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10. Large-time behavior of compressible polytropic fluids and nonlinear Schrödinger equation

Author

Rémi Carles, Kleber Carrapatoso, Matthieu Hillairet, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Centre de Mathématiques Laurent Schwartz (CMLS), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), ANR-18-CE40-0027,SingFlows,Ecoulements avec singularités : couches limites, filaments de vortex, interaction vague-structure(2018), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)

Subjects
Mathematics - Analysis of PDEs, Applied Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
Abstract

In this paper we analyze the large-time behavior of weak solutions to polytropic fluid models possibly including quantum and capillary effects. Formal a priori estimates show that the density of solutions to these systems should disperse with time. Scaling appropriately the system, we prove that, under a reasonable assumption on the decay of energy, the density of weak solutions converges in large times to an unknown profile. In contrast with the isothermal case, we also show that there exists a large variety of asymptotic profiles. We complement the study by providing existence of global-in-time weak solutions satisfying the required decay of energy. As a byproduct of our method, we also obtain results concerning the large-time behavior of solutions to nonlinear Schr{\"o}dinger equation, allowing the presence of a semi-classical parameter as well as long range nonlinearities., Comment: 22 pages. Some typos fixed, more explanations

Published
2022
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11. On ground state (in-)stability in multi-dimensional cubic-quintic Schrödinger equations

Author

Rémi Carles, Christian Klein, Christof Sparber, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics, Statistics and Computer Science [Chicago] (UIC), University of Illinois [Chicago] (UIC), University of Illinois System-University of Illinois System, NSF grant no. DMS-1348092Rennes Métropole (AIS)isite BFC project NAANoDEITAG project funded by the FEDER de Bourgogne, ANR-17-CE40-0035,ANuI,Approches analytiques, numériques et des systèmes intégrables pour les équations aux dérivées partielles dispersives nonlinéaires(2017), ANR-17-EURE-0002,EIPHI,Ingénierie et Innovation par les sciences physiques, les savoir-faire technologiques et l'interdisciplinarité(2017), ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), European Project: 778010,IPaDEGAN, AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), and Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB)

Subjects
[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Uncategorized
Abstract

We consider the nonlinear Schrödinger equation with a focusing cubic term and a defocusing quintic nonlinearity in dimensions two and three. The main interest of this article is the problem of orbital (in-)stability of ground state solitary waves. We recall the notions of energy minimizing versus action minimizing ground states and prove that, in general, the two must be considered as nonequivalent. We numerically investigate the orbital stability of least action ground states in the radially symmetric case, confirming existing conjectures or leading to new ones.

Published
2023
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19. On the Cauchy problem for the Hartree approximation in quantum dynamics

Author

Rémi Carles, Clotilde Fermanian Kammerer, Caroline Lasser, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Zentrum Mathematik [Munchen] (TUM), Technische Universität Munchen - Université Technique de Munich [Munich, Allemagne] (TUM), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), and Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers

Subjects
Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Applied Mathematics, FOS: Mathematics, General Physics and Astronomy, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], FOS: Physical sciences, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), Mathematical Physics, Analysis of PDEs (math.AP)
Abstract

We prove existence and uniqueness results for the time-dependent Hartree approximation arising in quantum dynamics. The Hartree equations of motion form a coupled system of nonlinear Schr{\"o}dinger equations for the evolution of product state approximations. They are a prominent example for dimension reduction in the context of the the time-dependent Dirac-Frenkel variational principle. We handle the case of Coulomb potentials thanks to Strichartz estimates. Our main result addresses a general setting where the nonlinear coupling cannot be considered as a perturbation. The proof uses a recursive construction that is inspired by the standard approach for the Cauchy problem associated to symmetric quasilinear hyperbolic equations., Comment: 25 pages

Published
2022
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20. Error estimates of local energy regularization for the logarithmic Schrodinger equation

Author

Chunmei Su, Rémi Carles, Qinglin Tang, Weizhu Bao, Department of Mathematics [Singapore], National University of Singapore (NUS), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Yau Mathematical Sciences Center, Tsinghua University, School of Mathematics, State Key Laboratory of Hydraulics and Mountain River Engineering, Chengdu University of Technology (CDUT), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS), and Technische Universität München [München] (TUM)

Subjects
Quadratic growth, Polynomial, Partial differential equation, Logarithm, Applied Mathematics, 010102 general mathematics, Numerical Analysis (math.NA), 16. Peace & justice, 01 natural sciences, Regularization (mathematics), 010101 applied mathematics, Nonlinear system, Rate of convergence, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Modeling and Simulation, FOS: Mathematics, Applied mathematics, Mathematics - Numerical Analysis, 0101 mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics, Energy functional
Abstract

The logarithmic nonlinearity has been used in many partial differential equations (PDEs) for modeling problems in various applications.Due to the singularity of the logarithmic function, it introducestremendous difficulties in establishing mathematical theories, as well asin designing and analyzing numerical methods for PDEs with such nonlinearity. Here we take the logarithmic Schr\"odinger equation (LogSE)as a prototype model. Instead of regularizing $f(\rho)=\ln \rho$ in theLogSE directly and globally as being done in the literature, we propose a local energy regularization (LER) for the LogSE byfirst regularizing $F(\rho)=\rho\ln \rho -\rho$ locally near $\rho=0^+$ with a polynomial approximation in the energy functional of the LogSE and then obtaining an energy regularized logarithmic Schr\"odinger equation (ERLogSE) via energy variation. Linear convergence is established between the solutions of ERLogSE and LogSE in terms of a small regularization parameter $0, Comment: 31 pages, 10 figures, final version. More explanations and some proofs are more detailed

Published
2022
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21. Dynamical approximations for composite quantum systems: Assessment of error estimates for a separable ansatz

