We prove that under a generic asymptotic condition on the charge, the small data solutions to the Vlasov-Maxwell system do not verify linear scattering. In other words, we show the non-L^1 asymptotic completeness of the system. The proof makes use of the Lorentz invariance of the equations.
. This work concerns the Vlasov-Poisson-Boltzmann system without angular cutoff and the Vlasov-Poisson-Landau system with Coulomb interactions in a specific bounded domain, namely a union of cubes with flat boundaries. We establish global stability and exponential large-time decay under the specular reflection boundary condition when the initial data is close to the global Maxwellian equilibrium. The flat boundaries do not create singularities at the boundary and ensure the existence of high-order Sobolev regularity. The proof is based on the compatibility of the specular boundary condition with high-order derivatives and employs the velocity-weighted energy method.
The pressureless Euler-Navier-Stokes system can be obtained formally from the Vlasov-Navier-Stokes system, under the assumption that the distribution function describing the density of particles is monokinetic. Its study has been the subject of several recent papers, which have established the global existence of solutions with high enough regularity, for small initial data. In this work, we demonstrate the global existence of strong solutions in the whole space case, without assuming the initial density to be small and regular: it suffices for it to be bounded and for the total mass to be finite. In passing, we obtain optimal decay estimates for the energy and dissipation functionals. As a corollary, we get a long-time description of the density. All these results are based on an elementary energy method, with no need of sophisticated Fourier analysis tools.
. This paper studies a Vlasov-Poisson system with radiation damping on the torus T3. Compared to the whole-space case, the lack of dispersion in the periodic setting presents essential difficulties. We establish the global existence of both classical and weak solutions and prove the propagation of velocity moments. Furthermore, by applying velocity averaging lemmas, we establish strong convergence of the macroscopic densities, which is essential for deriving the energy dissipation equality in the weak solution framework.
Diffusive limit of the non-cut off Vlasov-Poisson-Boltzmann system in perturbation framework still remains open. By employing a new weight function and some novel treatments, we solve this issue completely for the full range of potentials gamma > -3 and 0 < s < 1. This result marks the first comprehensive coverage of the full range of potentials in diffusive limit of the non-cutoff Boltzmann type equations incorporating electric or electromagnetic field. Uniform weighted estimate with respect to the Knudsen number e is an element of (0, 1] is established globally in time, which eventually establishes global solutions to the Vlasov-Poisson-Boltzmann system and hydrodynamic limit to the twofluid incompressible Navier-Stokes-Fourier-Poisson system with Ohm's law for the full potential range gamma > -3 and 0 < s < 1.
We introduce a novel linear transport equation that models the evolution of a one-particle distribution subject to free transport and two distinct scattering mechanisms: one affecting the particle's speed and the other its direction. These scattering processes occur at different time scales and with different intensities, leading to a kinetic equation where the total scattering operator is the sum of two separate operators. Each of them depends not only on the kernel characterizing the corresponding scattering mechanism, but also explicitly on the marginal distribution of either the speed or the direction. Therefore, unlike classical settings, the gain terms in our operators are not tied to a fixed equilibrium distribution but evolve in time through the marginals. As a result, typical analytical tools from kinetic theory, such as equilibrium characterization, entropy methods, spectral analysis in Hilbert spaces, and Fredholm theory, are not applicable in a standard fashion. In this work, we rigorously analyze the properties of this new class of scattering operators, including the structure of their non-standard pseudo-inverses and their asymptotic behavior. We also derive macroscopic (hydrodynamic) limits under different regimes of scattering frequencies, revealing new effective equations and highlighting the interplay between speed and directional relaxation.
We propose a BGK-type kinetic model for relativistic reactive gas mixtures. This model serves as a computationally tractable yet physically consistent alternative to the corresponding Boltzmann equation. The relaxation operator is constructed to ensure that the model correctly satisfies the conservation laws and relaxes to the proper equilibrium: a Ju & uml;ttner distribution characterized by a common temperature, velocity, and chemical potentials that obey the law of mass action. Furthermore, we prove that the model satisfies an H-theorem with the same entropy functional as the original Boltzmann equation. Finally, numerical simulations are presented, which confirm that the model preserves the conserved quantities and exhibits entropy decay towards the proper Ju & uml;ttner equilibrium.
