Hydrodynamic limit of the incompressible Navier-Stokes-Fourier-Maxwell System with Ohm's Law from the Vlasov-Maxwell-Boltzmann system: Hilbert expansion approach
We prove a global-in-time limit from the two-species Vlasov-Maxwell-Boltzmann system to the two-fluid incompressible Navier-Stokes-Fourier-Maxwell system with Ohm's law. Besides the techniques developed for the classical solutions to the Vlasov-Maxwell-Boltzmann equations in the past years, such as the nonlinear energy method and micro-macro decomposition are employed, key roles are played by the decay properties of both the electric field and the wave equation with linear damping of the divergence free magnetic field. This is a companion paper of [N. Jiang and Y.-L. Luo, \emph{Ann. PDE} 8 (2022), no. 1, Paper No. 4, 126 pp] in which Hilbert expansion is not employed.
In this paper, we justify the convergence from the two-species Vlasov-Poisson-Boltzmann (in briefly,VPB) system to the two-fluid incompressible Navier-Stokes-Fourier-Poisson (in briefly, NSFP) system with Ohm's law in the context of classical solutions. We prove the uniform estimates with respect to the Knudsen number $\varepsilon$ for the solutions to the two-species VPB system near equilibrium by treating the strong interspecies interactions. Consequently, we prove the convergence to the two-fluid incompressible NSFP as $\varepsilon$ go to 0.
For the two-species Vlasov-Maxwell-Boltzmann (VMB) system with the scaling under which the moments of the fluctuations to the global Maxwellians formally converge to the two-fluid incompressible Navier-Stokes-Fourier-Maxwell (NSFM) system with Ohm's law, we prove the uniform estimates with respect to Knudsen number $\eps$ for the fluctuations. As consequences, the existence of the global in time classical solutions of VMB with all $\eps \in (0,1]$ is established. Furthermore, the convergence of the fluctuations of the solutions of VMB to the classical solutions of NSFM with Ohm's law is rigoro…
The hydrodynamic limit of the Vlasov-Maxwell-Boltzmann equations is considered for weak solutions. Using relative entropy estimate about an absolute Maxwellian, an incompressible Electron-Magnetohydrodynamics-Fourier limit for solutions of the Vlasov-Maxwell-Blotzmann equations over any periodic spatial domain in $\R^3$ is studied. It is shown that any properly scaled sequence of renormalized solutions of the Vlasov-Maxwell-Boltzmann equations has fluctuations that (in the weak $L^1$ topology) converge to an infinitesimal Maxwellian with fluid variables that satisfy the incompressibility and B…
The local existence and uniqueness of strong solutions for Cauchy problem of three-dimensional inhomogeneous incompressible Navier-Stokes-Vlasov equations
For the one-species Vlasov-Poisson-Boltzmann (VPB) system in the scaling under which the moments of the fluctuations formally converge to the incompressible Navier-Stokes-Fourier-Poisson (NSFP) system, we prove the uniform estimates with respect to the Knudsen number $ε$ for the fluctuations. As a consequence, the existence of the global-in-time classical solutions of VPB with all $ε\in (0,1]$ is established in whole space under small size of initial data, and the convergence to incompressible NSFP as $ε$ go to 0 is rigorously justified.
For the Vlasov-Poisson-Boltzmann equations with random uncertainties from the initial data or collision kernels, we proved the sensitivity analysis and energy estimates uniformly with respect to the Knudsen number in the diffusive scaling using hypocoercivity method. As a consequence, we also justified the incompressible Navier-Stokes-Poisson limit with random inputs. In particular, for the first time, we obtain the precise convergence rate {\em without} employing any results based on Hilbert expansion. We not only generalized the previous deterministic Navier-Stokes-Poisson limits to random i…
The incompressible Navier-Stokes-Fourier-Maxwell system limits of the Vlasov-Maxwell-Boltzmann system for soft potentials: the noncutoff cases and cutoff cases
The global well-posedness for the incompressible Navier-Stokes-Vlasov equations in two spatial dimensions is established by a priori estimates, the characteristic method and the semigroup analysis.
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…
The goal of this article is to give a formal derivation of Ohm's law of Magnetohydrodynamics (MHD) starting from the Vlasov-Maxwell-Boltzmann system. The derivation is based on various physical scalings and the moment methods when the Knudsen number goes to zero. We also give a derivation of the so-called Hall effect as well as other limit models such as the Navier-Stokes-Maxwell system. Our results include both the compressible and incompressible MHD models.