Author

Irene Burghardt, Rémi Carles, Clotilde Fermanian Kammerer, Benjamin Lasorne, Caroline Lasser, Institute of Physical and Theoretical Chemistry [Frankfurt am Main], Goethe-Universität Frankfurt am Main, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Institut Charles Gerhardt Montpellier - Institut de Chimie Moléculaire et des Matériaux de Montpellier (ICGM ICMMM), Université Montpellier 1 (UM1)-Université Montpellier 2 - Sciences et Techniques (UM2)-Ecole Nationale Supérieure de Chimie de Montpellier (ENSCM)-Institut de Chimie du CNRS (INC)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Zentrum Mathematik, Technische Universität München, Technische Universität Munchen - Université Technique de Munich [Munich, Allemagne] (TUM), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institut Charles Gerhardt Montpellier - Institut de Chimie Moléculaire et des Matériaux de Montpellier (ICGM), Institut de Chimie du CNRS (INC)-Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-Ecole Nationale Supérieure de Chimie de Montpellier (ENSCM), Université de Montpellier (UM), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, Ecole Nationale Supérieure de Chimie de Montpellier (ENSCM)-Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-Université Montpellier 1 (UM1)-Université Montpellier 2 - Sciences et Techniques (UM2)-Institut de Chimie du CNRS (INC), Technische Universität München [München] (TUM), INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES), and Ecole Nationale Supérieure de Chimie de Montpellier (ENSCM)-Institut de Chimie du CNRS (INC)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Chemical Physics (physics.chem-ph), composite quantum systems, Quantum Physics, scale separation, dimension reduction, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], General Physics and Astronomy, FOS: Physical sciences, Statistical and Nonlinear Physics, [CHIM.THEO]Chemical Sciences/Theoretical and/or physical chemistry, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Modeling and Simulation, Physics - Chemical Physics, quantum dynamics, system-bath theory, [PHYS.PHYS.PHYS-CHEM-PH]Physics [physics]/Physics [physics]/Chemical Physics [physics.chem-ph], Quantum Physics (quant-ph), Mathematical Physics, quantum tunneling
Abstract

Numerical studies are presented to assess error estimates for a separable (Hartree) approximation for dynamically evolving composite quantum systems which exhibit distinct scales defined by their mass and frequency ratios. The relevant error estimates were formally described in our previous work Burghardt et al (2021 J. Phys. A: Math. Theor. 54 414002). Specifically, we consider a representative two-dimensional tunneling system where a double well and a harmonic coordinate are cubically coupled. The time-dependent Hartree approximation is compared with a fully correlated solution, for different parameter regimes. The impact of the coupling and the resulting correlations are quantitatively assessed in terms of a time-dependent reaction probability along the tunneling coordinate. We show that the numerical error is correctly predicted on moderate time scales by a theoretically derived error estimate.

Published
2021
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22. Logarithmic Schrödinger equation with quadratic potential

Author

Rémi Carles, Guillaume Ferriere, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), AIS, Rennes Metropole, Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST, Institut Montpelliérain Alexander Grothendieck (IMAG), and Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Kullback–Leibler divergence, Logarithmic Schrödinger equation, General Physics and Astronomy, Harmonic (mathematics), 01 natural sciences, Solitons, Quadratic equation, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, 010306 general physics, Mathematical Physics, Mathematics, Mathematical physics, Harmonic potential, Applied Mathematics, 010102 general mathematics, Statistical and Nonlinear Physics, Dispersion, Nonlinear system, Logarithmic sobolev inequality, Sign (mathematics)
Abstract

We analyze dynamical properties of the logarithmic Schr{\"o}dinger equation under a quadratic potential. The sign of the nonlinearity is such that it is known that in the absence of external potential, every solution is dispersive, with a universal asymptotic profile. The introduction of a harmonic potential generates solitary waves, corresponding to generalized Gaussons. We prove that they are orbitally stable, using an inequality related to relative entropy, which may be thought of as dual to the classical logarithmic Sobolev inequality. In the case of a partial confinement, we show a universal dispersive behavior for suitable marginals. For repulsive harmonic potentials, the dispersive rate is dictated by the potential, and no universal behavior must be expected., Comment: More explanations

Published
2021
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24. Nonuniqueness and nonlinear instability of Gaussons under repulsive harmonic potential

Author

Rémi Carles, Chunmei Su, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Yau Mathematical Sciences Center, Tsinghua University, Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Yau Mathematical Science Center and Beijing Institute of Mathematical Sciences and Applications, Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Université de Rennes 1 (UR1), and Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST

Subjects
instability, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Applied Mathematics, Nonlinear Schrödinger equation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], ground states, logarithmic nonlinearity, Analysis, Mathematical Physics
Abstract

We consider the Schr{\"o}dinger equation with a nondispersive logarithmic nonlinearity and a repulsive harmonic potential. For a suitable range of the coefficients, there exist two positive stationary solutions, each one generating a continuous family of solitary waves. These solutions are Gaussian, and turn out to be orbitally unstable. We also discuss the notion of ground state in this setting: for any natural definition, the set of ground states is empty., Comment: 14 pages, 3 figures

Published
2021
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25. Separation of scales: Dynamical approximations for complex quantum systems

