arXiv subjects

Chuqi Cao

Publications and source records attributed to Chuqi Cao.

15 recordsLinked to original sources

arXiv2026

Global stability of vacuum for the relativistic Vlasov-Maxwell-Boltzmann system

We consider the three-dimensional relativistic Vlasov-Maxwell-Boltzmann system, where the speed of light $c$ is an arbitrary constant no less than 1, and we establish global existence and nonlinear stability of the vacuum for small initial data, with bounds that are uniform in $c$. The analysis is based on the vector field method combined with the Glassey-Strauss decomposition of the electromagnetic field, and does not require any compact support assumption on the initial data. A key ingredient of the proof is the derivation of a chain rule for the relativistic Boltzmann collision operator that is compatible with the commutation properties of the vector fields. These tools allow us to control the coupled kinetic and electromagnetic equations and to obtain global stability near vacuum.

math.AP↗

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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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.

math.AP↗

arXiv2024

The Vlasov-Poisson-Boltzmann/Landau system with polynomial perturbation near Maxwellian

In this work, we consider the Vlasov-Poisson-Boltzmann system without angular cutoff and the Vlasov-Poisson-Landau system with Coulomb potential near a global Maxwellian $μ$. We establish the global existence, uniqueness and large time behavior for solutions in a polynomial-weighted Sobolev space $H^2_{x, v}( \langle v \rangle^k)$ for some constant $k >0$. The proof is based on extra dissipation generated from semigroup method and energy estimates on electrostatic field.

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arXiv2024

Hydrodynamic limit for the non-cutoff Boltzmann equation

This work deals with the non-cutoff Boltzmann equation for all type of potentials, in both the torus $\mathbf{T}^3$ and in the whole space $\mathbf{R}^3$, under the incompressible Navier-Stokes scaling. We first establish the well-posedness and decay of global mild solutions to this rescaled Boltzmann equation in a perturbative framework, that is for solutions close to the Maxwellian, obtaining in particular integrated-in-time regularization estimates. We then combine these estimates with spectral-type estimates in order to obtain the strong convergence of solutions to the non-cutoff Boltzmannn equation towards the incompressible Navier-Stokes-Fourier system.

math.AP↗

arXiv2024

Time-velocity decay of solutions to the non-cutoff Boltzmann equation in the whole space

In this paper, we consider the perturbed solutions with polynomial tail in large velocities for the non-cutoff Boltzmann equation near global Maxwellians in the whole space. The global in time existence is proved in the weighted Sobolev spaces and the almost optimal time decay is obtained in Fourier transform based low-regularity spaces. The result shows a time-velocity decay structure of solutions that can be decomposed into two parts. One part allows the slow polynomial tail in large velocities, carries the initial data and enjoys the exponential or arbitrarily large polynomial time decay. The other part, with zero initial data, is dominated by the non-negative definite symmetric dissipation and has the exponential velocity decay but only the slow polynomial time decay.

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arXiv2024

The global well-posedness and Newtonian limit for the relativistic Boltzmann equation in a periodic box

In this paper, we study the Newtonian limit for relativistic Boltzmann equation in a periodic box $\mathbb{T}^3$. We first establish the global-in-time mild solutions of relativistic Boltzmann equation with uniform-in-$\mathfrak{c}$ estimates and time decay rate. Then we rigorously justify the global-in-time Newtonian limits from the relativistic Boltzmann solutions to the solution of Newtonian Boltzmann equation in $L^1_pL^{\infty}_x$. Moreover, if the initial data of Newtonian Boltzmann equation belong to $W^{1,\infty}(\mathbb{T}^3\times\mathbb{R}^3)$, based on a decomposition and $L^2-L^\infty$ argument, the global-in-time Newtonian limit is proved in $L^{\infty}_{x,p}$. The convergence rates of Newtonian limit are obtained both in $L^1_pL^{\infty}_x$ and $L^{\infty}_{x,p}$.

