279--1293},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {10},
     year = {2021},
     doi = {10.5802/crmath.272},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/crmath.272/}
}

TY  - JOUR
AU  - Dolbeault, Jean
AU  - Frank, Rupert L.
AU  - Jeanjean, Louis
TI  - Logarithmic estimates for mean-field models in dimension two and the Schrödinger–Poisson system
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 1279
EP  - 1293
VL  - 359
IS  - 10
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.272/
DO  - 10.5802/crmath.272
LA  - en
ID  - CRMATH_2021__359_10_1279_0
ER  -

%0 Journal Article
%A Dolbeault, Jean
%A Frank, Rupert L.
%A Jeanjean, Louis
%T Logarithmic estimates for mean-field models in dimension two and the Schrödinger–Poisson system
%J Comptes Rendus. Mathématique
%D 2021
%P 1279-1293
%V 359
%N 10
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.272/
%R 10.5802/crmath.272
%G en
%F CRMATH_2021__359_10_1279_0

Dolbeault, Jean; Frank, Rupert L.; Jeanjean, Louis. Logarithmic estimates for mean-field models in dimension two and the Schrödinger–Poisson system. Comptes Rendus. Mathématique, Tome 359 (2021) no. 10, pp. 1279-1293. doi : 10.5802/crmath.272. http://www.numdam.org/articles/10.5802/crmath.272/

Bibliographie
Cité par

[1] Beckner, William Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality, Ann. Math., Volume 138 (1993) no. 1, pp. 213-242 | DOI | MR | Zbl

[2] Beckner, William Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality, Ann. Math., Volume 138 (1993) no. 1, pp. 213-242 | DOI | MR | Zbl

[3] Bellazzini, Jacopo; Siciliano, Gaetano Scaling properties of functionals and existence of constrained minimizers, J. Funct. Anal., Volume 261 (2011) no. 9, pp. 2486-2507 | DOI | MR | Zbl

[4] Ben Abdallah, Naoufel; Méhats, Florian On a Vlasov–Schrödinger–Poisson model, Commun. Partial Differ. Equations, Volume 29 (2004) no. 1-2, pp. 173-206 | DOI | MR | Zbl

[5] Białynicki-Birula, Iwo; Mycielski, Jerzy Nonlinear wave mechanics, Ann. Phys., Volume 100 (1976) no. 1-2, pp. 62-93 | DOI | MR

[6] Blanchet, Adrien; Dolbeault, Jean; Perthame, Benoît Two-dimensional Keller–Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differ. Equ., Volume 2006 (2006) no. 44, pp. 1-32 | MR | Zbl

[7] Campos, Juan F.; Dolbeault, Jean A functional framework for the Keller–Segel system: logarithmic Hardy–Littlewood–Sobolev and related spectral gap inequalities, C. R. Math. Acad. Sci. Paris, Volume 350 (2012) no. 21-22, pp. 949-954 | DOI | MR | Zbl

[8] Campos, Juan F.; Dolbeault, Jean Asymptotic Estimates for the Parabolic-Elliptic Keller–Segel Model in the Plane, Commun. Partial Differ. Equations, Volume 39 (2014) no. 5, pp. 806-841 | DOI | MR | Zbl

[9] Carlen, Eric A. Superadditivity of Fisher’s information and logarithmic Sobolev inequalities, J. Funct. Anal., Volume 101 (1991) no. 1, pp. 194-211 | DOI | MR | Zbl

[10] Carlen, Eric A.; Carrillo, José A.; Loss, Michael Hardy–Littlewood–Sobolev inequalities via fast diffusion flows, Proc. Natl. Acad. Sci. USA, Volume 107 (2010) no. 46, pp. 19696-19701 | DOI | MR | Zbl

[11] Carlen, Eric A.; Loss, Michael Competing symmetries, the logarithmic HLS inequality and Onofri’s inequality on $S^{n}$ , Geom. Funct. Anal., Volume 2 (1992) no. 1, pp. 90-104 | DOI | MR | Zbl

[12] Carles, Rémi; Gallagher, Isabelle Universal dynamics for the defocusing logarithmic Schrödinger equation, Duke Math. J., Volume 167 (2018) no. 9, pp. 1761-1801 | DOI | Zbl

[13] Castella, François $L^{2}$ solutions to the Schrödinger–Poisson system: existence, uniqueness, time behaviour, and smoothing effects, Math. Models Methods Appl. Sci., Volume 7 (1997) no. 8, pp. 1051-1083 | DOI | MR | Zbl

[14] Cazenave, Thierry Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal., Theory Methods Appl., Volume 7 (1983) no. 10, pp. 1127-1140 | DOI | MR | Zbl

[15] Cazenave, Thierry Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, 2003, xiv+323 pages | DOI | MR

[16] Cazenave, Thierry; Haraux, Alain Équations d’évolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse, Math., Volume 2 (1980) no. 1, pp. 21-51 | DOI | MR | Zbl

[17] Cingolani, Silvia; Jeanjean, Louis Stationary Waves with Prescribed $L^{2}$ -Norm for the Planar Schrödinger–Poisson System, SIAM J. Math. Anal., Volume 51 (2019) no. 4, pp. 3533-3568 | DOI | Zbl

