Specular Reflection Boundary Condition Research Articles

The Landau and non-cutoff Boltzmann equations in union of cubes

The existence and stability of collisional kinetic equations, especially non-cutoff Boltzmann equation, in a bounded domain with physical boundary condition is a longstanding open problem. This work proves the global stability of the Landau equation and non-cutoff Boltzmann equation in union of cubes with the specular reflection boundary condition when an initial datum is near Maxwellian. Moreover, the solution enjoys exponential large-time decay in the union of cubes. Our method is based on the fact that normal derivatives in cubes are also derivatives along the axis, which allows us to obtain high-order derivative estimates.

Numerical approximation of kinetic Fokker–Planck equations with specular reflection boundary conditions

This work is devoted to the analysis of a numerical approximation to a general multi-dimensional kinetic Fokker–Planck (FP) equation with reaction and source terms and subject to specular reflection boundary conditions. This numerical approximation is based on splitting the kinetic FP model into a transport equation in space and a FP diffusive model in the velocity coordinates. The former is discretized by a Kurganov-Tadmor finite-volume scheme, while the latter is approximated by a generalized Chang & Cooper finite-volume method. Time integration is performed by a strong stability-preserving Runge-Kutta method where the reaction and source terms are accommodated with a Strang splitting technique and the use of a Magnus integrator. It is proved that the resulting numerical solution method is conservative and positive preserving, in the case where the continuous model has these properties, and it is second-order accurate in time and in phase space in the L1-norm, subject to a CFL condition. Results of numerical experiments are reported that validate these theoretical results.

The Fokker–Planck–Boltzmann equation in the finite channel

In this paper, we establish the existence of small-amplitude unique solutions near the Maxwellian for the Fokker–Planck–Boltzmann equation in a finite channel with specular reflection boundary conditions. The solution space we consider is denoted as Lk̄1LT∞Lx1,v2, introduced in Duan et al. [Commun. Pure Appl. Math. 74(5), 932–1020 (2021)]. In addition, we investigate the long-time behavior of solutions for both hard and soft potentials.

The Boltzmann Equation with a Class of Large-Amplitude Initial Data and Specular Reflection Boundary Condition

The Boltzmann Equation with a Class of Large-Amplitude Initial Data and Specular Reflection Boundary Condition

Numerical investigation on radiative heat loss of a direct absorption solar collector using the discrete ordinates method

Direct-absorption solar collectors have the potential to be highly efficient and exhibit a uniform temperature distribution. However, the standard numerical model simplifies radiation and heat loss, which require further consideration at high temperatures. Thus, a novel self-programming model for photothermal conversion in a direct-absorption solar collector was developed based on the discrete ordinates method. This novel model contains further details on radiative transfer and specular reflection boundary conditions. This study compared the results of the two models at high temperatures. With an inlet temperature of 500℃ and a solar concentration ratio of 100, the standard model overestimated outlet temperature and conversion efficiency by 27.6 ℃ and 18.9%, respectively. This study investigated radiative heat loss at different temperatures, solar concentration ratios, and incident angles. The results showed that radiative heat loss will exceed 5% with a fluid temperature exceeding 600 ℃. Furthermore, efficiency decreases significantly at incident angles exceeding 30°. The proposed model expands the range of applications of numerical models of solar collectors to high temperatures. These results provide a reference for improving the conversion efficiency of solar collectors at high temperatures, which is regarded as an urgent problem.

About Lanford's theorem in the half-space with specular reflection

<p style='text-indent:20px;'>The present article proposes a rigorous derivation of the Boltzmann equation in the half-space. We show an analog of the Lanford's theorem in this domain, with specular reflection boundary condition, stating the convergence in the low density limit of the first marginal of the density function of a system of N hard spheres towards the solution of the Boltzmann equation associated to the initial data corresponding to the initial state of the one-particle-density function. The original contributions of this work consist in two main points: the rigorous definition of the collision operator and of the functional space in which the BBGKY hierarchy is solved in a strong sense; and the adaptation to the case of the half-space of the control of the recollisions performed by Gallagher, Saint- Raymond and Texier, which is a crucial step to obtain a quantitative version of the Lanford's theorem.</p>

Stability of rarefaction waves for the non-cutoff Vlasov–Poisson–Boltzmann system with physical boundary

In this paper, we are concerned with the Vlasov–Poisson–Boltzmann (VPB) system in three-dimensional spatial space without angular cutoff in a rectangular duct with or without physical boundary conditions. Near a local Maxwellian with macroscopic quantities given by rarefaction wave solution of one-dimensional compressible Euler equations, we establish the time-asymptotic stability of planar rarefaction wave solutions for the Cauchy problem to VPB system with periodic or specular-reflection boundary condition. In particular, we successfully introduce physical boundaries, namely, specular-reflection boundary, to the models describing wave patterns of kinetic equations. Moreover, we treat the non-cutoff collision kernel instead of the cutoff one. As a simplified model, we also consider the stability and large time behavior of the rarefaction wave solution for the Boltzmann equation.

