In this paper we discuss the connections between a Vlasov-Fokker-Planck equation and an underlying microscopic particle system, and we interpret those connections in the context of the GENERIC framework (Öttinger 2005). This interpretation provides (a) a variational formulation for GENERIC systems, (b) insight into the origin of this variational formulation, and (c) an explanation of the origins of the conditions that GENERIC places on its constitutive elements, notably the so-called degeneracy or non-interaction conditions. This work shows how the general connection between large-deviation principles on one hand and gradient-flow structures on the other hand extends to non-reversible particle systems.
Convergence of a system of particles, interacting with a fluid, to Navier-Stokes-Vlasov-Fokker-Planck system is studied. The interaction between particles and fluid is described by Stokes drag force. The empirical measure of particles is proved to converge to the Vlasov-Fokker-Planck component of the system and the velocity of the fluid coupled with the particles converges in the uniform topology to the the Navier-Stokes component. A new uniqueness result for the PDE system is added.
This work is concerned with the existence of mild solutions to non-linear Fokker-Planck equations with fractional Laplace operator $(-Δ)^s$ for $s\in\left(\frac12,1\right)$. The uniqueness of Schwartz distributional solutions is also proved under suitable assumptions on diffusion and drift terms. As applications, weak existence and uniqueness of solutions to McKean-Vlasov equations with Lévy-Noise, as well as the Markov property for their laws are proved.
This note adapts a probabilistic approach to establish a quantified estimate of the overdamped limit for the Vlasov-Fokker-Planck equation towards the aggregation-diffusion equation, which in particular includes cases of the Newtonian type singular forces. The proofs are based on the investigation of the weak convergence of the corresponding stochastic differential equations (SDEs) of Mckean type in the continuous path space. We show that one can obtain the same convergence rate as in [10] under the same assumptions.
We consider a particle system in 1D, interacting via repulsive or attractive Coulomb forces. We prove the trajectorial propagation of molecular chaos towards a nonlinear SDE associated to the Vlasov-Poisson-Fokker-Planck equation. We obtain a quantitative estimate of convergence in expectation, with an optimal convergence rate of order $N^{-1/2}$. We also prove some exponential concentration inequalities of the associated empirical measures. A key argument is a weak-strong stability estimate on the (nonlinear) VPFP equation, that we are able to adapt for the particle system in some sense.
We consider a Vlasov-Fokker-Planck equation governing the evolution of the density of interacting and diffusive matter in the space of positions and velocities. We use a probabilistic interpretation to obtain convergence towards equilibrium in Wasserstein distance with an explicit exponential rate. We also prove a propagation of chaos property for an associated particle system, and give rates on the approximation of the solution by the particle system. Finally, a transportation inequality for the distribution of the particle system leads to quantitative deviation bounds on the approximation of…
In this paper we study second order stochastic differential equations with measurable and density-distribution dependent coefficients. Through establishing a maximum principle for kinetic Fokker-Planck-Kolmogorov equations with distribution-valued inhomogeneous term, we show the existence of weak solutions under mild assumptions. Moreover, by using the Hölder regularity estimate obtained recently in \cite{GIMV19}, we also show the well-posedness of generalized martingale problems when diffusion coefficients only depend on the position variable (not necessarily continuous). Even in the non dens…
We consider the Vlasov-Manev-Fokker-Planck (VMFP) system in three dimensions, which differs from the Vlasov-Poisson-Fokker-Planck in that it has the gravitational potential of the form $-1/r - 1/r^2$ instead of the Newtonian one. For the VMFP system, we establish the global-in-time existence of weak solutions. The proof extends to several related kinetic systems.
We establish the exponential time decay rate of smooth solutions of small amplitude to the Vlasov-Poisson-Fokker-Planck equations to the Maxwellian both in the whole space and in the periodic box via the uniform-in-time energy estimates and also the macroscopic equations.
In this paper, we study the long-time behaviour of solutions to the Vlasov-Fokker-Planck equation where the confining potential is non-convex. This is a nonlocal nonlinear partial differential equation describing the time evolution of the probability distribution of a particle moving under the influence of a non-convex potential, an interaction potential, a friction force and a stochastic force. Using the free-energy approach, we show that under suitable assumptions solutions of the Vlasov-Fokker-Planck equation converge to an invariant probability.