We consider a nonlinear Fokker-Planck equation derived from a Cucker-Smale model for flocking with noise. There is a known phase transition depending on the noise between a regime with a unique stationary solution which is isotropic (symmetry) and a regime with a continuum of polarized stationary solutions (symmetry breaking). If the value of the noise is larger than the threshold value, the solution of the evolution equation converges to the unique radial stationary solution. This solution is linearly unstable in the symmetry-breaking range, while polarized stationary solutions attract all solutions with sufficiently low entropy. We prove that the convergence measured in a weighted $L^2$ norm occurs with an exponential rate and that the average speed also converges with exponential rate to a unique limit which determines a single polarized stationary solution.
Cross-diffusion systems are formally derived from multispecies kinetic models in the diffusion limit. The first limit in the multispecies BGK model of Gross and Krook leads to a variant of the non-isothermal Maxwell-Stefan equations. The second limit in a BGK model with Brinkman-type force term yields generalized Busenberg-Travis equations, which reduce for constant temperature to the classical Busenberg-Travis system for segregating population species. Entropy equalities are derived for the kinetic and cross-diffusion equations.
. The Green's function of the Boltzmann equation plays an important role in studying both nonlinear initial value problems and initial-boundary value problems [8, 11]. In this paper, we obtain a refined estimate of Green's function to facilitate a quantitative study of the Boltzmann equation. Specifically, more detailed microscopic velocity information is revealed in our refined estimates, which proves useful in the initial-boundary value problems. The proof relies on an enhanced version of the mixture lemma.
We consider the 1D Vlasov-Poisson system on the real line, and establish the time-global solvability for arbitrarily large initial data f0 +/-. We assume that the support of the difference between the initial data f0 +/- and f & lowast;,0 +/- is compact in the phase space, where f & lowast;,0 +/- is the initial value of traveling waves of the 1D Vlasov-Poisson system.
We propose a BGK-type kinetic model for a binary gas mixture, designed to serve as a kinetic formulation of compressible two-phase fluid dynamics. The model features species-dependent adiabatic exponents, and the relaxation operator is constructed by solving an entropy minimization problem under moments constraints. Starting from this model, we derive the compressible two-phase Euler equations via a formal Chapman--Enskog expansion and identify dissipative corrections of Navier--Stokes type. We then rigorously justify the Euler limit using the relative entropy method, establishing quantitative convergence estimates under appropriate regularity assumptions. Finally, we present numerical experiments based on an implicit-explicit Runge--Kutta method, which confirm the asymptotic preserving property and demonstrate the convergence from the BGK model to the isentropic two-phase Euler system in the hydrodynamic regime.
In this paper, we study the spatially homogeneous inelastic Boltzmann equation for angular cutoff pseudo-Maxwell molecules with an additional term of linear deformation. We establish the existence of non-Maxwellian selfsimilar profiles under the assumption of small deformation in the nearly elastic regime, and also obtain weak convergence to these self-similar profiles for global-in-time solutions with initial data that have finite mass and finite p-th order moment for any 2 < p <= 4. Our results confirm the competition between shear heating and inelastic cooling that governs the long-time behavior of temperature. Specifically, temperature increases to infinity if shear heating dominates, decreases to zero if inelastic cooling prevails, and converges to a positive constant if the two effects are balanced. In the balanced scenario, the corresponding self-similar profile aligns with the steady solution.
This article is devoted to the study of a model of thick sprays which combines the Vlasov equation for the particles and the barotropic compressible Euler equations to describe the fluid, coupled through the gradient of the pressure of the fluid. We prove that sound waves interact with particles of nearby velocities, which results in a damping or an amplification of these sound waves, depending on the sign of the derivative of the distribution function at the sound speed. This mechanism is very similar to the classical Landau damping which occurs in the Vlasov-Poisson system. If the sound waves are amplified then the thick spray model is linearly ill-posed in Sobolev spaces, even locally in time. We also show that such Landau damping type phenomena naturally arise when we couple an hyperbolic system of conservation laws with the Vlasov equation.
