Search Results (42)

In this article, we establish the existence of global mild solutions to the Vlasov–Fokker–Planck equation with local alignment forces under specular reflection boundary conditions in the low-regularity function space $L_k^1{L}_T^{\infty }{L}_v^2$. A key difficulty is that the macroscopic averaged velocity u does not directly possess a dissipative structure in the equation. To overcome this, we rely on the dissipation $u-b$ from the linear part, combined with the dissipation of the macroscopic component b derived from the associated macroscopic equation. Moreover, since no direct energy functional is available for u, we fully exploit the dissipative mechanisms of both $u-b$ and b when handling the estimates for the nonlinear terms. Full article

A simple analytic procedure for the linear static analysis of short thin-walled beams with monosymmetric open cross-sections subjected to eccentric axial loading is presented. Under eccentric compressive loading, the beam is subjected to compression/extension, to torsion with influence of shear with respect to the principal pole and to bending with influence of shear in two principal planes. The approximate closed-form solutions for displacements consist of the general Vlasov’s solutions and additional displacements due to shear according to the theory of torsion with the influence of shear, as well as the theory of bending with the influence of shear. The internal forces and displacements for beams clamped at one end and simply supported on the other end, where eccentric loading is acting, are calculated using the method of initial parameters. The shear coefficients for the monosymmetric cross-sections introduced in these equations are provided. Solutions for normal stress and total displacements according to Vlasov’s general thin-walled beam theory, and those obtained with the proposed method taking shear influence into account, are compared with shell finite element solutions analyzing isotropic and orthotropic I-section beams. According to the results for normal stress relative differences, and Euclidean norm for displacements, it has been demonstrated that shear effects must be accounted for in the analysis of such structural problems. Full article

This paper studies the large deviation principle (LDP) of a class of Hilfer fractional stochastic McKean–Vlasov differential equations with multiplicative noise. Firstly, by making use of the Laplace transform and its inverse transform, the solution of the equation is derived. Secondly, considering the equivalence between the LDP and the Laplace principle (LP), the weak convergence method is employed to prove that the equation satisfies the LDP. Finally, through specific example, it is elaborated how to utilize the LDP to analyze the behavioral characteristics of the system under small noise perturbation. Full article

This paper focuses on McKean-Vlasov stochastic differential equations under local Lipschitz conditions. We first introduce the stochastic interacting particle system and prove the propagation of chaos. Then we establish a truncated stochastic theta scheme to approximate the interacting particle system and obtain the strong convergence of the continuous-time truncated stochastic theta scheme to the non-interacting particle system. Furthermore, we study the asymptotical mean square stability of the interacting particle system and the truncated stochastic theta method. Finally, we give one numerical example to verify our theoretical results. Full article

The interaction between thermal fields and mechanical loads in thin-walled cylindrical shells introduces complex dynamic behaviors relevant to aerospace and mechanical engineering applications. This study investigates the axial stress wave propagation in a circular cylindrical shell subjected to combined thermal gradients and time-dependent harmonic compression. A semi-analytical model based on Donnell–Mushtari–Vlasov (DMV) shells theory is developed to derive the governing equations, incorporating elastic, inertial, and thermal expansion effects. Modal solutions are obtained to evaluate displacement and stress distributions across varying thermal and mechanical excitation conditions. Empirical Mode Decomposition (EMD) and Instantaneous Frequency (IF) analysis are employed to extract time–frequency characteristics of the dynamic response. Complementary Finite Element Analysis (FEA) is conducted to assess modal deformations, stress wave amplification, and the influence of thermal softening on resonance frequencies. Results reveal that increasing thermal gradients leads to significant reductions in natural frequencies and amplifies stress responses at critical excitation frequencies. The combination of analytical and numerical approaches captures the coupled thermomechanical effects on shell dynamics, providing an understanding of resonance amplification, modal energy distribution, and thermal-induced stiffness variation under axial harmonic excitation across thin-walled cylindrical structures. Full article

This paper examines temporal symmetry breaking and structural duality in a class of time-changed McKean–Vlasov neutral stochastic differential equations. The system features super-linear drift coefficients satisfying a one-sided local Lipschitz condition and incorporates a fundamental duality: one drift component evolves under a random time change $E_t$, while the other progresses in regular time t. Within the symmetric framework of mean-field interacting particle systems, where particles exhibit permutation invariance, we establish strong convergence of the tamed Euler–Maruyama method over finite time intervals. By replacing the one-sided local condition with a globally symmetric Lipschitz assumption, we derive an explicit convergence rate for the numerical scheme. Two numerical examples validate the theoretical results. Full article

The mechanisms by which rotation influences zonal flows (ZFs) in plasma are incompletely understood, presenting a significant challenge in the study of plasma dynamics. This research addresses this gap by investigating the role of non-inertial effects—specifically centrifugal and Coriolis forces—on Geodesic Acoustic Modes (GAMs) and ZFs in rotating tokamak plasmas. While previous studies have linked centrifugal convection to plasma toroidal rotation, they often overlook the Coriolis effects or inconsistently incorporate non-inertial terms into magneto-hydrodynamic (MHD) equations. In this work, we derive self-consistent drift-ordered two-fluid equations from the collisional Vlasov equation in a non-inertial frame, and we modify the Hermes cold ion code to simulate the impact of rotation on GAMs and ZFs. Our simulations reveal that toroidal rotation enhances ZF amplitude and GAM frequency, with Coriolis convection playing a critical role in GAM propagation and the global structure of ZFs. Analysis of simulation outcomes indicates that centrifugal drift drives parallel velocity growth, while Coriolis drift facilitates radial propagation of GAMs. This work may provide valuable insights into momentum transport and flow shear dynamics in tokamaks, with implications for turbulence suppression and confinement optimization. Full article