Author

Irene Burghardt, Caroline Lasser, Benjamin Lasorne, Clotilde Fermanian Kammerer, Rémi Carles, Institute of Physical and Theoretical Chemistry [Frankfurt am Main], Goethe-Universität Frankfurt am Main, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Institut Charles Gerhardt Montpellier - Institut de Chimie Moléculaire et des Matériaux de Montpellier (ICGM ICMMM), Université Montpellier 2 - Sciences et Techniques (UM2)-Université Montpellier 1 (UM1)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Ecole Nationale Supérieure de Chimie de Montpellier (ENSCM), Zentrum Mathematik, M5 Technische Universität München, CNRS 80|prime, Université Montpellier 1 (UM1)-Université Montpellier 2 - Sciences et Techniques (UM2)-Ecole Nationale Supérieure de Chimie de Montpellier (ENSCM)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institut Charles Gerhardt Montpellier - Institut de Chimie Moléculaire et des Matériaux de Montpellier (ICGM), Ecole Nationale Supérieure de Chimie de Montpellier (ENSCM)-Institut de Chimie du CNRS (INC)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Zentrum Mathematik, Technische Universität München, Technische Universität Munchen - Université Technique de Munich [Munich, Allemagne] (TUM), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Ecole Nationale Supérieure de Chimie de Montpellier (ENSCM)-Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-Université Montpellier 1 (UM1)-Université Montpellier 2 - Sciences et Techniques (UM2)-Institut de Chimie du CNRS (INC), Technische Universität München [München] (TUM), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Université Montpellier 1 (UM1)-Université Montpellier 2 - Sciences et Techniques (UM2)-Ecole Nationale Supérieure de Chimie de Montpellier (ENSCM)-Institut de Chimie du CNRS (INC)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, FOS: Physical sciences, General Physics and Astronomy, Context (language use), 01 natural sciences, symbols.namesake, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Physics - Chemical Physics, 0103 physical sciences, FOS: Mathematics, Taylor series, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Statistical physics, 010306 general physics, 10. No inequality, Wave function, Mathematical Physics, Mathematics, Ansatz, Chemical Physics (physics.chem-ph), Collocation, 010304 chemical physics, Statistical and Nonlinear Physics, Observable, Mathematical Physics (math-ph), Function (mathematics), Action (physics), ddc, [CHIM.THEO]Chemical Sciences/Theoretical and/or physical chemistry, Modeling and Simulation, symbols, [PHYS.PHYS.PHYS-CHEM-PH]Physics [physics]/Physics [physics]/Chemical Physics [physics.chem-ph], Analysis of PDEs (math.AP)
Abstract

We consider composite quantum-dynamical systems that can be partitioned into weakly interacting subsystems, similar to system-bath type situations. Using a factorized wave function ansatz, we mathematically characterize dynamical scale separation.Specifically, we investigate a coupling r{\'e}gime that is partially flat, i.e., slowly varying with respect to one set of variables, for example, those of the bath. Further, we study the situation where one of the sets of variables is semiclassically scaled and derive a quantum-classical formulation. In both situations, we propose two schemes of dimension reduction: one based on Taylor expansion (collocation) and the other one based on partial averaging (mean-field). We analyze the error for the wave function and for the action of observables, obtaining comparable estimates for both approaches. The present study is the first step towards a general analysis of scale separation in the context of tensorized wavefunction representations., Comment: Title changed, Section 3.3 reorganized, new Section 6 (numerical example)

Published
2021
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26. Global weak solutions for quantum isothermal fluids

Author

Rémi Carles, Kleber Carrapatoso, Matthieu Hillairet, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Centre de Mathématiques Laurent Schwartz (CMLS), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM), Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Université de Rennes 1 (UR1), and Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST

Subjects
Algebra and Number Theory, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, FOS: Physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Geometry and Topology, Mathematical Physics (math-ph), Mathematical Physics, Analysis of PDEs (math.AP)
Abstract

We construct global weak solutions to isothermal quantum Navier-Stokes equations, with or without Korteweg term, in the whole space of dimension at most three. Instead of working on the initial set of unknown functions, we consider an equivalent reformulation, based on a time-dependent rescaling, that we introduced in a previous paper to study the large time behavior, and which provides suitable a priori estimates, as opposed to the initial formulation where the potential energy is not signed. We proceed by working on tori whose size eventually becomes infinite. On each fixed torus, we consider the equations in the presence of drag force terms. Such equations are solved by regularization, and the limit where the drag force terms vanish is treated by resuming the notion of renormalized solution developed by I. Lacroix-Violet and A. Vasseur. We also establish global existence of weak solutions for the isothermal Korteweg equation (no viscosity), when initial data are well-prepared, in the sense that they stem from a Madelung transform., Comment: Assumptions of Proposition 1.7 improved. More references and comments

Published
2021
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27. Global dynamics below the ground states for NLS under partial harmonic confinement

Author

Rémi Carles, Alex H. Ardila, Departamento de Matemática [Minas Gerais] (DM - UFMG), Universidade Federal de Minas Gerais, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Rennes Metropole through AIS program, Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
Physics, Scattering, Applied Mathematics, General Mathematics, Dynamics (mechanics), Harmonic (mathematics), 01 natural sciences, Supercritical fluid, Virial theorem, 010101 applied mathematics, Nonlinear system, symbols.namesake, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Quantum electrodynamics, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Nonlinear Schrödinger equation, Mathematical Physics
Abstract

We are concerned with the global behavior of the solutions of the focusing mass supercritical nonlinear Schr{\"o}dinger equation under partial harmonic confinement. We establish a necessary and sufficient condition on the initial data below the ground states to determine the global behavior (blow-up/scattering) of the solution. Our proof of scattering is based on the varia-tional characterization of the ground states, localized virial estimates, linear profile decomposition and nonlinear profiles., Comment: 35 pages

Published
2021
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28. On ground states for the 2D Schrodinger equation with combined nonlinearities and harmonic potential

Author

Rémi Carles, Yavdat Il'yasov, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Institute of Mathematics, Ufa Federal Research Centre, RAS, Instituto de Matemática, Universidade Federal de Goiás [Goiânia] (UFG), Université de Rennes 1 (UR1), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, and Russian Academy of Sciences [Moscow] (RAS)

Subjects
Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Applied Mathematics, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], FOS: Physical sciences, Mathematical Physics (math-ph), Mathematical Physics, Analysis of PDEs (math.AP)
Abstract

