arXiv subjects

Kunlun Qi

Publications and source records attributed to Kunlun Qi.

17 recordsLinked to original sources

arXiv2026

The small Deborah number limit for the compressible fluid-particle flows

In this paper, we consider the hydrodynamic limit for the fluid-particle flows governed by the Vlasov-Fokker-Planck equation coupled with the compressible Navier-Stokes equation as the Deborah number tends to zero. The proof is based on a formal derivation via the Hilbert expansion around the limiting system, the rigorous justification of which is completed by the refined energy estimates involving the macro-micro decomposition. Compared with the existing results obtained by the relative entropy argument ([A. Mellet and A. F. Vasseur, Comm. Math. Phys., 281 (2008), pp. 573-596]), the present work extends to a pointwise convergence of the hydrodynamic limits with an explicit rate for the fluid-particle coupled model.

math.AP↗

arXiv2026

A fast Fourier spectral method for wave kinetic equation

The central object in wave turbulence theory is the wave kinetic equation (WKE), which is an evolution equation for wave action density and acts as the wave analog of the Boltzmann kinetic equations for particle interactions. Despite recent exciting progress in the theoretical aspects of the WKE, numerical developments have lagged behind. In this paper, we introduce a fast Fourier spectral method for solving the WKE. The key idea lies in reformulating the high-dimensional nonlinear wave kinetic operator as a spherical integral, analogous to the classical Boltzmann collision operator. The conservation of mass and momentum leads to a double convolution structure in Fourier space, which can be efficiently handled by the fast Fourier transform (FFT), reducing the computational cost from $O(N^{3d})$ to $O(M N^d \log N)$ with $N$-frequency nodes and $M \ll N^{2d-1}$ in $d$ dimensions. We demonstrate the accuracy and efficiency of the proposed method through several numerical tests in both 2D and 3D, revealing and conjecturing some interesting and unique features of this equation.

math.NA↗

arXiv2025

The Semiclassical Limit of the 2D Dirac--Hartree Equation with Periodic Potentials

We study the semiclassical limit of the two-dimensional Dirac--Hartree equation in the presence of a periodic external potential. The spinor dynamics are formulated using the matrix-valued Wigner transform together with spectral projectors onto the positive and negative energy bands. Under suitable assumptions on the initial data and the potentials, we rigorously derive Vlasov-type transport equations describing the evolution of the band-resolved phase-space densities in both the massive and massless regimes. In the massless case, the limiting dynamics propagate ballistically with constant speed, while in the massive case the velocity is relativistic. Our analysis justifies the emergence of relativistic Vlasov equations from Dirac--Hartree dynamics in the semiclassical regime. As a corollary, we recover the relativistic Vlasov--Poisson equation from the Dirac equation with a regularized Coulomb interaction when the regularization vanishes together with the semiclassical parameter.

math-ph↗

arXiv2025

Hydrodynamic limit of the Vlasov-Poisson-Fokker-Planck system in low-field regime

In this paper, we study the hydrodynamic limit of the scaled Vlasov-Poisson-Fokker-Planck (VPFP) system in the low-field regime. By employing the moment method, we formally derive the corresponding Drift-Diffusion-Poisson (DDP) system. Furthermore, we rigorously justify the pointwise convergence from the VPFP system to the DDP system through delicate high-order energy estimates based on the Macro-Micro decomposition. The main difficulty lies in controlling the nonlinear coupling between the kinetic and electrostatic fields and establishing uniform bounds with respect to the scaling parameter. These challenges are overcome by developing refined high-order energy methods that yield uniform energy estimates and ensure the global well-posedness of smooth solutions, without relying on any a priori assumptions for the limiting DDP system.

math.AP↗

arXiv2025

Solving the BGK Model and Boltzmann equation by Fourier Neural Operator with conservative constraints

The numerical approximation of the Boltzmann collision operator presents significant challenges arising from its high dimensionality, nonlinear structure, and nonlocal integral form. In this work, we propose a Fourier Neural Operator (FNO) based framework to learn the Boltzmann collision operator and its simplified BGK model across different dimensions. The proposed operator learning approach efficiently captures the mapping between the distribution functions in either sequence-to-sequence or point to point manner, without relying on fine grained discretization and large amount of data. To enhance physical consistency, conservation constraints are embedded into the loss functional to enforce improved adherence to the fundamental conservation laws of mass, momentum, and energy compared with the original FNO framework. Several numerical experiments are presented to demonstrate that the modified FNO can efficiently achieve the accurate and physically consistent results, highlighting its potential as a promising framework for physics constrained operator learning in kinetic theory and other nonlinear integro-differential equations.

math.NA↗

arXiv2025

Machine learning-based moment closure model for the linear Boltzmann equation with uncertainties

