arXiv subjects

Ling-Bing He

Publications and source records attributed to Ling-Bing He.

At least 19 recordsLinked to original sources

arXiv2026

Runaway avalanches in plasmas with external electric fields: spatially inhomogeneous case in a perturbation framework

We consider the Landau-Coulomb equation for a (hydrogen) plasma heated by an external electric field. In this setting, theoretical and experimental results in plasma physics show the emergence of so-called \emph{runaway electrons} which are linearly accelerating but only lead to a minimal increase of the plasma temperature. Runaway electrons are a major obstacle in nuclear fusion since they can overcome the confinement and damage the structure of the reactor. We rigorously prove the well-posedness of the underlying nonlinear \emph{open} Landau-Coulomb system in a perturbative setting and the conjectured growth bounds for the mean velocity and plasma temperature. We show that the mean velocity is linearly increasing in time, and capture the sharp logarithmic growth of the temperature. Furthermore, we prove that the electron distribution can be asymptotically described by a scattering-type Maxwellian. Due to the different nature of the electron-electron and electron-ion interactions, we recast the equation as a novel coupled system that allows us to isolate the dissipation structures of the two operators. For the coupled system, we perform a micro-macro decomposition to show convergence to the scattering-type Maxwellian.

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arXiv2026

A Fourier spectral method for the cutoff Boltzmann equation: Convergence analysis and numerical simulation

This work addresses a central challenge in the numerical analysis of the cutoff spatially homogeneous Boltzmann equation: the development of rigorously justified, accurate numerical schemes. We present (i) a novel Fourier spectral method for the equation with Maxwellian and hard potentials, (ii) the derivation of the first rigorous error estimates for the proposed schemes. Comprehensive numerical experiments validate the theory, confirming the predicted accuracy and illustrating the method's capability to capture solution dynamics, including the approach to equilibrium. The study thus provides a complete framework--from theoretical analysis to practical implementation--for the reliable computation of solutions to this foundational kinetic model.

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arXiv2025

Classical limit of the relativistic Vlasov-Maxwell-Landau system

The physical essence of the non-relativistic limit, from the relativistic Vlasov-Maxwell-Landau system to the Vlasov-Poisson-Landau system, lies in the transition from finite-speed electromagnetic waves to instantaneous Coulomb interactions, and from relativistic to Newtonian particle dynamics. We rigorously justify this limit (mathematically corresponding to the light speed $c \to \infty$) in a periodic box via three key technical advances: establishing a uniform-in-$c$ coercivity estimate for the relativistic Landau collision operator, constructing a novel weighted energy functional to overcome the weakening dissipation of the electromagnetic field at large $c$, and proving a corresponding global well-posedness result.

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arXiv2025

Numerical analysis of the homogeneous Landau equation: approximation, error estimates and simulation

We construct a numerical solution to the spatially homogeneous Landau equation with Coulomb potential on a domain $D_L$ with N retained Fourier modes. By deriving an explicit error estimate in terms of $L$ and $N$, we demonstrate that for any prescribed error tolerance and fixed time interval $[0, T ]$, there exist choices of $D_L$ and $N$ satisfying explicit conditions such that the error between the numerical and exact solutions is below the tolerance. Specifically, the estimate shows that sufficiently large $L$ and $N$ (depending on initial data parameters and $T$) can reduce the error to any desired level. Numerical simulations based on this construction are also presented. The results in particular demonstrate the mathematical validity of the spectral method proposed in the referenced literature.

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arXiv2025

Boltzmann boundary layer equation with Maxwell reflection boundary condition and applications to fluid limits

This paper investigates the Knudsen layer equation in half-space, arising from the hydrodynamic limit of the Boltzmann equation to fluid dynamics. We consider the Maxwell reflection boundary condition with accommodation coefficient $0<α<1$. We restrict our attention to hard sphere collisions with angular cutoff, proving the existence, uniqueness, and asymptotic behavior of the solution in $L^{\infty}_{x,v}$. Additionally, we demonstrate the application of our theorem to the hydrodynamic limit through a specific example. In this expample, we derive the boundary conditions of the fluid equations using our theorem and the symmetric properties of the Knudsen layer equation for $α\in(0,1]$ and $α=O(1)$. These derivations differs significantly from the cases of specular and almost specular reflection. This explicitly characterizes the {\em vanishing sources set} defined in \cite{jiang2024knudsenboundarylayerequations}

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arXiv2024

Existence, uniqueness and smoothing estimates for spatially homogeneous Landau-Coulomb equation in $H^{-\f12}$ space with polynomial tail

We demonstrate that the spatially homogeneous Landau-Coulomb equation exhibits global existence and uniqueness around the space $H^{-\frac12}_3\cap L^1_{7}\cap L\log L$. Additionally, we furnish several quantitative assessments regarding the smoothing estimates in weighted Sobolev spaces. The new ingredients of the proof lie in the localized techniques in both phase space and frequency space.