This paper is devoted to the rigorous derivation of the macroscopic limit of a Vlasov-Fokker-Planck equation in which the Laplacian is replaced by a fractional Laplacian. The evolution of the density is governed by a fractional heat equation with the addition of a convective term coming from the external force. The analysis is performed by a modified test function method and by obtaining a priori estimates from quadratic entropy bounds. In addition, we give the proof of existence and uniqueness of solutions to the Vlasov-fractional-Fokker-Planck equation.
We consider 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 $γ>3/2$, we establish the global existence of weak solutions to this system with arbitrary large initial and boundary data.
This paper is devoted to the linearized Vlasov-Poisson-Fokker-Planck system in presence of an external potential of confinement. We investigate the large time behaviour of the solutions using hypocoercivity methods and a notion of scalar product adapted to the presence of a Poisson coupling. Our framework provides estimates which are uniform in the diffusion limit. As an application in a simple case, we study the one-dimensional case and prove the exponential convergence of the nonlinear Vlasov-Poisson-Fokker-Planck system without any small mass assumption.
In the present paper, we study the diffusion limit of the classical solution to the Vlasov-Poisson-Fokker-Planck (VPFP) system with initial data near a global Maxwellian. We prove the convergence and establish the optimal convergence rate of the global strong solution to the VPFP system towards the solution to the drift-diffusion-Poisson system based on the spectral analysis with precise estimation on the initial layer.
We provide a quantitative asymptotic analysis for the nonlinear Vlasov--Poisson--Fokker--Planck system with a large linear friction force and high force-fields. The limiting system is a diffusive model with nonlocal velocity fields often referred to as aggregation-diffusion equations. We show that a weak solution to the Vlasov--Poisson--Fokker--Planck system strongly converges to a strong solution to the diffusive model. Our proof relies on the modulated macroscopic kinetic energy estimate based on the weak-strong uniqueness principle together with a careful analysis of the Poisson equation.
We establish a quantitfied 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. Introducing an intermediate system via a coarse-graining map, we quantitatively estimate the error between the spatial densities of the Vlasov-Fokker-Planck equation and the intermediate system in the Wasserstein distance of order 2. We then derive an evolution-variational-lik…
In this paper, we study the set of stationary solutions of the Vlasov-Fokker-Planck (VFP) equation. This equation describes the time evolution of the probability distribution of a particle moving under the influence of a double-well potential, an interaction potential, a friction force and a stochastic force. We prove, under suitable assumptions, that the VFP equation does not have a unique stationary solution and that there exists a phase transition. Our study relies on the recent results by Tugaut and coauthors regarding the McKean-Vlasov equation.
We study the "one and one-half" dimensional Vlasov-Maxwell-Fokker-Planck system and obtain the first results concerning well-posedness of solutions. Specifically, we prove the global-in-time existence and uniqueness in the large of classical solutions to the Cauchy problem and a gain in regularity of the distribution function in its momentum argument.
We prove a global existence result with initial data of low regularity, and prove the trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with small non linear term but with a possibly large exterior confining potential in dimension $d=2$ and $d=3$. The proof relies on a fixed point argument using sharp estimates (at short and long time scales) of the semi-group associated to the Fokker-Planck operator, which were obtained by the first author.
In a recent paper Calogero and Alcantara derived a Lorentz-invariant Fokker-Planck equation, which corresponds to the evolution of a particle distribution associated with relativistic Brownian Motion. We study the "one and one-half" dimensional version of this problem with nonlinear electromagnetic interactions - the relativistic Vlasov-Maxwell-Fokker-Planck system - and obtain the first results concerning well-posedness of solutions. Specifically, we prove the global-in-time existence and uniqueness of classical solutions to the Cauchy problem and a gain in regularity of the distribution func…
We study a hydrodynamic limit of a system of coupled kinetic and fluid equations under a strong local alignment force and a strong Brownian motion. More precisely, we consider the Vlasov-Fokker-Planck type equation and compressible Navier-Stokes equations with a density-dependent viscosity. Based on a relative entropy argument, by assuming the existence of weak solutions to that kinetic-fluid system, we rigorously derive a two-phase fluid model consisting of isothermal Euler equations and compressible Navier-Stokes equations with a density-dependent viscosity.