In this paper, we focus on mean-field stochastic differential equations driven by G-Brownian motion (G-MFSDEs for short) with a drift coefficient satisfying the local one-sided Lipschitz condition with respect to the state variable and the global Lipschitz condition with respect to the law. We are concerned with the well-posedness and the numerical approximation of the G-MFSDE. Probability uncertainty leads the resulting expectation usually to be the G-expectation, which means that we cannot apply the numerical approximation for McKean–Vlasov equations to G-MFSDEs directly. To numerically approximate the G-MFSDE, with the help of G-expectation theory, we use the sample average value to represent the law and establish the interacting particle system whose mean square limit is the G-MFSDE. After this, we introduce the modified stochastic theta method to approximate the interacting particle system and study its strong convergence and asymptotic mean square stability. Finally, we present an example to verify our theoretical results. Full article

This paper addresses the problem of optimal $H_2$-filtering for a class of continuous-time linear McKean–Vlasov stochastic differential equations under sampled measurements. The main tool used to solve the filtering problem is a forward jump matrix linear differential equation with a Riccati-type jumping operator. More specifically, the stabilizing solution of such a jump Riccati-type equation plays a key role. Full article

An exact solution of the collisionless time-dependent Vlasov equation is found. For the first time in a century, an analytical solution to the one-dimensional time-dependent Vlasov–Boltzmann equation has been found. It has been found that instead of the widely discussed damping, waves are subject to instability. By means of this solution, the behavior of the Langmuir waves in the nonlinear stage is considered. A symmetry method is found that allows us to establish the dependence on time of the desired quantity based on the dependence on the previous time. The analysis is restricted by the consideration of the first nonlinear approximation—keeping the second power of the electric strength. It is shown that in general the waves with finite amplitudes are not subjected to the damping. Conditions have been found under which waves can be unstable. Full article

We develop a unified analytical framework that systematically connects kinetic theory, optimal transport, and entropy dissipation through the novel integration of hypocoercivity methods with geometric structures. Building upon but distinctly extending classical hypocoercivity approaches, we demonstrate how geometric control, via commutators and curvature-like structures in probability spaces, resolves degeneracies inherent in kinetic operators. Centered around the Boltzmann and Fokker–Planck equations, we derive sharp exponential convergence estimates under minimal regularity assumptions, improving on prior methods by incorporating Wasserstein gradient flow techniques. Our framework is further applied to the study of hydrodynamic limits, collisional relaxation in magnetized plasmas, the Vlasov–Poisson system, and modern data-driven algorithms, highlighting the central role of entropy as both a physical and variational tool across disciplines. By bridging entropy dissipation, optimal transport, and geometric analysis, our work offers a new perspective on stability, convergence, and structure in high-dimensional kinetic models and applications. Full article

A mechanical model of a laterally loaded long pile in layered soils was established to accurately calculate the lateral load-bearing performance of the pile foundation, and attention was paid to the influence of the complete separation of the pile–soil contact surface in a certain part of the pile on its lateral load-bearing performance. Based on the modified Vlasov foundation model, the displacement equation of the laterally loaded long pile embedded in layered soils was derived by the separation variable method. Using the solution method presented in this study, the deformation and internal force of the free-fixed pile were obtained. Then, the effects of the slenderness ratio of the pile and the complete separation of the pile–soil contact surface on the lateral load-bearing performance of the long pile in layered soils were analyzed. The results show that the deformation of the pile body increases with the increase in the slenderness ratio under the lateral load. Meanwhile, the position of the maximum bending moment and the negative shear force moves upward along the pile as the slenderness ratio increases. When the contact surface of the pile–upper stratum is separated, the deformation of the pile top doubles, and the negative shear force increases by three times compared to the case without the effect of separation of the pile–soil contact surface. When the contact surface between the pile and the middle layer soil is separated, the deformation and bending moment of the pile increase by 25%, and the maximum negative shear force decreases. Full article

In this article, we study the asymptotic behavior of the two-dimensional Vlasov–Yukawa system with a point charge under a large external magnetic field. When the intensity of the magnetic field tends to infinity, we show that the kinetic system converges to the measure-valued Euler equation with a defect measure, which extends the results of Miot to the case of the Vlasov–Yukawa system. And compared with the Miot’s work, an important improvement is that our results do not require compact support conditions for spatial variables or uniform bound conditions for second-order spatial moments. In addition, the extra small condition for initial data is also not required. Full article

In this study, we investigate the Cauchy problem for the Vlasov-Maxwell-Fokker-Planck system near a global Maxwellian in low regularity space. We establish the existence of global mild solutions to the system by employing the energy method, provided that the perturbative initial data is sufficiently small. Moreover, despite the absence of zeroth-order dissipation for the magnetic field, we are able to derive exponential decay estimates for solutions in higher-order regularity space. This is achieved by leveraging the higher-order dissipation properties of the magnetic field, which are deduced from the Maxwell equation. Full article

This study explores the applications of extended Gauss–Hertz variational principles to determine the evolution of complex systems under the influence of impulse actions, coherent accelerations, and their application to electrophysical systems with fractal elements. Impulsive effects on systems initiate coherent accelerations (including higher-order accelerations, such as modes with intensification), leading to variations in connections, structure, symmetry, and inertia; the emergence of coherence; and the evolution of fractal elements in electrophysical circuits. The combination of results from the non-local Vlasov theory and modifications to the Gauss–Hertz principle allows for the formulation of a variational principle for the evolution of fractal systems. A key feature of this variational principle is the ability to simultaneously derive equations for both the system’s dynamics and the self-harmonizing evolution of its internal symmetry and structure (e.g., fractal parameters). Full article