We are concerned with large-time behaviors of solutions for Vlasov--Navier--Stokes equations in two dimensions and Vlasov-Stokes system in three dimensions including the effect of velocity alignment/misalignment. We first revisit the large-time behavior estimate for our main system and refine assumptions on the dimensions and a communication weight function. In particular, this allows us to take into account the effect of the misalignment interactions between particles. We then use a sharp heat kernel estimate to obtain the exponential time decay of fluid velocity to its average in $L^\infty$-…
In this paper, we develop an abstract framework to establish ill-posedness in the sense of Hadamard for some nonlocal PDEs displaying unbounded unstable spectra. We apply it to prove the ill-posedness for the hydrostatic Euler equations as well as for the kinetic incompressible Euler equations and the Vlasov-Dirac-Benney system.
In the present paper, we study the diffusion limit of the classical solution to the unipolar Vlasov-Poisson-Boltzmann (VPB) system with initial data near a global Maxwellian. We prove the convergence and establish the convergence rate of the global strong solution to the unipolar VPB system towards the solution to an incompressible Navier-Stokes-Poisson-Fourier system based on the spectral analysis with precise estimation on the initial layer.
A moderately thick spray can be described by a coupled system of equations consisting of the incompressible Navier-Stokes equations and the Vlasov-Boltzmann equation. We investigate this kind of mathematical model in this paper. In particular, we study the initial value problem for the Navier-Stokes-Vlasov-Boltzmann equations. The existence of global weak solutions is established by a weak convergence method. The interesting point of our main result is to handle the model with some breakup effects while the velocity of particles is in the whole space.
In this paper, we are interested in the dynamics of charged particles interacting with the incompressible viscous flow. More precisely, we consider the Vlasov-Poisson or Vlasov-Poisson-Fokker-Planck equation coupled with the incompressible Navier-Stokes system through the drag force. For the proposed kinetic-fluid model, we study the asymptotic regime corresponding to strong local alignment and diffusion forces. Under suitable assumptions on the initial data, we rigorously derive a coupled isothermal/pressureless Euler-Poisson system and incompressible Navier-Stokes system(in short, EPNS syste…
Diffusive limit of the Vlasov-Poisson-Boltzmann system with cutoff soft potentials $-3<γ<0$ in the perturbative framework around global Maxwellian still remains open. By introducing a new weighted $H_{x,v}^2$-$W_{x,v}^{2, \infty}$ approach with time decay, we solve this problem for the full range of cutoff potentials $-3<γ\leq 1$. The core of this approach lies in the interplay between the velocity weighted $H_{x,v}^2$ energy estimate with time decay and the time-velocity weighted $W_{x,v}^{2,\infty}$ estimate with time decay for the Vlasov-Poisson-Boltzmann system, which leads to the uniform …
A Quantitative Hydrodynamic Limit for the Vlasov-Fokker-Planck/Navier-Stokes System in the Fine Particle Regime
School of Mathematics and Information Science, Guangzhou University（广州大学数学与信息科学学院）；School of Mathematics, South China University of Technology（华南理工大学数学学院）；Department of Computational Mathematics, Science and Engineering and Department of Mathematics, Michigan State University（密歇根州立大学计算数学、科学与工程系及数学系）
Diffusion Limit with Optimal Convergence Rate of Classical Solutions to the modified Vlasov-Poisson-Boltzmann System
School of Mathematics and Information Sciences, Guangxi University（广西大学数学与信息科学学院）；Center for Applied Mathematical of Guangxi (Guangxi University)（广西应用数学中心（广西大学））；Post-doctoral Research Station of the First-level Discipline in Mathematics, Guangxi University（广西大学数学一级学科博士后科研流动站）
This work concerns the global existence of the weak solutions to a system of partial differential equations modeling the evolution of particles in the fluid. That system is given by a coupling between the standard isentropic compressible Navier-Stokes equations for the macroscopic description of a gas fluid flow, and a Vlasov-Boltzmann type equation governing the evolution of spray droplets modeled as particles with varying radius. We establish the existence of global weak solutions with finite energy, whose density of gas satisfies the renormalized mass equation. The proof, is partially motiv…
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…
We prove the global-in-time existence of large-data finite-energy weak solutions to an incompressible hybrid Vlasov-magnetohydrodynamic model in three space dimensions. The model couples three essential ingredients of magnetized plasmas: a transport equation for the probability density function, which models energetic rarefied particles of one species; the incompressible Navier--Stokes system for the bulk fluid; and a parabolic evolution equation, involving magnetic diffusivity, for the magnetic field. The physical derivation of our model is given. It is also shown that the weak solution, whos…
This article proposes a derivation of the Vlasov-Navier-Stokes system for spray/aerosol flows. The distribution function of the dispersed phase is governed by a Vlasov-equation, while the velocity field of the propellant satisfies the Navier-Stokes equations for incompressible fluids. The dynamics of the dispersed phase and of the propellant are coupled through the drag force exerted by the propellant on the dispersed phase. We present a formal derivation of this model from a multiphase Boltzmann system for a binary gaseous mixture, involving the droplets/dust particles in the dispersed phase …
This paper studies the Vlasov-Monge-Ampere system (VMA), a fully non-linear version of the Vlasov-Poisson system (VP) where the (real) Monge-Ampere equation substitutes for the usual Poisson equation. This system can be derived as a geometric approximation of the Euler equations of incompressible fluid mechanics in the spirit of Arnold and Ebin. Global existence of weak solutions and local existence of smooth solutions are obtained. Links between the VMA system, the VP system and the Euler equations are established through rigorous asymptotic analysis.
