Article
Mathematics, Applied
Hao-Guang Li, Chao-Jiang Xu
Summary: This study demonstrates the analytic regularizing effect of the Cauchy problem of the nonlinear Landau equation with hard potentials in a close-to-equilibrium framework. The evolution of the analytic radius is shown to be identical to that of heat equations.
SCIENCE CHINA-MATHEMATICS
(2022)
Article
Mathematics
Jin Woo Jang, Seok-Bae Yun
Summary: In this paper, the propagation of L-p upper bounds for the spatially homogeneous relativistic Boltzmann equation is proved for any 1 < p < infinity, with the case of relativistic hard ball with Grad's angular cutoff considered. The proof is based on a detailed study of the inter-relationship between the relative momenta, the regularity and the L-p estimates for the gain operator, the development of the relativistic Carleman representation, and several estimates on the relativistic hypersurface E-v'-v(v)*. Additionally, a Pythagorean theorem for the relative momenta g(v, v(*)), g(v, v'), and g(v', v(*)) is derived, playing a crucial role in reducing the momentum singularity.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Hao-guang Li, Chao-jiang Xu
Summary: In this paper, an improved new argument is presented to prove that the solution of the Cauchy problem for the nonlinear spatially homogeneous Landau equation with hard potentials with L-2(R-3) initial datum exhibits an analytic Gelfand-Shilov regularizing effect in the class S-1(1) (R-3), where the evolution of the analytic radius is similar to the heat equation.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2023)
Article
Chemistry, Physical
S. M. Tschopp, H. D. Vuijk, J. M. Brader
Summary: Superadiabatic dynamical density functional theory (superadiabatic-DDFT) is used to investigate the response of interacting Brownian particles to time-dependent external driving. The theory accurately predicts the time-evolution of the one-body density without the need for adjustable parameters or simulation input. Comparison with other methods shows that superadiabatic-DDFT gives reliable results.
JOURNAL OF CHEMICAL PHYSICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Yunan Yang, Denis Silantyev, Russel Caflisch
Summary: We develop an adjoint method for DSMC for the spatially homogeneous Boltzmann equation with a general collision law, extending previous results restricted to constant collision rate. The challenge lies in dealing with variable collision rates, requiring rejection sampling in the DSMC algorithm. Our method introduces a new term, the score function, and an adjoint Jacobian matrix to capture the dependence of collision parameters on velocities. This new approach applies to a wider range of collision models.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics
Ling-Bing He, Jie Ji
Summary: Departing from the weak solution, this paper proves the uniqueness and long-time behavior of the weak solution to the non-cutoff spatially homogeneous Boltzmann equation with moderate soft potentials. The proof relies on the development of the localized techniques in phase and frequency spaces and entropy methods.
INTERNATIONAL JOURNAL OF MATHEMATICS
(2022)
Article
Mathematics, Applied
Mario Pulvirenti, Sergio Simonella
Summary: The study continues to investigate the statistics of backward clusters in a gas of N hard spheres of small diameter epsilon, providing an estimate of the average cardinality of clusters with respect to the equilibrium measure under specific conditions in the system of hard spheres.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2021)
Article
Mathematics, Applied
Jianjun Huang, Zhenglu Jiang
Summary: We study the spatially homogeneous solutions for the relativistic kinetic equations and prove the existence of global weak solutions for the Cauchy problem of the relativistic Boltzmann and Landau equation with finite mass, energy, and entropy in the initial data. We also investigate the asymptotic behavior of grazing collisions in the relativistic Boltzmann equation and prove that solutions weakly converge to the solutions of the relativistic Landau equation when almost all collisions are grazing. These results extend Villani's work on the spatially homogeneous Boltzmann and Landau equations in the classical case.
