Article
Mathematics, Applied
Vincent Calvez, Jose Antonio Carrillo, Franca Hoffmann
Summary: In this study, a generalized Keller-Segel model with non-linear porous medium type diffusion and non-local attractive power law interaction is considered, with a focus on potentials more singular than Newtonian interaction. Uniqueness of stationary states in different regimes is shown, with the key result being a sharp functional inequality in the radial setting.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2021)
Article
Mathematics, Applied
Akos Nagy
Summary: We study stationary solutions to the Keller-Segel equation on curved planes and prove the necessity of mass being 8 pi with a sharp decay bound. We establish a correspondence between stationary solutions and positively curved Riemannian metrics on the sphere and show the nonexistence of solutions in certain situations. Additionally, we prove a curved version of the logarithmic Hardy-Littlewood-Sobolev inequality and demonstrate that the Keller-Segel free energy is bounded from below when the mass is 8 pi, even in the curved case.
PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS
(2023)
Article
Mathematics
Jaewook Ahn, Kyungkeun Kang, Jihoon Lee
Summary: The passage discusses a class of logarithmic Keller-Segel type systems modeling the spatio-temporal behavior of chemotactic cells or criminal activities in spatial dimensions two and higher. It establishes the existence of classical solutions globally in time under certain assumptions on parameter values and given functions. The text also introduces a new type of small initial data to obtain global classical solutions and discusses the long-time asymptotic behaviors of solutions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Ujjal Das
Summary: The text discusses weighted logarithmic Sobolev inequalities and logarithmic Hardy inequalities in various spaces and conditions, exploring their properties and related theorems.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Guanglan Wang, Yan Wu, Guoliang Li
Summary: This article investigates several sharp weighted Adams type inequalities in Lorentz-Sobolev spaces by utilizing symmetry, rearrangement, and the Riesz representation formula. Moreover, the sharpness of these inequalities is established through the construction of an appropriate test sequence.
Article
Mathematics
Jianxiong Wang
Summary: In this paper, we derive a double-weighted Hardy-Sobolev inequality with monomial weights on Euclidean space and establish the Gross' type logarithmic Sobolev inequality with monomial weights using product structure. We also study the Moser-Onofri-Beckner inequality with monomial weights and explicitly give some best constants and extremals.
POTENTIAL ANALYSIS
(2023)
Article
Mathematics
Jean Dolbeault, Rupert L. Frank, Louis Jeanjean
Summary: In two dimensions, we study the free energy and ground state energy of the Schrodinger-Poisson system coupled with a logarithmic nonlinearity, taking into account scaling invariances, external potential with minimal growth, and new logarithmic interpolation inequalities. This two-dimensional model serves as a limit case of more classical problems in higher dimensions, and considers both repulsive and attractive forces.
COMPTES RENDUS MATHEMATIQUE
(2021)
Article
Mathematics, Applied
Yansheng Shen
Summary: This article investigates the existence of nontrivial solutions to nonlocal elliptic problems in R-N involving fractional Laplacians and the Hardy-Sobolev-Maz'ya potential. Using variational methods, the attainability of the corresponding minimization problem is studied, leading to the existence of solutions. Another Choquard type equation involving the p-Laplacian and critical nonlinearities in R-N is also considered.
ADVANCED NONLINEAR STUDIES
(2021)
Article
Mathematics, Applied
Shutao Zhang, Yazhou Han
Summary: This paper studies the existence of extremal problems for the Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds without boundary using the Concentration-Compactness principle.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Physics, Mathematical
Nicholas LaRacuente
Summary: This paper presents a class of purely multiplicative comparisons of quantum relative entropy, including the relative entropy of a density with respect to its subalgebraic restriction. These inequalities approach known inequalities as the perturbation size approaches zero. Based on these results, a type of inequality known as quasi-factorization or approximate tensorization of relative entropy is obtained. It has applications in uncertainty-like relations and decay estimates of mixing processes.