We consider the nonlinear Schr{\"o}dinger equation with a harmonic potential in the presence of two combined energy-subcritical power nonlinearities. We assume that the larger power is defocusing, and the smaller power is focusing. Such a framework includes physical models, and ensures that finite energy solutions are global in time. We address the questions of the existence and the orbital stability of the set of standing waves. Given the mathematical features of the equation (external potential and inhomogeneous nonlinearity), the set of parameters for which standing waves exist in unclear. In the twodimensional case, we adapt the method of fundamental frequency solutions, introduced by the second author in the higher dimensional case without potential. This makes it possible to describe accurately the set of fundamental frequency standing waves and ground states, and to prove its orbital stability., Comment: 23 pages, final version. More comments, references and explanations, some typos fixed

Published
2021
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29. Orbital stability vs. scattering in the cubic-quintic Schrödinger equation

Author

Christof Sparber, Rémi Carles, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Department of Mathematics, Statistics and Computer Science [Chicago] (UIC), University of Illinois [Chicago] (UIC), University of Illinois System-University of Illinois System, Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), and DMS-1348092, NSF

Subjects
Mathematics::Analysis of PDEs, Orbital stability, FOS: Physical sciences, Space (mathematics), 01 natural sciences, Schrödinger equation, symbols.namesake, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Nonlinear Schrödinger equation, Nonlinear Sciences::Pattern Formation and Solitons, Mathematical Physics, Mathematical physics, Uncategorized, Physics, Scattering, Cubic nonlinearity, 010102 general mathematics, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), Quintic function, 010101 applied mathematics, symbols, Scattering theory, Analysis of PDEs (math.AP)
Abstract

We consider the cubic-quintic nonlinear Schr{\"o}dinger equation in space dimension up to three. The cubic nonlinearity is thereby focusing while the quintic one is defocusing, ensuring global well-posedness of the Cauchy problem in the energy space. The main goal of this paper is to investigate the interplay between dispersion and orbital (in-)stability of solitary waves. In space dimension one, it is already known that all solitons are orbitally stable. In dimension two, we show that if the initial data belong to the conformal space, and have at most the mass of the ground state of the cubic two-dimensional Schr{\"o}dinger equation, then the solution is asymptotically linear. For larger mass, solitary wave solutions exist, and we review several results on their stability. Finally, in dimension three, relying on previous results from other authors, we show that solitons may or may not be orbitally stable., Comment: Orbital stability clarified

Published
2021
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31. On an intercritical log-modified nonlinear Schrodinger equation in two spatial dimensions

Author

Rémi Carles, Christof Sparber, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Department of Mathematics, Statistics and Computer Science [Chicago] (UIC), University of Illinois [Chicago] (UIC), University of Illinois System-University of Illinois System, Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
Physics, Applied Mathematics, General Mathematics, Mathematical analysis, 16. Peace & justice, Space (mathematics), Dispersive partial differential equation, Nonlinear system, symbols.namesake, Mathematics - Analysis of PDEs, Homogeneous space, symbols, Initial value problem, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, Nonlinear Schrödinger equation, Schrödinger's cat, Uncategorized
Abstract

We consider a dispersive equation of Schr{\"o}dinger type with a non-linearity slightly larger than cubic by a logarithmic factor. This equation is supposed to be an effective model for stable two dimensional quantum droplets with LHY correction. Mathematically, it is seen to be mass supercritical and energy subcritical with a sign-indefinite nonlinearity. For the corresponding initial value problem, we prove global in-time existence of strong solutions in the energy space. Furthermore, we prove the existence and uniqueness (up to symmetries) of nonlinear ground states and the orbital stability of the set of energy minimizers. We also show that for the corresponding model in 1D a stronger stability result is available., Comment: 15 pages

Published
2020
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32. Norm inflation for nonlinear Schrodinger equations in Fourier-Lebesgue and modulation spaces of negative regularity

Author

Rémi Carles, Divyang G. Bhimani, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Rennes Métropole., ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
Modulation space, Partial differential equation, Applied Mathematics, General Mathematics, 010102 general mathematics, 020206 networking & telecommunications, 02 engineering and technology, Lebesgue integration, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, Schrödinger equation, symbols.namesake, Nonlinear system, Fourier transform, Mathematics - Analysis of PDEs, Fourier analysis, Norm (mathematics), 0202 electrical engineering, electronic engineering, information engineering, symbols, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Analysis, Mathematics
Abstract

We consider nonlinear Schr{\"o}dinger equations in Fourier-Lebesgue and modulation spaces involving negative regularity. The equations are posed on the whole space, and involve a smooth power nonlinearity. We prove two types of norm inflation results. We first establish norm inflation results below the expected critical regularities. We then prove norm inflation with infinite loss of regularity under less general assumptions. To do so, we recast the theory of multiphase weakly nonlinear geometric optics for nonlinear Schr{\"o}dinger equations in a general abstract functional setting., Comment: 27 pages. Two references added, some typos fixed

Published
2020
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33. A conservation law with spatially localized sublinear damping

Author

Rémi Carles, Sylvain Ervedoza, Christophe Besse, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), ANR-15-CE40-0010, Agence Nationale de la Recherche, ANR-15-CE40-0010,IFSMACS,Interaction Fluide-Structure : Modélisation, analyse, contrôle et simulation(2015), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse III - Paul Sabatier ( UPS ), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-PRES Université de Toulouse-Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Centre National de la Recherche Scientifique ( CNRS ), Institut Montpelliérain Alexander Grothendieck ( IMAG ), Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ), and ANR : IFSMACS,ANR-15-CE40-0010

Subjects
[ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], Sublinear function, Space (mathematics), 01 natural sciences, Critical point (mathematics), [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics - Optimization and Control, Mathematical Physics, Physics, Conservation law, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Zero (complex analysis), Function (mathematics), 010101 applied mathematics, Boundary layer, Optimization and Control (math.OC), Ordinary differential equation, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Analysis, Analysis of PDEs (math.AP)
Abstract

International audience; We consider a general conservation law on the circle, in the presence of a sublinear damping. If the damping acts on the whole circle, then the solution becomes identically zero in finite time, following the same mechanism as the corresponding ordinary differential equation. When the damping acts only locally in space, we show a dichotomy: if the flux function is not zero at the origin, then the transport mechanism causes the extinction of the solution in finite time, as in the first case. On the other hand, if zero is a non-degenerate critical point of the flux function, then the solution becomes extinct in finite time only inside the damping zone, decays algebraically uniformly in space, and we exhibit a boundary layer, shrinking with time, around the damping zone. Numerical illustrations show how similar phenomena may be expected for other equations.