APPLICABLE ANALYSIS
(2023)
Article
Mathematics, Applied
Daniel Heydecker
Summary: We investigate Kac's many-particle stochastic model of gas dynamics with hard potentials and moderate angular singularity. We show that the noncutoff particle system can be obtained as the limit of cutoff systems, regardless of the number of particles N. As a result, we establish the wellposedness of the corresponding Boltzmann equation and the propagation of chaos in the many-particle limit N -> infinity.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2022)
Article
Physics, Mathematical
Bocheng Liu, Xuguang Lu
Summary: In this paper, we prove the strong and time-averaged strong convergence to equilibrium for solutions of the spatially homogeneous Boltzmann equation for Fermi-Dirac particles with general initial data. The assumption on the collision kernel includes the Coulomb potential with a weaker angular cutoff. The proof is based on moment estimates, entropy dissipation inequalities, regularity of the collision gain operator, and a new observation regarding the comparison of collision kernels.
JOURNAL OF STATISTICAL PHYSICS
(2023)
Article
Physics, Mathematical
Irene M. M. Gamba, Milana Pavic-Colic
Summary: In this article, we study the Boltzmann equation for modeling polyatomic gases with the introduction of a microscopic internal energy variable. We establish the existence and uniqueness theory for the full non-linear case under an extended Grad-type assumption on transition probability rates. The Cauchy problem is solved using abstract ordinary differential equation (ODE) theory in Banach spaces, with various constraints on the initial data.
JOURNAL OF MATHEMATICAL PHYSICS
(2023)
Article
Physics, Mathematical
Renjun Duan, Dongcheng Yang, Hongjun Yu
Summary: This paper focuses on the large time asymptotics toward the viscous contact waves for solutions of the Landau equation with physically realistic Coulomb interactions. By introducing a new time-velocity weight function, it is proved that the solution tends toward a local Maxwellian in large time, which is the first result on the dynamical stability of contact waves for the Landau equation.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2022)
Article
Mathematics, Applied
O. O. Hukalov, V. D. Gordevskyy
Summary: The paper presents explicit approximate solutions of the Boltzmann equation for the hard spheres model in the form of function series of Maxwellians with coefficient functions of a spatial coordinate and time. It also establishes sufficient conditions for minimizing the uniform-integral error between the parts of the Boltzmann equation for the constructed distribution.
JOURNAL OF MATHEMATICAL PHYSICS ANALYSIS GEOMETRY
(2021)
Article
Mathematics, Applied
Haoguang Li, Hengyue Wang
Summary: This study demonstrates the spectral decomposition of linear and nonlinear radially symmetric homogeneous non-cutoff Landau operators under the hard potential gamma = 2 in a perturbation framework, proving the existence of solutions and the Gelfand-Shilov smoothing effect for the Cauchy problem of the symmetric homogenous Landau equation with small initial datum.
JOURNAL OF PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics
Daniele Cassani, Zhisu Liu, Giulio Romani
Summary: This article investigates the strongly coupled nonlinear Schrodinger equation and Poisson equation in two dimensions. The existence of solutions is proved using a variational approximating procedure, and qualitative properties of the solutions are established through the moving planes technique.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Giovanni Alessandrini, Romina Gaburro, Eva Sincich
Summary: This paper considers the inverse problem of determining the conductivity of a possibly anisotropic body Ω, subset of R-n, by means of the local Neumann-to-Dirichlet map on a curved portion Σ of its boundary. Motivated by the uniqueness result for piecewise constant anisotropic conductivities, the paper provides a Hölder stability estimate on Σ when the conductivity is a priori known to be a constant matrix near Σ.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Nuno Costa Dias, Cristina Jorge, Joao Nuno Prata
Summary: This article studies the time dependent Euler-Bernoulli beam equation with discontinuous and singular coefficients, and obtains an explicit formulation of the differential problem using an extension of the Hormander product of distributions. The dynamics of the Euler-Bernoulli beam model with discontinuous flexural stiffness and structural cracks are further explored, and the relationship between the characteristic frequencies of the beam and the singularities in the flexural stiffness is investigated.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Baoquan Zhou, Hao Wang, Tianxu Wang, Daqing Jiang
Summary: This paper is Part I of a two-part series that presents a mathematical framework for approximating the invariant probability measures and density functions of stochastic generalized Kolmogorov systems with small diffusion. It introduces two new approximation methods and demonstrates their utility in various applications.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Yun Li, Danhua Jiang, Zhi-Cheng Wang