This work is concerned with the existence and uniqueness of generalized (mild or distributional) solutions to (possibly degenerate) Fokker-Planck equations $ρ_t-Δβ(ρ)+{\rm div}(Db(ρ)ρ)=0$ in $(0,\infty)\times\mathbb{R}^d,$ $ρ(0,x) \equiv ρ_0(x)$. Under suitable assumptions on $β:\mathbb{R}\to\mathbb{R},\,b:\mathbb{R}\to\mathbb{R}$ and $D:\mathbb{R}^d\to\mathbb{R}^d$, $d\ge1$, this equation generates a unique flow $ρ(t)=S(t)ρ_0:[0,\infty)\to L^1(\mathbb{R}^d)$ as a mild solution in the sense of nonlinear semigroup theory. This flow is also unique in the class of $L^\infty((0,T)\times\mathbb{R}^…
The pointwise space-time behaviors of the Green's function and the global solution to the Vlasov-Poisson-Fokker-Planck (VPFP) system in spatial three dimension are studied in this paper. It is shown that the Green's function consists of the diffusion waves decaying exponentially in time but algebraically in space, and the singular kinetic waves which become smooth for all $(t,x,v)$ when $t>0.$ Furthermore, we establish the pointwise space-time behaviors of the global solution to the nonlinear VPFP system when the initial data is not necessarily smooth in terms of the Green's function.
In this paper, we construct unique, local-in-time strong solutions to the Vlasov-Poisson (VP) and Vlasov-Poisson-Fokker-Planck (VPFP) systems subjected to external, spatially regular, white-in-time electromagnetic fields in $\mathbb T^d \times \mathbb R^d$. Initial conditions are taken $H^σ$ with $σ> d/2 + 1$ (in addition to polynomial velocity weights). We additionally show that solutions to the VPFP are instantly $C^\infty_{x,v}$ due to hypoelliptic regularization if the external force fields are smooth. The external forcing arises in the kinetic equation as a stochastic transport in velocit…
We study the asymptotic regime of strong electric fields that leads from the Vlasov-Poisson system to the Incompressible Euler equations. We also deal with the Vlasov-Poisson-Fokker-Planck system which induces dissipative effects. The originality consists in considering a situation with a finite total charge confined by a strong external field. In turn, the limiting equation is set in a bounded domain, the shape of which is determined by the external confining potential. The analysis extends to the situation where the limiting density is non-homogeneous and where the Euler equation is replaced…
In this paper, a delay Vlasov-Fokker-Planck equation associated to a stochastic interacting particle system with delay is investigated analytically. Under certain restrictions on the parameters well-posedness and ergodicity of the mean-field equation are shown and an exponential rate of convergence towards the unique stationary solution is proven as long as the delay is finite. For infinte delay i.e., when all the history of the solution paths are taken into consideration polynomial decay of the solution is shown.
In this paper, we are concerned with the global well-posedness and time-asymptotic decay of the Vlasov-Fokker-Planck equation with local alignment forces. The equation can be formally derived from an agent-based model for self-organized dynamics which is called Motsch-Tadmor model with noises. We present the global existence and uniqueness of classical solutions to the equation around the global Maxwellian in the whole space. For the large-time behavior, we show the algebraic decay rate of solutions towards the equilibrium under suitable assumptions on the initial data. We also remark that the…
We derive a diffusion approximation for the kinetic Vlasov-Fokker-Planck equation in bounded spatial domains with specular reflection type boundary conditions. The method of proof involves the construction of a particular class of test functions to be chosen in the weak formulation of the kinetic model. This involves the analysis of the underlying Hamiltonian dynamics of the kinetic equation coupled with the reflection laws at the boundary. This approach only demands the solution family to be weakly compact in some weighted Hilbert space rather than the much tricky $\mathrm L^1$ setting.