Published
2020
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34. WKB analysis of non-elliptic nonlinear Schrodinger equations

Author

Rémi Carles, Clément Gallo, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
Work (thermodynamics), Applied Mathematics, General Mathematics, 010102 general mathematics, Mathematics::Analysis of PDEs, Order (ring theory), 010103 numerical & computational mathematics, Radius, 01 natural sciences, WKB approximation, Schrödinger equation, Sobolev space, symbols.namesake, Nonlinear system, Mathematics - Analysis of PDEs, Nonlinear Sciences::Exactly Solvable and Integrable Systems, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], symbols, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Order system, Nonlinear Sciences::Pattern Formation and Solitons, Mathematical Physics, Mathematics
Abstract

We justify the WKB analysis for generalized nonlinear Schr{\"o}dinger equations (NLS), including the hyperbolic NLS and the Davey-Stewartson II system. Since the leading order system in this analysis is not hyperbolic, we work with analytic regularity, with a radius of analyticity decaying with time, in order to obtain better energy estimates. This provides qualitative information regarding equations for which global well-posedness in Sobolev spaces is widely open., Comment: arXiv admin note: text overlap with arXiv:1612.04149

Published
2020
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35. Superexponential growth or decay in the heat equation with a logarithmic nonlinearity

Author

Matthieu Alfaro, Rémi Carles, Institut Montpelliérain Alexander Grothendieck (IMAG), and Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Logarithm, Applied Mathematics, Gaussian, 010102 general mathematics, Mathematical analysis, Double exponential function, Ode, 01 natural sciences, 010101 applied mathematics, Nonlinear system, symbols.namesake, Mathematics - Analysis of PDEs, Line (geometry), symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Heat equation, Uniqueness, 0101 mathematics, Analysis, Mathematics
Abstract

We consider the heat equation with a logarithmic nonlinearity, on thereal line. For a suitable sign in front of the nonlinearity, weestablish the existence and uniqueness of solutions of the Cauchyproblem, for a well-adapted class of initial data. Explicitcomputations in the case of Gaussian data lead to various scenariiwhich are richer than the mere comparison with the ODE mechanism,involving (like in the ODE case) double exponential growth or decayfor large time. Finally, we prove that such phenomena remain, in the case of compactlysupported initial data., Comment: 14 pages

Published
2017
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36. Norm-inflation with Infinite Loss of Regularity for Periodic NLS Equations in Negative Sobolev Spaces

Author

Thomas Kappeler and Rémi Carles

Subjects
General Mathematics, 010102 general mathematics, Mathematical analysis, 01 natural sciences, Schrödinger equation, 010101 applied mathematics, Sobolev space, symbols.namesake, Norm (mathematics), symbols, 0101 mathematics, Nonlinear Schrödinger equation, Mathematics, Mathematical physics
Abstract

In this paper we consider Schrodinger equations with nonlinearities of odd order 2σ + 1 on Td. We prove that for σd ≥2, they are strongly illposed in the Sobolev space Hs for any s < 0, exhibiting norm-inflation with infinite loss of regularity. In the case of the one-dimensional cubic nonlinear Schrödinger equation and its renormalized version we prove such a result for Hs with s < −2/3.

Published
2017
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37. Universal Dynamics for the Logarithmic Schrödinger Equations

Author

Rémi Carles, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Jean-Michel Coron, Tatsien Li, Wei-Min Wang, Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
Physics, Fokker-Planck operator, Logarithm, Dynamics (mechanics), Schrödinger equation, symbols.namesake, large time behaviour, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], symbols, Nonlinear Schrödinger equation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Euler equation, logarithmic nonlinearity, Mathematical physics
Abstract

International audience; We consider the nonlinear Schrödinger equation with a logarithmic nonlinearity, whose sign is such that no non-trivial stationary solution exists. Explicit computations show that in the case of Gaussian initial data, the presence of the nonlinearity affects the large time behaviour of the solution, on at least three aspects. The dispersion is faster than usual by a logarithmic factor in time. The positive Sobolev norms of the solution grow logarithmically in time. Finally, after rescaling in space by the dispersion rate, the modulus of the solution converges to a universal Gaussian profile (whose variance is independent of the initial variance). In the case of general initial data, we show that these properties remain, up to weakening the third point (weak convergence instead of strong convergence). One of the key steps of the proof for the last point consists in using the Madelung transform. It reduces the equation to a variant of the isothermal compressible Euler equation, whose large time behaviour turns out to be governed by a parabolic equation involving a Fokker-Planck operator.