JOURNAL OF MATHEMATICAL PHYSICS
(2022)
Article
Physics, Multidisciplinary
Ivan Bardet, Angela Capel, Li Gao, Angelo Lucia, David Perez-Garcia, Cambyse Rouze
Summary: This paper proves that spin chains weakly coupled to a large heat bath thermalize rapidly at any temperature for finite-range, translation-invariant commuting Hamiltonians, reaching equilibrium in a time which scales logarithmically with the system size. This generalizes to the quantum regime a seminal result of Holley and Stroock from 1989 for classical spin chains and represents an exponential improvement over previous bounds based on the nonclosure of the spectral gap. We discuss the implications in the context of dissipative phase transitions and in the study of symmetry protected topological phases.
PHYSICAL REVIEW LETTERS
(2023)
Article
Physics, Multidisciplinary
Ivan Bardet, Angela Capel, Cambyse Rouze
Summary: In this paper, a new generalisation of the strong subadditivity of entropy to von Neumann algebras is derived, introducing the concept of approximate tensorization of relative entropy. This concept provides a lower bound for the sum of relative entropies between a density and its projections onto different algebras, and is key in the modern proofs of the logarithmic Sobolev inequality for classical lattice spin systems, while also offering estimates on constants based on clustering conditions in quantum lattice spin systems.
ANNALES HENRI POINCARE
(2022)
Article
Mathematics, Applied
Sushmita Rawat, Konijeti Sreenadh
Summary: This paper investigates the existence, multiplicity, and regularity of positive weak solutions for the Kirchhoff-Choquard problem. The study shows that each positive weak solution is bounded and satisfies Holder regularity of order s, and the existence of two positive solutions is proved using variational methods and truncation arguments.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics
Justin Salez, Konstantin Tikhomirov, Pierre Youssef
Summary: For reversible Markov chains on finite state spaces, it is shown that the modified log-Sobolev inequality (MLSI) can be upgraded to a log-Sobolev inequality (LSI) with a slight degradation in the associated constant. The first log-Sobolev estimate for Zero-Range processes on arbitrary graphs is provided as an illustration. Additionally, the modified log-Sobolev constant of the Lamplighter chain on all bounded-degree graphs is determined, and it is used to provide negative answers to two open questions by Montenegro and Tetali (2006) [27] and Hermon and Peres (2018) [17]. The proof is based on the 'regularization trick' introduced by the authors. (c) 2023 Elsevier Inc. All rights reserved.
JOURNAL OF FUNCTIONAL ANALYSIS
(2023)
Article
Mathematics
Taiki Takeuchi
Summary: We prove the existence and uniqueness of local strong solutions of Keller-Segel system, as well as construct global strong solutions for small initial data. The proof is based on the maximal Lorentz regularity theorem of heat equations.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Physics, Mathematical
Denis Bonheure, Jean Dolbeault, Maria J. Esteban, Ari Laptev, Michael Loss
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2020)
Article
Mathematics
Jean Dolbeault, Xingyu Li
Summary: This paper focuses on logarithmic Hardy-Littlewood-Sobolev inequalities in the 2D Euclidean space with an external potential exhibiting logarithmic growth. The introduction of a new parameter through the coupling with the potential leads to two different regimes. The attractive regime reflects the standard logarithmic Hardy-Littlewood-Sobolev inequality, while the second regime results in a reverse inequality, enabling the bounding of the free energy of a drift-diffusion-Poisson system from below.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2021)
Article
Mathematics, Applied
Jean Dolbeault, Maria J. Esteban
ADVANCED NONLINEAR STUDIES
(2020)
Article
Physics, Mathematical
Lanoir Addala, Jean Dolbeault, Xingyu Li, M. Lazhar Tayeb
Summary: This paper focuses on the linearized Vlasov-Poisson-Fokker-Planck system in the presence of an external potential of confinement. Through hypocoercivity methods and a specific notion of scalar product adapted to the Poisson coupling, the large time behavior of the solutions is investigated with uniform estimates in the diffusion limit. As an application in a simple case, the one-dimensional case is studied and exponential convergence of the nonlinear Vlasov-Poisson-Fokker-Planck system is proven without any small mass assumption.