Stochastic differential equations with Levy motion arise the mathematical models for various phenomenon in geophysical and biochemical sciences. The Fokker Planck equation for such a stochastic differential equations is a nonlocal partial differential equations. We prove the existence and uniqueness of the weak solution for this equation.
We investigate stochastic differential equations with jumps and irregular coefficients, and obtain the existence and uniqueness of generalized stochastic flows. Moreover, we also prove the existence and uniqueness of $L^p$-solutions or measure-valued solutions for second order integro-differential equation of Fokker-Planck type.
We consider various sets of Vlasov-Fokker-Planck equations modeling the dynamics of charged particles in a plasma under the effect of a strong magnetic field. For each of them in a regime where the strength of the magnetic field is effectively stronger than that of collisions we first formally derive asymptotically reduced models. In this regime, strong anisotropic phenomena occur; while equilibrium along magnetic field lines is asymptotically reached our asymptotic models capture a non trivial dynamics in the perpendicular directions. We do check that in any case the obtained asymptotic model…
In this paper, we prove the uniqueness of weak solutions to the Vlasov-Poisson-Fokker-Planck system in $C([0,T]; L^p)$, by assuming the solution has a local bounded density which tends to infinite with a "reasonable" rate as $t\to 0$. And particularly as a corollary, we get the uniqueness of weak solutions with initial data $f_0$ satisfying $f_0|v|^2\in L^1$, which solves the uniqueness of weak solutions with finite energy. In addition, we prove that the moments with respect to the velocity propagate for any order higher than 2.
This paper is concerned with a kineitc-fluid model with random initial inputs in the fine particle regime, which is a system coupling the incompressible Navier-Stokes equations and the Vlasov-Fokker-Planck equations that model dispersed particles of different sizes. A uniform regularity for random initial data near the global equilibrium is established in some suitable Sobolev spaces by using energy estimates, and we also prove the energy decays exponentially in time by hypocoercivity arguments, which means that the long time behavior of the solution is insensitive to the random perturbation i…
The Einstein-Vlasov-Fokker-Planck system describes the kinetic diffusion dynamics of self-gravitating particles within the Einstein theory of general relativity. We study the Cauchy problem for spatially homogeneous and isotropic solutions and prove the existence of both global solutions and solutions that blow-up in finite time depending on the size of certain functions of the initial data. We also derive information on the large-time behavior of global solutions and toward the singularity for solutions which blow-up in fine time. Our results entail the existence of a phase of decelerated exp…
Modulated Gibbs Measure and Mean-Field Limit for 3D Vlasov--Poisson--Fokker--Planck Equations
School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）
We study an asymptotic analysis of a coupled system of kinetic and fluid equations. More precisely, we deal with the nonlinear Vlasov-Fokker-Planck equation coupled with the compressible isentropic Navier-Stokes system through a drag force in a bounded domain with the specular reflection boundary condition for the kinetic equation and homogeneous Dirichlet boundary condition for the fluid system. We establish a rigorous hydrodynamic limit corresponding to strong noise and local alignment force. The limiting system is a type of two-phase fluid model consisting of the isothermal Euler system and…
The Vlasov-Nordström-Fokker-Planck system describes the evolution of self-gravitating matter experiencing collisions with a fixed background of particles in the framework of a relativistic scalar theory of gravitation. We study the spatially-homogeneous system and prove global existence and uniqueness of solutions for the corresponding initial value problem in three momentum dimensions. Additionally, we study the long time asymptotic behavior of the system and prove that even in the absence of friction, solutions possess a non-trivial asymptotic profile. An exact formula for the long time limi…
We study the linearized Vlasov equations and the linearized Vlasov-Fokker-Planck equations in the weakly collisional limit in a uniform magnetic field. In both cases, we consider periodic confinement and Maxwellian (or close to Maxwellian) backgrounds. In the collisionless case, for modes transverse to the magnetic field, we provide a precise decomposition into a countably infinite family of standing waves for each spatial mode. These are known as Bernstein modes in the physics literature, though the decomposition is not an obvious consequence of any existing arguments that we are aware of. We…