Published
2019
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38. Error estimates of a regularized finite difference method for the logarithmic Schr\'odinger equation

Author

Rémi Carles, Chunmei Su, Weizhu Bao, Qinglin Tang, Department of Mathematics [Singapore], National University of Singapore (NUS), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Department of Mathematics, University of Innsbruck, MOE2015-T2-2-146, Ministry of Education - Singapore, YJ201807, Fundamental Research Funds for the Central Universities, Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), and Universität Innsbruck [Innsbruck]

Subjects
Numerical Analysis, Logarithm, Logarithmic Schrödinger equation, AMS subject classifications. 35Q40, 35Q55, 65M15, 81Q05, Applied Mathematics, Mathematical analysis, Finite difference method, 010103 numerical & computational mathematics, regularized logarithmic Schrödinger equation, 01 natural sciences, Computational Mathematics, Nonlinear system, convergence rate, 35Q40, 35Q55, 65M15, 81Q05, Rate of convergence, error estimates, Mathematics - Numerical Analysis, 0101 mathematics, semi-implicit finite difference method, logarithmic nonlinearity, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

We present a regularized finite difference method for the logarithmic Schr\"odinger equation (LogSE) and establish its error bound. Due to the blow-up of the logarithmic nonlinearity, i.e. $\ln \rho\to -\infty$ when $\rho\rightarrow 0^+$ with $\rho=|u|^2$ being the density and $u$ being the complex-valued wave function or order parameter, there are significant difficulties in designing numerical methods and establishing their error bounds for the LogSE. In order to suppress the round-off error and to avoid blow-up, a regularized logarithmic Schr\"odinger equation (RLogSE) is proposed with a small regularization parameter $0, Comment: 23 pages, 4 figures

Published
2018
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39. On semi-classical limit of nonlinear quantum scattering

Author

Rémi Carles, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), ANR-12-JS01-0005,SchEq,Equations de Schrödinger et applications(2012), and ANR-12-MONU-0007,BECASIM,Simulation numérique avancée pour les condensats de Bose-Einstein : Modèles numériques déterministes et stochastiques, Calcul haute performance, Simulation d'expériences physiques(2012)

Subjects
Scattering, General Mathematics, 010102 general mathematics, Scattering length, Planck constant, 01 natural sciences, Classical limit, 010101 applied mathematics, Nonlinear system, symbols.namesake, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Quantum mechanics, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Coherent states, Scattering theory, Statistical physics, 0101 mathematics, Nonlinear Schrödinger equation, Mathematical Physics, Mathematics
Abstract

We consider the nonlinear Schr{\"o}dinger equation with a short-range external potential, in a semi-classical scaling. We show that for fixed Planck constant, a com-plete scattering theory is available, showing that both the potential and the nonlinearity are asymptotically negligible for large time. Then, for data under the form of coherent state, we show that a scattering theory is also available for the approximate envelope of the propagated coherent state, which is given by a nonlinear equation. In the semi-classical limit, these two scattering operators can be compared in terms of classical scattering the-ory, thanks to a uniform in time error estimate. Finally, we infer a large time decoupling phenomenon in the case of finitely many initial coherent states., Comment: 41 pages

Published
2016
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40. Complementary study of the standing wave solutions of the Gross-Pitaevskii equation in dipolar quantum gases

Author

Hichem Hajaiej, Rémi Carles, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), King Saud University [Riyadh] (KSU), ANR-12-JS01-0005,SchEq,Equations de Schrödinger et applications(2012), and ANR-13-BS01-0009,BoND,Frontières, numérique, dispersion.(2013)

Subjects
Condensed Matter::Quantum Gases, Condensed Matter::Other, General Mathematics, 010102 general mathematics, Condensation, 01 natural sciences, Symmetry (physics), 010101 applied mathematics, Standing wave, Gross–Pitaevskii equation, Dipole, Mathematics - Analysis of PDEs, Classical mechanics, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, 0101 mathematics, Orbit (control theory), Ground state, Mathematical Physics, Mathematics
Abstract

We study the stability of the standing wave solutions of a Gross-Pitaevskii equation describing Bose-Einstein condensation of dipolar quantum gases and characterize their orbit. As an intermediate step, we consider the corresponding constrained minimization problem and establish existence, symmetry and uniqueness of the ground state solutions., Comment: 10 pages

Published
2015
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41. Explicit solutions in evolutionary genetics and applications

Author

Rémi Carles and Matthieu Alfaro

Subjects
Human evolutionary genetics, Calculus, Applied mathematics, Heat equation, General Medicine, Finite time, Mathematics
Abstract

We show that the solution to a nonlocal reaction–diffusion equation, present in evolutionary genetics, can be related explicitly to the solution of the heat equation with the same initial data. As a consequence, we show different possible scenario for the solution: it can be either well-defined for all time, or become extinct in finite time, or even be defined for no positive time. In the former case, we give the leading-order asymptotic behavior of the solution for large time, which is universal.

Published
2015
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42. Rigidity results in generalized isothermal fluids

Author

Rémi Carles, Kleber Carrapatoso, Matthieu Hillairet, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), ANR-13-BS01-0003,DYFICOLTI,DYnamique des Fluides, Couches Limites, Tourbillons et Interfaces(2013), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
Physics, 010102 general mathematics, Mathematical analysis, Regular polygon, 01 natural sciences, Isothermal process, 010101 applied mathematics, symbols.namesake, Mathematics - Analysis of PDEs, Compact space, Bounded function, FOS: Mathematics, Euler's formula, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Navier–Stokes equations, Quantum, Scaling, Analysis of PDEs (math.AP)
Abstract

We investigate the long-time behavior of solutions to the isothermal Euler, Korteweg or quantum Navier Stokes equations, as well as generalizations of these equations where the convex pressure law is asymptotically linear near vacuum. By writing the system with a suitable time-dependent scaling we prove that the densities of global solutions display universal dispersion rate and asymptotic profile. This result applies to weak solutions defined in an appropriate way. In the exactly isothermal case, we establish the compactness of bounded sets of such weak solutions, by introducing modified entropies adapted to the new unknown functions., Comment: 34 pages

Published
2018
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43. On the Cauchy problem for the Hartree type equation in the Wiener algebra

Author

Lounes Mouzaoui and Rémi Carles

Subjects
Pure mathematics, Integrable system, Applied Mathematics, General Mathematics, 010102 general mathematics, Wiener algebra, Space (mathematics), 01 natural sciences, 010101 applied mathematics, symbols.namesake, Fourier transform, Square-integrable function, Hartree equation, Kernel (statistics), symbols, Standard probability space, 0101 mathematics, Mathematics
Abstract

We consider the mass-subcritical Hartree equation with a homogeneous kernel, in the space of square integrable functions whose Fourier transform is integrable. We prove a global well-posedness result in this space. On the other hand, we show that the Cauchy problem is not even locally well-posed if we simply work in the space of functions whose Fourier transform is integrable. Similar results are proven when the kernel is not homogeneous, and is such that its Fourier transform belongs to some Lebesgue space.