Summary: In this study, a nonlocal reaction-diffusion equation is used to model the growth of phytoplankton species in a vertical water column with changing-sign advection. The species relies solely on light for metabolism. The paper primarily focuses on the concentration phenomenon of phytoplankton under conditions of large advection amplitude and small diffusion rate. The findings show that the phytoplankton tends to concentrate at certain critical points or the surface of the water column under these conditions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Monica Conti, Stefania Gatti, Alain Miranville
Summary: The aim of this paper is to study a perturbation of the Cahn-Hilliard equation with nonlinear terms of logarithmic type. By proving the existence, regularity and uniqueness of solutions, as well as the (strong) separation properties of the solutions from the pure states, we finally demonstrate the convergence to the Cahn-Hilliard equation on finite time intervals.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Qi Qiao
Summary: This paper investigates a volume-filling chemotaxis model with a small cell diffusion coefficient and chemotactic sensitivity. By using the geometric singular perturbation theory, the existence of a positive traveling wave connecting two constant steady states is confirmed. The monotonicity of the wave is analyzed for different parameter ranges, and spectral instability is observed in some exponentially weighted spaces.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Xiaolong He
Summary: This article employs the CWB method to construct quasi-periodic solutions for nonlinear delayed perturbation equations, and combines the techniques of Green's function estimate and the reducibility method in KAM theory to solve the linear equation, thus extending the applicability of the CWB method. As an application, it studies the positive quasi-periodic solutions for a class of Lotka-Volterra equations with quasi-periodic coefficients and time delay.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Nicolas Camps, Louise Gassot, Slim Ibrahim
Summary: In this paper, we consider the probabilistic local well-posedness problem for the Schrodinger half-wave equation with a cubic nonlinearity in quasilinear regimes. Due to the lack of probabilistic smoothing in the Picard's iterations caused by high-low-low nonlinear interactions, we need to use a refined ansatz. The proof is an adaptation of Bringmann's method on the derivative nonlinear wave equation [6] to Schrodinger-type equations. In addition, ill-posedness results for this equation are discussed.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Elie Abdo, Mihaela Ignatova
Summary: In this study, we investigate the Nernst-Planck-Navier-Stokes system with periodic boundary conditions and prove the exponential nonlinear stability of constant steady states without constraints on the spatial dimension. We also demonstrate the exponential stability from arbitrary large data in the case of two spatial dimensions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Peter De Maesschalck, Joan Torregrosa
Summary: This paper provides the best lower bound for the number of critical periods of planar polynomial centers known up to now. The new lower bound is obtained in the Hamiltonian class and considering a single period annulus. The key idea is the perturbation of a vector field with many cusp equilibria, which is constructed using elements of catastrophe theory.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Leyi Jiang, Taishan Yi, Xiao-Qiang Zhao
Summary: This paper studies the propagation dynamics of a class of integro-difference equations with a shifting habitat. By transforming the equation using moving coordinates and establishing the spreading properties of solutions and the existence of nontrivial forced waves, the paper contributes to the understanding of the propagation properties of the original equation.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Mckenzie Black, Changhui Tan
Summary: This article investigates a family of nonlinear velocity alignments in the compressible Euler system and shows the asymptotic emergent phenomena of alignment and flocking. Different types of nonlinearity and nonlocal communication protocols are studied, resulting in a variety of different asymptotic behaviors.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Lorenzo Cavallina
Summary: In this paper, the concept of variational free boundary problem is introduced, and a unified functional-analytical framework is provided for constructing families of solutions. The notion of nondegeneracy of a critical point is extended to this setting.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Ying-Chieh Lin, Kuan-Hsiang Wang, Tsung-Fang Wu
Summary: In this study, we investigate a linearly coupled Schrodinger system and establish the existence of positive ground states under suitable assumptions and by using variational methods. We also relax some of the conditions and provide some results on the existence of positive ground states to a linearly coupled Schrodinger system in a bounded domain.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)