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arXiv2023

The Vlasov-Maxwell-Boltzmann/Landau system with polynomial perturbation near Maxwellian

In this paper, we study the Vlasov-Maxwell-Boltzmann system without angular cutoff and the Vlasov-Maxwell-Landau/Boltzmann system with polynomial perturbation $F=μ+f$ near global Maxwellian. In particular, we prove the global existence, uniqueness and large time behavior for solutions in a polynomial weighted space $H^N_{x,v}(\langle v\rangle^k)$. The method is based on Duhamel's principle with the crucial time-decay analysis on the particle distribution $f$ and the electromagnetic field $(E,B)$.

math.AP↗

arXiv2022

De Giorgi Argument for non-cutoff Boltzmann equation with soft potentials

In this paper, we consider the global well-posedness to the non-cutoff Boltzmann equation with soft potential in the $L^\infty$ setting. We show that when the initial data is close to equilibrium and the perturbation is small in $L^2 \cap L^\infty$ polynomial weighted space, the Boltzmann equation has a global solution in the weighted $L^2 \cap L^\infty$ space. The ingredients of the proof lie in strong averaging lemma, new polynomial weighted estimate for the non-cutoff Boltzmann equation and the $L^2$ level set Di Giorgi iteration method developed in \cite{AMSY2}. The convergence to the equilibrium state in both $L^2$ and $L^\infty$ spaces is also proved.

math.AP↗

arXiv2022

Cutoff Boltzmann equation with polynomial perturbation near Maxwellian

In this paper, we consider the cutoff Boltzmann equation near Maxwellian, we proved the global existence and uniqueness for the cutoff Boltzmann equation in polynomial weighted space for all $γ\in (-3, 1]$. We also proved initially polynomial decay for the large velocity in $L^2$ space will induce polynomial decay rate, while initially exponential decay will induce exponential rate for the convergence. Our proof is based on newly established inequalities for the cutoff Boltzmann equation and semigroup techniques. Moreover, by generalizing the $L_x^\infty L^1_v \cap L^\infty_{x, v}$ approach, we prove the global existence and uniqueness of a mild solution to the Boltzmann equation with bounded polynomial weighted $L^\infty_{x, v}$ norm under some small condition on the initial $L^1_x L^\infty_v$ norm and entropy so that this initial data allows large amplitude oscillations.

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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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arXiv2021

Large Time Asymptotic Behaviors of Two Types of Fast Diffusion Equations

We consider two types of non linear fast diffusion equations in R^N:(1) External drift type equation with general external potential. It is a natural extension of the harmonic potential case, which has been studied in many papers. In this paper we can prove the large time asymptotic behavior to the stationary state by using entropy methods.(2) Mean-field type equation with the convolution term. The stationary solution is the minimizer of the free energy functional, which has direct relation with reverse Hardy-Littlewood-Sobolev inequalities. In this paper, we prove that for some special cases, it also exists large time asymptotic behavior to the stationary state.

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arXiv2020

The Kinetic Fokker-Planck Equation with General Force

We consider the kinetic Fokker-Planck equation with a class of general force. We prove the existence and uniqueness of a positive normalized equilibrium (in the case of a general force) and establish some exponential rate of convergence to the equilibrium (and the rate can be explicitly computed). Our results improve results about classical force to general force case. Our result also improve the rate of convergence for the Fitzhugh-Nagumo equation from non-quantitative to quantitative explicit rate.

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arXiv2019

Hypocoercivity of linear kinetic equations via Harris's Theorem

We study convergence to equilibrium of the linear relaxation Boltzmann (also known as linear BGK) and the linear Boltzmann equations either on the torus $(x,v) \in \mathbb{T}^d \times \mathbb{R}^d$ or on the whole space $(x,v) \in \mathbb{R}^d \times \mathbb{R}^d$ with a confining potential. We present explicit convergence results in total variation or weighted total variation norms (alternatively $L^1$ or weighted $L^1$ norms). The convergence rates are exponential when the equations are posed on the torus, or with a confining potential growing at least quadratically at infinity. Moreover, we give algebraic convergence rates when subquadratic potentials considered. We use a method from the theory of Markov processes known as Harris's Theorem.

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arXiv2018

The kinetic Fokker-Planck equation with weak confinement force

We consider the kinetic Fokker-Planck equation with weak confinement force. We proved some (polynomial and sub-exponential) rate of convergence to the equilibrium (depending on the space to which the initial datum belongs). Our results generalized the result in [4, 5, 21, 11, 10, 9, 1, 14] to weak confinement case.

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