Published
2014
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44. Splitting methods for the nonlocal Fowler equation

Author

Rémi Carles, Afaf Bouharguane, Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), and ANR-08-BLAN-0301,MathOcean,Analyse mathématique en océanographie et applications(2008)

Subjects
Discrete-time Fourier transform, Numerical time integration, 010103 numerical & computational mathematics, 01 natural sciences, symbols.namesake, FOS: Mathematics, Mathematics - Numerical Analysis, Nonlocal operator, Operator splitting, Pseudo-spectral method, 65M15, 35K59, 86A05, 0101 mathematics, Mathematics, Fourier transform on finite groups, Algebra and Number Theory, Applied Mathematics, Mathematical analysis, Fourier inversion theorem, Finite difference method, Numerical Analysis (math.NA), 010101 applied mathematics, Split-step method, Split-step Fourier method, Computational Mathematics, Fourier transform, Error analysis, Fourier analysis, symbols, Stability, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

We consider a nonlocal scalar conservation law proposed by Andrew C. Fowler to describe the dynamics of dunes, and we develop a numerical procedure based on splitting methods to approximate its solutions. We begin by proving the convergence of the well-known Lie formula, which is an approximation of the exact solution of order one in time. We next use the split-step Fourier method to approximate the continuous problem using the fast Fourier transform and the finite difference method. Our numerical experiments confirm the theoretical results., Comment: 20 pages, 3 figures. Presentation modified, some errors fixed

Published
2013
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45. On Fourier time-splitting methods for nonlinear Schrodinger equations in the semi-classical limit II. Analytic regularity

Author

Clément Gallo, Rémi Carles, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), ANR-12-MONU-0007,BECASIM,Simulation numérique avancée pour les condensats de Bose-Einstein : Modèles numériques déterministes et stochastiques, Calcul haute performance, Simulation d'expériences physiques(2012), and ANR-12-JS01-0005,SchEq,Equations de Schrödinger et applications(2012)

Subjects
Discretization, Applied Mathematics, Mathematical analysis, 010103 numerical & computational mathematics, 16. Peace & justice, Planck constant, 01 natural sciences, Classical limit, WKB approximation, Schrödinger equation, 010101 applied mathematics, Computational Mathematics, Nonlinear system, symbols.namesake, Mathematics - Analysis of PDEs, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), Mathematics - Numerical Analysis, 0101 mathematics, Nonlinear Schrödinger equation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

We consider the time discretization based on Lie-Trotter splitting, for the nonlinear Schrodinger equation, in the semi-classical limit, with initial data under the form of WKB states. We show that both the exact and the numerical solutions keep a WKB structure, on a time interval independent of the Planck constant. We prove error estimates, which show that the quadratic observables can be computed with a time step independent of the Planck constant. The functional framework is based on time-dependent analytic spaces, in order to overcome a previously encountered loss of regularity phenomenon., Comment: 21 pages

Published
2017
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46. WKB analysis of generalized derivative nonlinear Schrodinger equations without hyperbolicity

Author

Clément Gallo, Rémi Carles, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), ANR-12-MONU-0007,BECASIM,Simulation numérique avancée pour les condensats de Bose-Einstein : Modèles numériques déterministes et stochastiques, Calcul haute performance, Simulation d'expériences physiques(2012), and ANR-13-BS01-0009,BoND,Frontières, numérique, dispersion.(2013)

Subjects
Applied Mathematics, 010102 general mathematics, Mathematical analysis, Derivative, Space (mathematics), 01 natural sciences, WKB approximation, Euler equations, Schrödinger equation, 010101 applied mathematics, symbols.namesake, Nonlinear system, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Modeling and Simulation, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), 0101 mathematics, Nonlinear Schrödinger equation, Mathematical Physics, Mathematics
Abstract

We consider the semi-classical limit of nonlinear Schrodinger equations in the presence of both a polynomial nonlinearity and thederivative in space of a polynomial nonlinearity. By working in a class of analytic initial data, we do not have to assume any hyperbolicstructure on the (limiting) phase/amplitude system. The solution, its approximation, and the error estimates are considered in time dependent analyticregularity., Comment: 13 pages

Published
2016
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47. Replicator-mutator equations with quadratic fitness

Author

Matthieu Alfaro, Rémi Carles, Institut Montpelliérain Alexander Grothendieck (IMAG), and Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Work (thermodynamics), Applied Mathematics, General Mathematics, Gaussian, 010102 general mathematics, Harmonic potential, 01 natural sciences, Term (time), 010101 applied mathematics, symbols.namesake, Quadratic equation, Mathematics - Analysis of PDEs, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, Heat equation, 0101 mathematics, Finite time, Mathematics
Abstract

This work completes our previous analysis on models arising in evolutionary genetics. We consider the so-called replicator-mutator equation, when the fitness is quadratic. This equation is a heat equation with a harmonic potential, plus a specific nonlocal term. We give an explicit formula for the solution, thanks to which we prove that when the fitness is non-positive (harmonic potential), solutions converge to a universal stationary Gaussian for large time, whereas when the fitness is non-negative (inverted harmonic potential), solutions always become extinct in finite time., Comment: 12 pages