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arXiv2024

The $L^p$ estimate for the gain term of the Boltzmann collision operator and its application

We prove the Hardy-Littlewood-Sobolev type $L^p$ estimates for the gain term of the Boltzmann collision operator including Maxwellian molecule, hard potential and hard sphere models. Combining with the results of Alonso et al. [2] for the soft potential and Maxwellian molecule models, we provide an unified form of $L^p$ estimates for all cutoff models which are sharp in the sense of scaling. The most striking feature of our new estimates for the hard potential and hard sphere models is that they do not increase the moment, the same as Maxwellian molecule and soft potential models. Based on these novelties, we prove the global existence and scattering of the non-negative unique mild solution for the Cauchy problem of the Boltzmann equation when the positive initial data is small in the weighted $L^3_{x,v}$ space.

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arXiv2024

On the Boltzmann equation with soft potentials: Existence, uniqueness and smoothing effect of mild solutions

We consider the spatially inhomogeneous Boltzmann equation without angular cutoff for soft potentials. For any given initial datum such that the mass, energy and entropy densities are bounded and the mass is away from vacuum, we establish the local-in-time existence and uniqueness of mild solutions, and further provide the first result on sharp smoothing effect in analytic space or Gevrey space for soft potentials.

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arXiv2024

On Landau equation with harmonic potential: nonlinear stability of time-periodic Maxwell-Boltzmann distributions

We provide the first and rigorous confirmations of the hypotheses by Ludwig Boltzmann in his seminal paper \cite{Boltzmann} within the context of the Landau equation in the presence of a harmonic potential. We prove that (i) Each {\it entropy-invariant solution} can be identified as a {\it time-periodic Maxwell-Boltzmann distribution}. Moreover, these distributions can be characterized by thirteen conservation laws, which sheds light on the global dynamics. (ii) Each {\it time-periodic Maxwell-Boltzmann distribution} is nonlinearly stable, including neutral asymptotic stability and Lyapunov stability. Furthermore, the convergence rate is entirely reliant on the thirteen conservation laws and is optimal when compared to the linear scenario.

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arXiv2023

Global stability and scattering theory for non-cutoff Boltzmann equation with soft potentials in the whole space: weak collision regime}

A Traveling Maxwellian $\mathcal{M} = \mathcal{M}(t, x, v)$ represents a traveling wave solution to the Boltzmann equation in the whole space $\R^3_x$(for the spatial variable). The primary objective of this study is to investigate the global-in-time stability of $\mathcal{M}$ and its associated scattering theory in $L^1_{x,v}$ space for the non-cutoff Boltzmann equation with soft potentials when the dissipative effects induced by collisions are {\it weak}. We demonstrate the following results: (i) $\mathcal{M}$ exhibits Lyapunov stability; (ii) The perturbed solution, which is assumed to satisfy the same conservation law as $\mathcal{M}$, scatters in $L^1_{x,v}$ space towards a particular traveling wave (with an explicit convergence rate), which may not necessarily be $\mathcal{M}$. The key elements in the proofs involve the formulation of the {\it Strichartz-Scaled Boltzmann equation}(achieved through the Strichartz-type scaling applied to the original equation) and the propagation of analytic smoothness.

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arXiv2023

On semi-classical limit of spatially homogeneous quantum Boltzmann equation: asymptotic expansion

We continue our previous work [Ling-Bing He, Xuguang Lu and Mario Pulvirenti, Comm. Math. Phys., 386(2021), no. 1, 143223.] on the limit of the spatially homogeneous quantum Boltzmann equation as the Planck constant $ε$ tends to zero, also known as the semi-classical limit. For general interaction potential, we prove the following: (i). The spatially homogeneous quantum Boltzmann equations are locally well-posed in some weighted Sobolev spaces with quantitative estimates uniformly in $ε$. (ii). The semi-classical limit can be further described by the following asymptotic expansion formula: $$ f^ε(t,v)=f_L(t,v)+O(ε^{\vartheta}).$$ This holds locally in time in Sobolev spaces. Here $f^ε$ and $f_L$ are solutions to the quantum Boltzmann equation and the Fokker-Planck-Landau equation with the same initial data.The convergent rate $0<\vartheta \leq 1$ depends on the integrability of the Fourier transform of the particle interaction potential. Our new ingredients lie in a detailed analysis of the Uehling-Uhlenbeck operator from both angular cutoff and non-cutoff perspectives.

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arXiv2023

Quantitative global well-posedness of Boltzmann-Bose-Einstein equation and incompressible Navier-Stokes-Fourier limit

In the diffusive scaling and in the whole space, we prove the global well-posedness of the scaled Boltzmann-Bose-Einstein (briefly, BBE) equation with high temperature in the low regularity space $H^2_xL^2$. In particular, we quantify the fluctuation around the Bose-Einstein equilibrium $\mathcal{M}_{λ,T}(v)$ with respect to the parameters $λ$ and temperature $T$. Furthermore, the estimate for the diffusively scaled BBE equation is uniform to the Knudsen number $ε$. As a consequence, we rigorously justify the hydrodynamic limit to the incompressible Navier-Stokes-Fourier equations. This is the first rigorous fluid limit result for BBE.