We study the diffusive expansion for solutions around Maxwellian equilibrium and in a periodic box to the Vlasov-Maxwell-Boltzmann system, the most fundamental model for an ensemble of charged particles. Such an expansion yields a set of dissipative new macroscopic PDE's, the incompressible Vlasov-Navier-Stokes-Fourier system and its higher order corrections for describing a charged fluid, where the self-consistent electromagnetic field is present. The uniform estimate on the remainders is established via a unified nonlinear energy method and it guarantees the global in time validity of such a…
A Rigorous Derivation of the Vlasov-Navier-Stokes Model from Multicomponent Boltzmann Equations
Hunan University（湖南大学）；School of Mathematics, Hunan University（湖南大学数学学院）；Hunan Provincial Key Laboratory of Intelligent Information Processing and Applied Mathematics, Hunan University（湖南大学智能信息处理与应用数学湖南省重点实验室）
We reformulate a general class of classical bead-spring-chain models for dilute polymeric fluids, with Hookean spring potentials, as McKean-Vlasov diffusion. This results in a coupled system of partial differential equations involving the unsteady incompressible linearized Navier-Stokes equations, referred to as the Oseen system, for the velocity and the pressure of the fluid, with a source term which is a nonlinear function of the probability density function, and a second-order degenerate parabolic Fokker-Planck equation, whose transport terms depend on the velocity field, for the probabilit…
We discuss several geometric PDEs and their relationship with Hydrodynamics and classical Electrodynamics. We start from the Euler equations of ideal incompressible fluids that, geometrically speaking, describe geodesics on groups of measure preserving maps with respect to the $L^2$ metric. Then, we introduce a geometric approximation of the Euler equation, which involves the Monge-Ampère equation and the Monge-Kantorovich optimal transportation theory. This equation can be interpreted as a fully nonlinear correction of the Vlasov-Poisson system that describes the motion of electrons in a unif…
We deal with a three dimensional model based on the use of barycentric velocity that describes unsteady flows of a heat conducting electrically charged multicomponent chemically reacting non-Newtonian fluid. We show that under certain assumptions on data and the constitutive relations for such a fluid there exists a global in time and large data weak solution. The paper has two key novelties. The first one is that we present a model that is thermodynamically and mechanically consistent and that is able to describe the cross effects in a generality never considered before, i.e., we cover the so…
We study the local and global wellposedness of a full system of Magneto-Hydro-Dynamic equations. The system is a coupling of the forced (Lorentz force) incompressible Navier-Stokes equations with the Maxwell equations through Ohm's law for the current. We show the local existence of mild solutions for arbitrarily large data in a space similar to the scale invariant spaces classically used for Navier-Stokes. These solutions are global if the initial data are small enough.
In \cite{ChCa}, Califano and Chiuderi conjectured that the energy of incompressible Magnetic hydrodynamical system is dissipated at a rate that is independent of the ohmic resistivity. The goal of this paper is to mathematically justify this conjecture in three space dimension provided that the initial magnetic field and velocity is a small perturbation of the equilibrium state $(e_3,0).$ In particular, we prove that for such data, 3-D incompressible MHD system without magnetic diffusion has a unique global solution. Furthermore, the velocity field and the difference between the magnetic field…
Vlasov equations can be formally derived from N-body dynamics in the mean-field limit. In some suitable singular limits, they may themselves converge to fluid dynamics equations. Motivated by this heuristic, we introduce natural scalings under which the incompressible Euler equations can be rigorously derived from N-body dynamics with repulsive Coulomb interaction. Our analysis is based on the modulated energy methods of Brenier and Serfaty.
This course explains how the usual mean field evolution partial differential equations (PDEs) in Statistical Physics - such as the Vlasov-Poisson system, the vorticity formulation of the two-dimensional Euler equation for incompressible fluids, or the time-dependent Hartree equation in quantum mechanics - can be rigorously derived from first principles, i.e. from the fundamental microscopic equations that govern the evolution of large, interacting particle systems. The emphasis is put on the mathematical methods used in these derivations, such as Dobrushin's stability estimate in the Monge-Kan…