The Boltzmann equation, a fundamental equation in kinetic theory, serves as a bridge between microscopic particle dynamics and macroscopic continuum mechanics. However, deriving closed macroscopic moment systems from the Boltzmann equation remains a long-standing challenge due to the intrinsic non-closure of the moment hierarchy. In this paper, we propose a machine learning (ML)-based moment closure model for the linear Boltzmann equation, addressing both the deterministic and stochastic settings. Our approach leverages neural networks to learn the spatial gradient of the unclosed highest-order moment, enabling effective training through natural output normalization. For the deterministic problem, to ensure global hyperbolicity and stability, we derive and apply the constraints that enforce symmetrizable hyperbolicity of the system. For the stochastic problem, we adopt the generalized polynomial chaos (gPC)-based stochastic Galerkin method to discretize the random variables, resulting in a system for which the approach in the deterministic case can be used similarly. Several numerical experiments are shown to demonstrate the effectiveness and accuracy of our ML-based moment closure model for the linear Boltzmann equation with or without uncertainties.

math.NA↗

arXiv2025

Continuous data assimilation for hydrodynamics: consistent discretization and application to moment recovery

Motivated by the challenge of moment recovery in hydrodynamic approximation in kinetic theory, we propose a data-driven approach for the hydrodynamic models. Inspired by continuous data assimilation, our method introduces a relaxation-based nudging system coupled with a novel discretization technique. This approach facilitates the simultaneous recovery of both the force term and a high-resolution solution from sparsely observed data. To address potential numerical artifacts, we use kernel regression to fit the observed data. We also analyze the convergence of the proposed nudging system under both full and partial data scenarios. When applied to moment systems, the source term involves the derivative of higher-order moments, our approach serves as a crucial step for data preparation in machine-learning based moment closure models. Multiple numerical experiments demonstrate the effectiveness of our algorithm, and we discuss its potential extension to high-dimensional systems.

math.NA↗

arXiv2024

Radiative transport in a periodic structure with band crossings

We use the Wigner transformation and asymptotic analysis to systematically derive the semi-classical model for the Schrödinger equation in arbitrary spatial dimensions, with any periodic structure. Our particular emphasis lies in addressing the \textit{diabatic} effect, i.e., the impact of Bloch band crossings. We consider both deterministic and random scenarios. In the former case, we derive a coupled Liouville system, revealing lower-order interactions among different Bloch bands. In the latter case, a coupled system of radiative transport equations emerges, with the scattering cross-section induced by the random inhomogeneities. As a specific application, we deduce the effective dynamics of a wave packet in graphene with randomness.

math-ph↗

arXiv2024

On the kinetic description of the objective molecular dynamics

In this paper, we develop a multiscale hierarchy framework for objective molecular dynamics (OMD), a reduced order molecular dynamics with a certain symmetry, that connects it to the statistical kinetic equation, and the macroscopic hydrodynamic model. In the mesoscopic regime, we exploit two interaction scalings that lead, respectively, to either a mean-field type or to a Boltzmann type equation. It turns out that, under the special symmetry of OMD, the mean-field scaling results in vastly simplified dynamics that extinguishes the underlying molecular interaction rule, whereas the Boltzmann scaling yields a meaningful reduced model called the homo-energetic Boltzmann equation. At the macroscopic level, we derive the corresponding Euler and Navier-Stokes systems by conducting a detailed asymptotic analysis. The symmetry again significantly reduces the complexity of the resulting hydrodynamic systems.

math.AP↗

arXiv2024

From the Boltzmann equation for gas mixture to the two-fluid incompressible hydrodynamic system

In this paper, we study the hydrodynamic limit transition from the Boltzmann equation for gas mixtures to the two-fluid macroscopic system. Employing a meticulous dimensionless analysis, we derive several novel hydrodynamic models via the moments' method. For a certain class of scaled Boltzmann equations governing gas mixtures of two species, we rigorously establish the two-fluid incompressible Navier-Stokes-Fourier system as the hydrodynamic limit. This validation is achieved through the Hilbert expansion around the global Maxwellian and refined energy estimates based on the Macro-Micro decomposition.

math.AP↗

arXiv2024

Convergence of the Fourier-Galerkin spectral method for the Boltzmann equation with uncertainties

It is well-known that the Fourier-Galerkin spectral method has been a popular approach for the numerical approximation of the deterministic Boltzmann equation with spectral accuracy rigorously proved. In this paper, we will show that such a spectral convergence of the Fourier-Galerkin spectral method also holds for the Boltzmann equation with uncertainties arising from both collision kernel and initial condition. Our proof is based on newly-established spaces and norms that are carefully designed and take the velocity variable and random variables with their high regularities into account altogether. For future studies, this theoretical result will provide a solid foundation for further showing the convergence of the full-discretized system where both the velocity and random variables are discretized simultaneously.

math.NA↗

arXiv2024

Spectral convergence of a semi-discretized numerical system for the spatially homogeneous Boltzmann equation with uncertainties