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arXiv2023

Regularity estimates for the non-cutoff soft potential Boltzmann equation with typical rough and slowly decaying data

In the present work, we investigate estimates of regularity for weak solutions to the non-cutoff Boltzmann equation with soft potentials. We restrict our focus to the so-called "typically rough and slowly decaying data", which is constructed to satisfy typical properties: low regularity and having exact polynomial decay in high velocity regimes. By exploring the degenerate and non-local properties of the collision operator, we demonstrate that (i) such data induce only finite smoothing effects for weak solutions in Sobolev spaces; (ii) this finite smoothing property implies that the Leibniz rule does not hold for high derivatives of the collision operator (even in the weak sense). Moreover, we can also prove that the average of the solution or the average of the collision operator on a special domain in $\mathbb{R}^3_v$ will induce discontinuity in the $x$ variable. These facts present major obstacles to proving the conjecture that solutions to the equation will instantly become infinitely smooth for both spatial and velocity variables at any positive time if the initial data has only polynomial decay in high velocity regimes.

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arXiv2022

Global strong solutions of 3D Compressible Navier-Stokes equations with short pulse type initial data

Short pulse initial datum is referred to the one supported in the ball of radius $δ$ and with amplitude $δ^{\frac12}$ which looks like a pulse. It was first introduced by Christodoulou to prove the formation of black holes for Einstein equations and also to catch the shock formation for compressible Euler equations. The aim of this article is to consider the same type initial data, which allow the density of the fluid to have large amplitude $δ^{-\fracαγ}$ with $δ\in(0,1],$ for the compressible Navier-Stokes equations. We prove the global well-posedness and show that the initial bump region of the density with large amplitude will disappear within a very short time. As a consequence, we obtain the global dynamic behavior of the solutions and the boundedness of $\|\nabla u\|_{L^1([0,\infty);L^\infty)}$. The key ingredients of the proof lie in the new observations for the effective viscous flux and new decay estimates for the density via the Lagrangian coordinate.

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arXiv2022

Uniqueness and global dynamics of spatially homogeneous non cutoff Boltzmann equation with moderate soft potentials

Departing from the weak solution, we prove the uniqueness, smoothing estimates and the global dynamics for the non cutoff spatially homogeneous Boltzmann equation with moderate soft potentials. Our results show that the behavior of the solution(including the production of regularity and the longtime behavior) can be {\it characterized quantitatively} by the initial data at the large velocities, i.e.(i). initially polynomial decay at the large velocities in $L^1$ space will induce the finite smoothing estimates in weighted Sobolev spaces and the polynomial convergence rate (including the lower and upper bounds) to the equilibrium; (ii). initially the exponential decay at the large velocities in $L^1$ space will induce $C^\infty$ regularization effect and the stretched exponential convergence rate. The new ingredients of the proof lie in the development of the localized techniques in phase and frequency spaces and the propagation of the exponential momentum.

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arXiv2022

Propagation of moments and sharp convergence rate for inhomogeneous non-cutoff Boltzmann equation with soft potentials

We prove the well-posedness for the non-cutoff Boltzmann equation with soft potentials when the initial datum is close to the {\it global Maxwellian} and has only polynomial decay at the large velocities in $L^2$ space. As a result, we get the {\it propagation of the exponential moments} and the {\it sharp rates} of the convergence to the {\it global Maxwellian} which seems the first results for the original equation with soft potentials. The new ingredients of the proof lie in localized techniques, the semigroup method as well as the propagation of the polynomial and exponential moments in $L^2$ space.

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arXiv2022

On the Cauchy problem for the cutoff Boltzmann equation with small initial data

We prove the global existence of the non-negative unique mild solution for the Cauchy problem of the cutoff Boltzmann equation for soft potential model $-1\leq γ< 0$ with the small initial data in three dimensional space. Thus our result fixes the gap for the case $γ=-1$ in three dimensional space in the authors' previous work where the estimate for the loss term was improperly used. The other gap there for the case $γ=0$ in two dimensional space is recently fixed by Chen, Denlinger and Pavlović. The initial data $f_{0}$ is non-negative, small in weighted $L^{3}_{x,v}$ and finite in weighted $L^{15/8}_{x,v}$. We also show that the solution scatters with respect to the kinetic transport operator. The novel contribution of this work lies in the exploration of the symmetric property of the gain term in terms of weighted estimate. It is the key ingredient for solving the model $-1<γ<0$ when applying the Strichartz estimates.

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arXiv2021

Asymptotic analysis of the linearized Boltzmann collision operator from angular cutoff to non-cutoff

We give quantitative estimates on the asymptotics of the linearized Boltzmann collision operator and its associated equation from angular cutoff to non cutoff. On one hand, the results disclose the link between the hyperbolic property resulting from the Grad's cutoff assumption and the smoothing property due to the long-range interaction. On the other hand, with the help of the localization techniques in the phase space, we observe some new phenomenon in the asymptotic limit process. As a consequence, we give the affirmative answer to the question that there is no jump for the property that the collision operator with cutoff does not have the spectrum gap but the operator without cutoff does have for the moderate soft potentials.

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