Published
2016
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48. Universal dynamics for the defocusing logarithmic Schrodinger equation

Author

Isabelle Gallagher, Rémi Carles, Institut Montpelliérain Alexander Grothendieck ( IMAG ), Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques de Jussieu ( IMJ ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), Département de Mathématiques et Applications - ENS Paris ( DMA ), École normale supérieure - Paris ( ENS Paris ) -Centre National de la Recherche Scientifique ( CNRS ), ANR-13-BS01-0009,BoND,Frontières, numérique, dispersion. ( 2013 ), ANR-12-MONU-0007,BECASIM,Simulation numérique avancée pour les condensats de Bose-Einstein : Modèles numériques déterministes et stochastiques, Calcul haute performance, Simulation d'expériences physiques ( 2012 ), Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Département de Mathématiques et Applications - ENS Paris (DMA), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), ANR-13-BS01-0009,BoND,Frontières, numérique, dispersion.(2013), ANR-12-MONU-0007,BECASIM,Simulation numérique avancée pour les condensats de Bose-Einstein : Modèles numériques déterministes et stochastiques, Calcul haute performance, Simulation d'expériences physiques(2012), École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Fokker-Planck operator, Logarithm, General Mathematics, Gaussian, global attractor, 01 natural sciences, symbols.namesake, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Analysis of PDEs, 35Q40, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Euler equation, 0101 mathematics, nonlinear Schrödinger equation, Nonlinear Schrödinger equation, logarithmic nonlinearity, Mathematics, Operator (physics), 010102 general mathematics, Mathematical analysis, Euler equations, 010101 applied mathematics, Sobolev space, Fokker–Planck operator, 35Q55, Nonlinear system, Sobolev norms, symbols, Compressibility
Abstract

We consider the nonlinear Schrodinger equation with a logarithmic nonlinearity in a dispersive regime. We show that the presence of the nonlinearity affects the large time behavior of the solution: the dispersion is faster than usual by a logarithmic factor in time and the positive Sobolev norms of the solution grow logarithmically in time. Moreover, after rescaling in space by the dispersion rate, the modulus of the solution converges to a universal Gaussian profile. These properties are suggested by explicit computations in the case of Gaussian initial data, and remain when an extra power-like nonlinearity is present in the equation. One of the key steps of the proof consists in using the Madelung transform to reduce the equation to a variant of the isothermal compressible Euler equation, whose large time behavior turns out to be governed by a parabolic equation involving a Fokker-Planck operator., Comment: Final version

Published
2016
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49. Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials

Author

Rémi Carles, Christof Sparber, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Department of Mathematics, Statistics and Computer Science [Chicago] (UIC), University of Illinois [Chicago] (UIC), University of Illinois System-University of Illinois System, and ANR-08-JCJC-0124,R.A.S.,Régimes Asymptotiques pour l'équation de Schrödinger(2008)

Subjects
Applied Mathematics, Wave packet, 010102 general mathematics, Semiclassical physics, 01 natural sciences, Periodic potential, Schrödinger equation, symbols.namesake, Mathematics - Analysis of PDEs, Classical mechanics, Effective mass (solid-state physics), [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Discrete Mathematics and Combinatorics, 010307 mathematical physics, 0101 mathematics, Adiabatic process, MSC : 81Q20, 35A35, 35Q40, 81Q05, Mathematical Physics, Mathematics
Abstract

We consider semiclassically scaled Schrodinger equations with an external potential and a highly oscillatory periodic potential. We construct asymptotic solutions in the form of semiclassical wave packets. These solutions are concentrated (both, in space and in frequency) around the effective semiclassical phase-space flow obtained by Peierl's substitution, and involve a slowly varying envelope whose dynamics is governed by a homogenized Schrodinger equation with time-dependent effective mass. The corresponding adiabatic decoupling of the slow and fast degrees of freedom is shown to be valid up to Ehrenfest time scales., Comment: 15 pages. More explanations. An error fixed in the main result, to include the Berry phase

Published
2012
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50. Geometric optics and instability for NLS and Davey–Stewartson models

Author

Christof Sparber, Rémi Carles, Eric Dumas, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Institut Fourier (IF ), Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Centre National de la Recherche Scientifique (CNRS), Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge [UK] (CAM), ANR-08-JCJC-0124,R.A.S.,Régimes Asymptotiques pour l'équation de Schrödinger(2008), Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Department of Applied Mathematics and Theoretical Physics / Centre for Mathematical Sciences (DAMTP/CMS), and ANR-08-JCJC-0124-01,R.A.S,Régimes Asymptotiques pour l'équation de Schrödinger(2008)

Subjects
Physics, Geometrical optics, Applied Mathematics, General Mathematics, 010102 general mathematics, Mathematical analysis, 01 natural sciences, Constructive, Instability, Schrödinger equation, 010101 applied mathematics, Sobolev space, symbols.namesake, Nonlinear system, Mathematics - Analysis of PDEs, Norm (mathematics), FOS: Mathematics, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Initial value problem, 0101 mathematics, Analysis of PDEs (math.AP)
Abstract

We study the interaction of (slowly modulated) high frequency waves for multi-dimensional nonlinear Schrodinger equations with gauge invariant power-law nonlinearities and non-local perturbations. The model includes the Davey--Stewartson system in its elliptic-elliptic and hyperbolic-elliptic variant. Our analysis reveals a new localization phenomenon for non-local perturbations in the high frequency regime and allows us to infer strong instability results on the Cauchy problem in negative order Sobolev spaces, where we prove norm inflation with infinite loss of regularity by a constructive approach., Comment: 33 pages

Published
2012
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