The Boltzmann equation, a fundamental model in kinetic theory, describes the evolution of particle distribution functions through a nonlinear, high-dimensional collision operator. However, its numerical solution remains computationally demanding, particularly for inelastic collisions and high-dimensional velocity domains. In this work, we propose the Fourier Neural Spectral Network (FourierSpecNet), a hybrid framework that integrates the Fourier spectral method with deep learning to approximate the collision operator in Fourier space efficiently. FourierSpecNet achieves resolution-invariant learning and supports zero-shot super-resolution, enabling accurate predictions at unseen resolutions without retraining. Beyond empirical validation, we establish a consistency result showing that the trained operator converges to the spectral solution as the discretization is refined. We evaluate our method on several benchmark cases, including Maxwellian and hard-sphere molecular models, as well as inelastic collision scenarios. The results demonstrate that FourierSpecNet offers competitive accuracy while significantly reducing computational cost compared to traditional spectral solvers. Our approach provides a robust and scalable alternative for solving the Boltzmann equation across both elastic and inelastic regimes.
We present a modified simulated annealing method with a dynamical choice of the cooling temperature. The latter is determined via a closed-loop control and is proven to yield exponential decay of the entropy of the particle system. The analysis is carried out through kinetic equations for interacting particle systems describing the simulated annealing method in an extended phase space. Decay estimates are derived under the quasi-invariant scaling of the resulting system of Boltzmann-type equations to assess the consistency with their mean-field limit. Numerical results are provided to illustrate and support the theoretical findings.
. We consider the kinetic transport equation that arises in the BoltzmannGrad limit of the two-dimensional periodic Lorentz gas. This equation was obtained by extending the phase space of positions and velocities through the introduction of two new variables, representing the time to the next collision and the corresponding impact parameter. Here, we mostly focus on the case of periodic boundary conditions on the position space; we prove that, under suitable conditions, the time evolution of a probability density on the extended phase space converges to the equilibrium state with respect to the Lp norm (& lowast;-weakly if p = infinity), if such initial density is Lp. If p = 2, or if the initial datum does not depend on the position, we also get more precise estimates about the rate of convergence to the equilibrium. Our proof is based on the analysis of the long-time behavior of the Fourier coefficients of the solution.
In the present work, we propose a novel method for reconstruction of multi-dimensional kinetic distributions, based on their representation as a mixture of Dirac delta functions. The representation is found as a solution of an optimization problem. Different target functionals are considered, with a focus on sparsity-promoting regularization terms. The proposed algorithm guarantees non-negativity of the distribution by construction, and avoids an exponential dependence of the computational cost on the dimensionality of the problem. Numerical comparisons with other classical methods for reconstruction of kinetic distributions are provided for model problems, and the role of the different parameters governing the optimization problem is studied.
We study sequences of solutions to the inhomogeneous Landau-Fermi-Dirac equation with Coulomb potential in which the quantum parameter converges to zero. Our main result establishes the compactness of these sequences, which allows us to show that, up to a subsequence, these solutions converge to a renormalized solution of the classical Landau equation with a defect measure, as defined by Villani. To do this, we work in the class of solutions that are obtained through approximation procedures. For these solutions, we were able to show compactness in the vanishing quantum parameter limit through a diagonal argument, which combines techniques from the study of Cauchy problems for both the classical Landau and the Landau-Fermi-Dirac equations.
Solving non-convex minimization problems using multi-particle metaheuristic derivative-free optimization methods is still an active area of research. Popular methods are Particle Swarm Optimization (PSO) methods, that iteratively update a population of particles according to dynamics inspired by social interactions between individuals. We present a modification to include constrained minimization problems using exact penalization. Additionally, we utilize the hierarchical structure of PSO to introduce a micro-macro decomposition of the algorithm. The probability density of particles is written as a convex combination of microscopic and macroscopic contributions, and both parts are propagated separately. The decomposition is dynamically updated based on heuristic considerations. Numerical examples compare the results obtained using the algorithm in the microscopic scale, in the macroscopic scale, and, using the new micro-macro decomposition.