In this paper, we study the Boltzmann equation with uncertainties and prove that the spectral convergence of the semi-discretized numerical system holds in a combined velocity and random space, where the Fourier-spectral method is applied for approximation in the velocity space whereas the generalized polynomial chaos (gPC)-based stochastic Galerkin (SG) method is employed to discretize the random variable. Our proof is based on a delicate energy estimate for showing the well-posedness of the numerical solution as well as a rigorous control of its negative part in our well-designed functional space that involves high-order derivatives of both the velocity and random variables. This paper rigorously justifies the statement proposed in [Remark 4.4, J. Hu and S. Jin, J. Comput. Phys., 315 (2016), pp. 150-168].

math.NA↗

arXiv2023

Moments creation for the inelastic Boltzmann equation for hard potentials without angular cutoff

This paper is concerned with the inelastic Boltzmann equation without angular cutoff. We work in the spatially homogeneous case. We establish the global-in-time existence of measure-valued solutions under the generic hard potential long-range interaction on the collision kernel. In addition, we provide a rigorous proof for the creation of polynomial moments of the measure-valued solutions, which is a special property that can only be expected from hard potential collisional cross-sections. The proofs rely crucially on the establishment of a refined Povzner-type inequality for the inelastic Boltzmann equation without angular cutoff. The class of initial data that we require is general in the sense that we only require the boundedness of $(2+κ)$-moment for $κ>0$ and do not assume any entropy bound.

math.AP↗

arXiv2021

Measure Valued Solution to the Spatially Homogeneous Boltzmann Equation with Inelastic Long-Range Interactions

This paper is to study the inelastic Boltzmann equation without Grad's angular cutoff assumption, where the well-posedness theory of the solution to the initial value problem is established for the Maxwellian molecules in a space of probability measure defined by Cannone-Karch in [Comm. Pure. Appl. Math. 63 (2010), 747-778] via Fourier transform and the infinite energy solutions are not a priori excluded as well. Meanwhile, the geometric relation of the inelastic collision mechanism is introduced to handle the strong singularity of the non-cutoff collision kernel. Moreover, we extend the self-similar solution to the Boltzmann equation with infinite energy shown by Bobylev-Cercignani in [J. Stat. Phy. 106 (2002), 1039-1071] to the inelastic case by a constructive approach, which is also proved to be the large-time asymptotic steady solution with the help of asymptotic stability result in a certain sense.

math-ph↗

arXiv2020

On the Measure Valued Solution to the Inelastic Boltzmann Equation with Soft Potentials

The goal of this paper is to extend the existence result of measure-valued solution to the Boltzmann equation in elastic interaction, given by Morimoto-Wang-Yang, to the inelastic Boltzmann equation with moderately soft potentials, also as an extensive work of our preceding result in the inelastic Maxwellian molecules case. We prove the existence and uniqueness of measure-valued solution under Grad's angular cutoff assumption, as well as the existence of non-cutoff solution, for both finite and infinite energy initial datum, by a delicate compactness argument. In addition, the moments propagation and energy dissipation properties are justified for the obtained measure-valued solution as well.

math.AP↗

arXiv2020

A fast Fourier spectral method for the homogeneous Boltzmann equation with non-cutoff collision kernels

We introduce a fast Fourier spectral method for the spatially homogeneous Boltzmann equation with non-cutoff collision kernels. Such kernels contain non-integrable singularity in the deviation angle which arise in a wide range of interaction potentials (e.g., the inverse power law potentials). Albeit more physical, the non-cutoff kernels bring a lot of difficulties in both analysis and numerics, hence are often cut off in most studies (the well-known Grad's angular cutoff assumption). We demonstrate that the general framework of the fast Fourier spectral method can be extended to handle the non-cutoff kernels, achieving the accuracy/efficiency comparable to the cutoff case. We also show through several numerical examples that the solution to the non-cutoff Boltzmann equation enjoys the smoothing effect, a striking property absent in the cutoff case.

physics.comp-ph↗

arXiv2020

A new stability and convergence proof of the Fourier-Galerkin spectral method for the spatially homogeneous Boltzmann equation

Numerical approximation of the Boltzmann equation is a challenging problem due to its high-dimensional, nonlocal, and nonlinear collision integral. Over the past decade, the Fourier-Galerkin spectral method has become a popular deterministic method for solving the Boltzmann equation, manifested by its high accuracy and potential of being further accelerated by the fast Fourier transform. Albeit its practical success, the stability of the method is only recently proved by Filbet, F. & Mouhot, C. in [$ Trans. Amer. Math. Soc.$ 363, no. 4 (2011): 1947-1980.] by utilizing the "spreading" property of the collision operator. In this work, we provide a new proof based on a careful $L^2$ estimate of the negative part of the solution. We also discuss the applicability of the result to various initial data, including both continuous and discontinuous functions.

math.NA↗