[18] Cingolani, Silvia; Weth, Tobias On the planar Schrödinger–Poisson system, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 33 (2016) no. 1, pp. 169-197 | DOI | Zbl

[19] Costa, Max H. M. A new entropy power inequality, IEEE Trans. Inf. Theory, Volume 31 (1985) no. 6, pp. 751-760 | DOI | MR | Zbl

[20] d’Avenia, Pietro; Montefusco, Eugenio; Squassina, Marco On the logarithmic Schrödinger equation, Commun. Contemp. Math., Volume 16 (2014) no. 2, 1350032, 15 pages | DOI | MR | Zbl

[21] Dolbeault, Jean Sobolev and Hardy–Littlewood–Sobolev inequalities: duality and fast diffusion, Math. Res. Lett., Volume 18 (2011) no. 6, pp. 1037-1050 | DOI | MR | Zbl

[22] Dolbeault, Jean; Esteban, Maria J.; Jankowiak, Gaspard The Moser–Trudinger–Onofri inequality, Chin. Ann. Math., Ser. B, Volume 36 (2015) no. 5, pp. 777-802 | DOI | MR | Zbl

[23] Dolbeault, Jean; Jankowiak, Gaspard Sobolev and Hardy–Littlewood–Sobolev inequalities, J. Differ. Equations, Volume 257 (2014) no. 6, pp. 1689-1720 | DOI | MR | Zbl

[24] Dolbeault, Jean; Li, Xingyu Generalized Logarithmic Hardy–Littlewood–Sobolev Inequality, Int. Math. Res. Not., Volume 12 (2019), pp. 1-13 | DOI

[25] Dolbeault, Jean; Perthame, Benoît Optimal critical mass in the two-dimensional Keller–Segel model in $ℝ^{2}$ , C. R. Math. Acad. Sci. Paris, Volume 339 (2004) no. 9, pp. 611-616 | DOI | MR | Zbl

[26] Dolbeault, Jean; Toscani, Giuseppe Stability results for logarithmic Sobolev and Gagliardo–Nirenberg inequalities, Int. Math. Res. Not. (2016) no. 2, pp. 473-498 | DOI | MR | Zbl

[27] Federbush, Paul Partially alternate derivation of a result of Nelson, J. Math. Phys., Volume 10 (1969), pp. 50-52 | DOI | Zbl

[28] Frank, Rupert L.; Lieb, Elliott H. Spherical reflection positivity and the Hardy–Littlewood–Sobolev inequality, Concentration, functional inequalities and isoperimetry (Contemporary Mathematics), Volume 545, American Mathematical Society, 2011, pp. 89-102 | DOI | MR | Zbl

[29] Gross, Leonard Logarithmic Sobolev inequalities, Am. J. Math., Volume 97 (1975) no. 4, pp. 1061-1083 | DOI | MR

[30] Lieb, Elliott H.; Loss, Michael Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, 2001, xxii+346 pages | DOI | MR

[31] Lommel, Eugen Über eine mit den Bessel’schen Functionen verwandte Function, Math. Ann., Volume 9 (1875) no. 3, pp. 425-444 | DOI | Zbl

[32] Lommel, Eugen Zur Theorie der Bessel’schen Functionen, Math. Ann., Volume 16 (1880) no. 2, pp. 183-208 | DOI | MR | Zbl

[33] Lopes, Orlando Uniqueness and radial symmetry of minimizers for a nonlocal variational problem, Commun. Pure Appl. Anal., Volume 18 (2019) no. 5, pp. 2265-2282 | DOI | MR

[34] López, José Luis; Montejo-Gámez, Jesús On a rigorous interpretation of the quantum Schrödinger–Langevin operator in bounded domains with applications, J. Math. Anal. Appl., Volume 383 (2011) no. 2, pp. 365-378 | DOI | Zbl

[35] Onofri, Enrico On the positivity of the effective action in a theory of random surfaces, Commun. Math. Phys., Volume 86 (1982) no. 3, pp. 321-326 | DOI | MR | Zbl

[36] Sánchez, Óscar; Soler, Juan Asymptotic decay estimates for the repulsive Schrödinger–Poisson system, Math. Methods Appl. Sci., Volume 27 (2004) no. 4, pp. 371-380 | DOI | MR | Zbl

[37] Stam, Aart J. Some inequalities satisfied by the quantities of information of Fisher and Shannon, Inform. and Control, Volume 2 (1959), pp. 101-112 | DOI | MR | Zbl

[38] Villani, Cédric A short proof of the “concavity of entropy power”, IEEE Trans. Inf. Theory, Volume 46 (2000) no. 4, pp. 1695-1696 | DOI | MR | Zbl

[39] Weinstein, Michael I. Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., Volume 87 (1982/83) no. 4, pp. 567-576 | DOI | MR | Zbl

[40] Weissler, Fred B. Logarithmic Sobolev inequalities for the heat-diffusion semigroup, Trans. Am. Math. Soc., Volume 237 (1978), pp. 255-269 | DOI | MR | Zbl

Cité par Sources :