Comptes Rendus
Volume articles
Partial differential equations
Logarithmic estimates for mean-field models in dimension two and the Schrödinger–Poisson system
Jean Dolbeault1; Rupert L. Frank2; Louis Jeanjean3
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, Volume 359 (2021) no. 10, pp. 1279-1293.

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.

Received:
Accepted:
Published online:
DOI: 10.5802/crmath.272
Classification: 35J50, 35Q55, 35J47

Jean Dolbeault 1; Rupert L. Frank 2; Louis Jeanjean 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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{CRMATH_2021__359_10_1279_0,
     author = {Jean Dolbeault and Rupert L. Frank and Louis Jeanjean},
     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},
}
TY  - JOUR
AU  - Jean Dolbeault
AU  - Rupert L. Frank
AU  - Louis Jeanjean
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
DO  - 10.5802/crmath.272
LA  - en
ID  - CRMATH_2021__359_10_1279_0
ER  - 
%0 Journal Article
%A Jean Dolbeault
%A Rupert L. Frank
%A Louis Jeanjean
%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
%R 10.5802/crmath.272
%G en
%F CRMATH_2021__359_10_1279_0
Jean Dolbeault; Rupert L. Frank; Louis Jeanjean. Logarithmic estimates for mean-field models in dimension two and the Schrödinger–Poisson system. Comptes Rendus. Mathématique, Volume 359 (2021) no. 10, pp. 1279-1293. doi : 10.5802/crmath.272. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.272/

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

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

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

[6] Adrien Blanchet; Jean Dolbeault; Benoît Perthame 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] Juan F. Campos; Jean Dolbeault 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] Juan F. Campos; Jean Dolbeault 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] Eric A. Carlen Superadditivity of Fisher’s information and logarithmic Sobolev inequalities, J. Funct. Anal., Volume 101 (1991) no. 1, pp. 194-211 | DOI | MR | Zbl

[10] Eric A. Carlen; José A. Carrillo; Michael Loss 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] Eric A. Carlen; Michael Loss 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] Rémi Carles; Isabelle Gallagher Universal dynamics for the defocusing logarithmic Schrödinger equation, Duke Math. J., Volume 167 (2018) no. 9, pp. 1761-1801 | DOI | Zbl

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

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

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

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

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

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

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

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

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

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

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

[28] Rupert L. Frank; Elliott H. Lieb 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] Leonard Gross Logarithmic Sobolev inequalities, Am. J. Math., Volume 97 (1975) no. 4, pp. 1061-1083 | DOI | MR

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

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

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

[33] Orlando Lopes 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] José Luis López; Jesús Montejo-Gámez 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] Enrico Onofri 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] Óscar Sánchez; Juan Soler 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] Aart J. Stam 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] Cédric Villani 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] Michael I. Weinstein Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., Volume 87 (1982/83) no. 4, pp. 567-576 | DOI | MR | Zbl

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

Cited by Sources:

Comments - Policy