Article
Mathematics, Applied
Antonio Esposito, Francesco S. Patacchini, Andre Schlichting, Dejan Slepcev
Summary: The research explores dynamics driven by interaction energies on graphs and introduces graph analogues of the continuum nonlocal-interaction equation as gradient flows with respect to a graph Wasserstein distance. The study focuses on the existence theory for solutions of the nonlocal nonlocal-interaction equation on graphs, showing convergence of solutions as the empirical measures of the set of vertices converge weakly, establishing a valuable discrete-to-continuum convergence result.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2021)
Article
Mathematics
Viorel Barbu, Michael Rockner
Summary: This article studies the properties of the omega-limit set corresponding to the orbit of a solution to a nonlinear Fokker-Planck equation in Banach spaces. The main conclusion is that the equation has an equilibrium state and the omega-limit set is a nonempty, compact subset of L-1(R-d).
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Biochemical Research Methods
Qi Jiang, Shuo Zhang, Lin Wan
Summary: This study introduces a novel nonlinear model and dynamic inference framework based on graphs, which can reconstruct cell state transitions and unveil nonlinear cell-cell interactions from single-cell RNA data, showing promising applications in understanding cellular dynamics.
PLOS COMPUTATIONAL BIOLOGY
(2022)
Article
Mathematics, Applied
Li Chen, Fucai Li, Yue Li, Nicola Zamponi
Summary: This paper considers the compressible Vlasov-Poisson-Fokker-Planck-Navier-Stokes system in a three-dimensional bounded domain with nonhomogeneous Dirichlet boundary conditions. The system describes the evolution of charged particles ensemble dispersed in an isentropic fluid. For the adiabatic coefficient gamma>3/2, the global existence of weak solutions to this system with arbitrary large initial and boundary data is established.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics
Reem Abdullah Aljethi, Adem Kilicman
Summary: This paper proposes a generalized fractional Fokker-Planck equation based on a stable Levy stochastic process. By using the Levy process instead of the Brownian motion, this fractional equation provides a better description of heavy tails and skewness. The analytical solution is used to solve the fractional equation and is expressed using the H-function to demonstrate the indicator entropy production rate. Market data is modeled using a stable distribution to show the relationships between tails and the new fractional Fokker-Planck model, and an R code is developed for drawing figures from real data.
Article
Computer Science, Interdisciplinary Applications
Ibrahim Almuslimani, Nicolas Crouseilles
Summary: In this work, numerical schemes are constructed for the Vlasov-Fokker-Planck system coupled with Poisson or Ampere equation. The schemes efficiently preserve mass, momentum, and total energy, while utilizing splitting methods and a stabilized Runge-Kutta-Chebyshev integrator. The conservation properties are proved, and numerical experiments demonstrate the efficiency of the schemes for Landau damping and bump-on-tail instability phenomena.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Physics, Mathematical
Cristyan Pinheiro, Gabriela Planas
Summary: This study focuses on the alpha-Navier-Stokes equations coupled with a Vlasov type equation to model the flow of incompressible fluid with small particles. It establishes the existence of global weak solutions under periodic boundary conditions and investigates the regularity and uniqueness of these solutions. The study also proves the convergence of solutions to the Navier-Stokes-Vlasov equations as alpha tends to zero, and extends the results to models involving spray diffusion.
JOURNAL OF MATHEMATICAL PHYSICS
(2021)
Article
Mathematics, Applied
Chenghua Duan, Wenbin Chen, Chun Liu, Xingye Yue, Shenggao Zhou
Summary: In this work, novel structure-preserving numerical schemes are developed for a class of nonlinear Fokker-Planck equations with nonlocal interactions. The schemes uniquely solve the equations, respect mass conservation and positivity, and preserve steady states at the fully discrete level. Extensive numerical simulations demonstrate the effectiveness and robustness of the proposed schemes, even in degenerate cases, with accurate convergence order.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Ferdinando Auricchio, Giuseppe Toscani, Mattia Zanella
Summary: In this study, equilibration rates for a one-dimensional nonlocal Fokker-Planck equation are obtained, which models the relaxation process of a large swarm of robots that sense each other's distances. The equation involves time-dependent diffusion coefficient and drift, and aims to achieve a steady profile characterized by uniform spreading over a finite interval of the line. The result is obtained by combining entropy methods and the properties of the quasi-stationary profile.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mathematics, Interdisciplinary Applications
Lihua Tan, Yingzhe Fan
Summary: Motivated by a previous study, this research proves the global existence of mild solutions to the Vlasov-Poisson-Fokker-Planck system near a global Maxwellian when small-amplitude initial data is considered. Additionally, exponential time decay of the solution is obtained.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Physics, Fluids & Plasmas
Leonardo Santos
Summary: This study presents an approach to describe the effective microscopic dynamics of nonlinear Fokker-Planck equations, using a nonextensive generalization of the Wiener process. The formalism provides simple analytical results and physical insights, including analytical solutions for nonextensive versions of Brownian free-particle and Ornstein-Uhlenbeck processes, as well as explanations for anomalous diffusive behaviors in terms of memory effects in nonextensive Gaussian white noise. The developed formalism is also applied to model thermal noise in electric circuits.
Article
Computer Science, Interdisciplinary Applications
Yanyan Shi, Yajuan Sun
Summary: This paper develops a numerical discretization for solving the Vlasov-Poisson-Fokker-Planck system, studies the geometric structure of the corresponding stochastic differential equation, constructs contact methods for numerical solution, and analyzes the time accuracy order of the proposed numerical schemes. Numerical experiments are conducted to validate the methods.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Yue Li
Summary: In this paper, the global existence of weak solutions to a kinetic-fluid model with nonhomogeneous Dirichlet boundary data in a 3D bounded domain is established. The model consists of a Vlasov-Fokker-Planck equation coupled with the compressible Navier-Stokes equations via a friction force. This result is proved for the isentropic fluid with large initial data, and large velocity and density at the inflow boundary.
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK
(2021)
Article
Mathematics
Young-Pil Choi, Oliver Tse
Summary: We establish a quantified overdamped limit for kinetic Vlasov-Fokker-Planck equations with nonlocal interaction forces. We provide explicit bounds on the error between solutions of that kinetic equation and the limiting equation, which is known under the names of aggregation-diffusion equation or McKean-Vlasov equation. Our strategy allows us to quantify the error between the intermediate system and the corresponding limiting equation by deriving an evolution-variational-like inequality for Wasserstein gradient flows. The importance of this article lies in its establishment of a quantitative bound for the overdamped limit of kinetic Vlasov-Fokker-Planck equations and its applicability to various interaction potentials.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Automation & Control Systems
Viorel Barbu
Summary: This work focuses on the existence of optimal controllers for the Bolza optimal control problem governed by the nonlinear Fokker-Planck equation in L1(Rd) with control input in the drift term. The solution to the control state system is a weak (mild) solution obtained from a vanishing viscosity approximation scheme. In particular, we prove the existence of a stochastic Markov optimal controller in feedback form for the stochastic optimal control problem governed by McKean-Vlasov SDEs.
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
(2023)