JOURNAL OF STATISTICAL PHYSICS
(2021)
Article
Mathematics, Applied
Kleber Carrapatoso, Jean Dolbeault, Frederic Herau, Stephane Mischler, Clement Mouhot
Summary: We prove functional inequalities on vector fields u : R-d -> R-d when R-d is equipped with a bounded measure e(-phi) dx that satisfies a Poincare inequality, and study associated self-adjoint operators. The weighted Korn inequality compares the differential matrix Du, once projected orthogonally to certain finite-dimensional spaces, with its symmetric part D(s)u and, in an improved form of the inequality, an additional term del phi.u. We also consider Poincare-Korn inequalities for estimating a projection of u by D(s)u and zeroth-order versions of these inequalities obtained using the Witten-Laplace operator. The constants depend on geometric properties of the potential phi and the estimates are quantitative and constructive. These inequalities are motivated by kinetic theory and related with the Korn inequality (1906) in mechanics, which compares Du and D(s)u on a bounded domain.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2022)
Article
Physics, Mathematical
Emeric Bouin, Jean Dolbeault, Laurent Lafleche
Summary: This paper investigates the large time behavior of kinetic equations without confinement, focusing on collision operators with fat tailed local equilibria and their anomalous diffusion limit. The study develops an L-2-hypocoercivity approach at the kinetic level to establish a decay rate compatible with the fractional diffusion limit.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2022)
Article
Mathematics, Applied
Jean Dolbeault
Summary: Interpolation inequalities play a crucial role in analysis, mathematical physics, nonlinear PDEs, and other areas of Science. Research interests have shifted towards qualitative questions and the use of entropy methods, which provide insights into optimal rates of decay and stability in various inequalities. The framework of entropy methods allows for the study of nonlinear regimes and their linearized counterparts, leading to optimality results and symmetry properties.
MILAN JOURNAL OF MATHEMATICS
(2021)
Article
Mathematics, Applied
Jean Dolbeault, An Zhang
Summary: The carre du champ method is a powerful technique for proving interpolation inequalities with explicit constants in presence of a non-trivial metric on a manifold. The method applies to some classical Gagliardo-Nirenberg-Sobolev inequalities on the sphere, with optimal constants. Very nonlinear regimes close to the critical Sobolev exponent can be covered using nonlinear parabolic flows of porous medium or fast diffusion type.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2022)
Correction
Mathematics
Jean Dolbeault, Maria J. Esteban, Eric Sere
Summary: This corrigendum addresses some overlooked closability issues in [1].
JOURNAL OF FUNCTIONAL ANALYSIS
(2023)
Article
Mathematics
Jean Dolbeault, Rupert L. Frank, Louis Jeanjean
Summary: In two dimensions, we study the free energy and ground state energy of the Schrodinger-Poisson system coupled with a logarithmic nonlinearity, taking into account scaling invariances, external potential with minimal growth, and new logarithmic interpolation inequalities. This two-dimensional model serves as a limit case of more classical problems in higher dimensions, and considers both repulsive and attractive forces.
COMPTES RENDUS MATHEMATIQUE
(2021)
Review
Physics, Mathematical
Denis Bonheure, Jean Dolbeault, Maria J. Esteban, Ari Laptev, Michael Loss
Summary: This paper explores nonlinear interpolation inequalities associated with Schrodinger operators involving Aharonov-Bohm magnetic potentials, focusing on symmetry and considering various cases in different geometric settings. The emphasis is on new results and methods, particularly in the presence of a magnetic field, with the most significant applications being new magnetic Hardy inequalities in dimensions 2 and 3.
REVIEWS IN MATHEMATICAL PHYSICS
(2021)
Article
Mathematics
E. Bouin, J. Dolbeault, L. Lafleche, C. Schmeiser
Summary: Hypocoercivity methods are used in linear kinetic equations with sub-exponential decay of local equilibria, deriving global rates of decay through Nash type estimates. The method is applicable to Fokker-Planck and scattering collision operators, utilizing weighted Poincare inequality and norms with various weights. Weighted Poincare inequalities are advantageous in describing convergence rates to local equilibrium without the need for extra regularity assumptions, covering transitions from super-exponential and exponential to sub-exponential local equilibria.
MONATSHEFTE FUR MATHEMATIK
(2021)
Article
Mathematical & Computational Biology
Jean Dolbeault, Gabriel Turinici
MATHEMATICAL MODELLING OF NATURAL PHENOMENA
(2020)
Article
Mathematics, Applied
Emeric Bouin, Jean Dolbeault, Christian Schmeiser
KINETIC AND RELATED MODELS
(2020)
Article
Mathematics, Applied
Jean Dolbeault, Marta Garcia-Huidobro, Raul Manasevich
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2020)