arXiv subjects

Linjie Xiong

Publications and source records attributed to Linjie Xiong.

6 recordsLinked to original sources

arXiv2026

A Rigorous Derivation of the Vlasov-Navier-Stokes Model from Multicomponent Boltzmann Equations

The rigorous justification of the Vlasov-Navier-Stokes system remains an outstanding open problem, both as a mean-field limit of an $N$-particle fluid-interacting system and as a hydrodynamic limit derived from multiphase Boltzmann equations. Inspired by the work of Bernard, Desvillettes, Golse, and Ricci [{\it Commun. Math. Sci.}, {\bf 15}(6), 1703-1741, 2017], we present a rigorous derivation of the incompressible Vlasov-Navier-Stokes system from the two-component Boltzmann system for elastic hard-sphere collisions. To this end, we establish the well-posedness of the rescaled multicomponent Boltzmann system for small initial data. Specifically, we obtain estimates for the solution that hold uniformly with respect to both the thermal speed ratio $\varepsilon$ and the mass ratio $\eta$. Furthermore, under the Vlasov-Navier-Stokes scaling assumption $O(1)\varepsilon^3 \leqslant \eta \leqslant o(1)\varepsilon^2$ as $\varepsilon \rightarrow 0$, we establish the weak convergence of the solution to the fluid-kinetic coupled limit system.

math.AP↗

arXiv2026

Acoustic limit of Boltzmann equations for gas mixture

In this paper, we study the hydrodynamic and acoustic limit from Boltzmann equations for two species gas mixture with potential $\gamma \in \left(-3, 1\right]$. % in the whole space $(x \in \mathbb{R}^3)$.Here the particle masses are different which derives to the loss of symmetry to the linearized collision operator. %This paper resolves it precisely by using a framework based on vector-valued functions. We construct the hydrodynamic limit for two species based on the Hilbert expansion method when the Knudsen number is small. The key observation is the precise properties of the linearized collision operators, including the extra operators due to the different particle masses $(m^A \neq m^B)$. In additional, the acoustic limit of the Boltzmann equations for gas mixtures is rigorously justified by assuming the strength of the initial data depends on the Knudsen number.

math.AP↗

arXiv2021

The Incompressible Navier-Stokes-Fourier Limit from Boltzmann-Fermi-Dirac Equation

We study Boltzmann-Fermi-Dirac equation when quantum effects are taken into account in dilute gas dynamics. By employing new estimates on trilinear terms of collision kernels, we prove the global existence of the classical solution to Boltzmann-Fermi-Dirac equation near equilibrium. Furthermore, the limit from Boltzmann-Fermi-Dirac equation to incompressible Navier-Stokes-Fourier equations is justified rigorously. The corresponding formal analysis was given in the thesis of Zakrevskiy \cite{Zakrevskiy}

math.AP↗

arXiv2015

Hydrodynamic limits of the kinetic self-organized models

The self-organized hydrodynamic models can be derived from the kinetic version of the Vicsek model. The formal derivations and local well-posedness of the macroscopic equations are done by Degond and his collaborators. In this paper, we rigorously justify this hydrodynamic limit.

math.AP↗

arXiv2014

The Vlasov-Poisson-Boltzmann system for the whole range of cutoff soft potentials

The dynamics of dilute electrons can be modeled by the fundamental one-species Vlasov-Poisson-Boltzmann system which describes mutual interactions of the electrons through collisions in the self-consistent electrostatic field. For cutoff intermolecular interactions, although there are some progress on the construction of global smooth solutions to its Cauchy problem near Maxwellians recently, the problem for the case of very soft potentials remains unsolved. By introducing a new time-velocity weighted energy method and based on some new optimal temporal decay estimates on the solution itself and some of its derivatives with respect to both the spatial and the velocity variables, it is shown in this manuscript that the Cauchy problem of the one-species Vlasov-Poisson-Boltzmann system for all cutoff soft potentials does exist a unique global smooth solution for general initial perturbation which is unnecessary to satisfy the neutral condition imposed in [13] for the case of cutoff moderately soft potentials but is assumed to be small in certain weighted Sobolev spaces. Our approach applies also to the case of cutoff hard potentials and thus provides a satisfactory global well-posedness theory to the one-species Vlasov-Poisson-Boltzmann system near Maxwellians for the whole range of cutoff intermolecular interactions in the perturbative framework.

math.AP↗

arXiv2013

Global Existence and Decay of Solutions to the Fokker-Planck-Boltzmann Equation

The Cauchy problem to the Fokker-Planck-Boltzmann equation under Grad's angular cut-off assumption is investigated. When the initial data is a small perturbation of an equilibrium state, global existence and optimal temporal decay estimates of classical solutions are established. Our analysis is based on the coercivity of the Fokker-Planck operator and an elementary weighted energy method.

math.AP↗