Axisymmetric lattice Boltzmann model for liquid flows with super-hydrophobic cylindrical surfaces

The lattice Boltzmann (LB) method has been extensively applied to the simulation of various fluid flows. Nonetheless, the standard LB model in rectangular coordinates faces some challenges when attempts are made to use it to simulate single-phase liquid flows involving super-hydrophobic cylindrical surfaces with a large slip length. Therefore, in this paper, boundary conditions for an axisymmetric LB model are proposed for the simulation of liquid slip at both convex and concave cylindrical surfaces. Based on the Navier slip boundary conditions, an equilibrium–nonequilibrium boundary scheme is developed to obtain the liquid slip in the azimuthal direction, with a combined bounce-back and specular-reflection boundary condition being developed to obtain the liquid slip in the axial direction. These boundary schemes are verified by simulating several established cases, namely, cylindrical Couette flow, harmonic oscillatory cylindrical Couette flow, Hagen–Poiseuille flow, and annular Poiseuille flow. The results are in excellent agreement with analytical results. Furthermore, some more complex flows involving super-hydrophobic cylindrical surfaces, such as stepwise oscillatory cylindrical Couette flow, are also simulated and studied using the proposed LB model. The axisymmetric LB model presented here overcomes the inability of standard LB models to accurately and efficiently simulate single-phase liquid flows involving super-hydrophobic cylindrical surfaces with a large slip length.

The Vlasov–Poisson–Landau System with the Specular-Reflection Boundary Condition

We consider the Vlasov–Poisson–Landau system, a classical model for a dilute collisional plasma interacting through Coulombic collisions and with its self-consistent electrostatic field. We establish global stability and well-posedness near the Maxwellian equilibrium state with decay in time and some regularity results for small initial perturbations, in any general bounded domain (including a torus as in a tokamak device), in the presence of specular reflection boundary condition. We provide a new improved \(L^{2}\rightarrow L^{\infty }\) framework: \(L^{2}\) energy estimate combines only with \(S_{ }^{p}\) estimate for the ultra-parabolic equation.

The initial boundary value problem for the Vlasov–Poisson–Fokker–Planck system

This paper is devoted to the initial boundary value problem of the Vlasov–Poisson–Fokker–Planck system in a bounded domain. The global existence of a unique strong solution and its exponential convergence rate to the equilibrium state are proved under the specular reflection boundary condition. The proof is based on an elementary energy method and the SLp theory developed by Guo et al. [Arch. Ration. Mech. Anal. 236(3), 1389–1454 (2020)] for kinetic models.

Hypocoercivity for kinetic linear equations in bounded domains with general Maxwell boundary condition

We establish the convergence to the equilibrium for various linear collisional kinetic equations (including linearized Boltzmann and Landau equations) with physical local conservation laws in bounded domains with general Maxwell boundary condition. Our proof consists in establishing an hypocoercivity result for the associated operator, in other words, we exhibit a convenient Hilbert norm for which the associated operator is coercive in the orthogonal of the global conservation laws. Our approach allows us to treat general domains with all type of boundary conditions in a unified framework. In particular, our result includes the case of vanishing accommodation coefficient and thus the specific case of the specular reflection boundary condition.

Rarefied gas dynamics with external fields under specular reflection boundary condition

We consider the Boltzmann equation with external fields in strictly convex domains with the specular reflection boundary condition. We construct classical $C^1$ solutions away from the grazing set under the assumption that the external field is $C^2$ and the normal derivative of the field is positive and bounded away from $0$.

Collisional damping of the surface ion-acoustic mode in semi-bounded plasmas

The dissipation of ion-acoustic surface waves in a fully ionized and semi-bounded plasma is investigated by solving the dispersion integral formulated by employing the specular reflection boundary condition. Non-zero ion temperature and ion-ion collisions are considered in the dielectric permittivity. The calculated damping rate is composed of the collisionless Landau damping and the collisional dissipation. The distinctions between them are addressed here. It is found that the collisionless damping is significant in the region where the wavelength is larger than the electron Debye length. The collisional dynamics, however, as well as the ion temperature, dominates the wave damping when the wavelength is smaller than the electron Debye length.

Asymptotic analysis for a Vlasov–Fokker–Planck/Navier–Stokes system in a bounded domain

We study an asymptotic analysis of a coupled system of kinetic and fluid equations. More precisely, we deal with the nonlinear Vlasov–Fokker–Planck equation coupled with the compressible isentropic Navier–Stokes system through a drag force in a bounded domain with the specular reflection boundary condition for the kinetic equation and homogeneous Dirichlet boundary condition for the fluid system. We establish a rigorous hydrodynamic limit corresponding to strong noise and local alignment force. The limiting system is a type of two-phase fluid model consisting of the isothermal Euler system and the compressible Navier–Stokes system. Our main strategy relies on the relative entropy argument based on the weak–strong uniqueness principle. For this, we provide a global-in-time existence of weak solutions for the coupled kinetic-fluid system. We also show the existence and uniqueness of strong solutions to the limiting system in a bounded domain with the kinematic boundary condition for the Euler system and Dirichlet boundary condition for the Navier–Stokes system.

The model reduction of the Vlasov–Poisson–Fokker–Planck system to the Poisson–Nernst–Planck system via the Deep Neural Network Approach

The model reduction of a mesoscopic kinetic dynamics to a macroscopic continuum dynamics has been one of the fundamental questions in mathematical physics since Hilbert’s time. In this paper, we consider a diagram of the diffusion limit from the Vlasov–Poisson–Fokker–Planck (VPFP) system on a bounded interval with the specular reflection boundary condition to the Poisson–Nernst–Planck (PNP) system with the no-flux boundary condition. We provide a Deep Learning algorithm to simulate the VPFP system and the PNP system by computing the time-asymptotic behaviors of the solution and the physical quantities. We analyze the convergence of the neural network solution of the VPFP system to that of the PNP system via the Asymptotic-Preserving (AP) scheme. Also, we provide several theoretical evidence that the Deep Neural Network (DNN) solutions to the VPFP and the PNP systems converge to the a priori classical solutions of each system if the total loss function vanishes.

Hilbert Expansion of the Boltzmann Equation with Specular Boundary Condition in Half-Space

Boundary effects play an important role in the study of hydrodynamic limits in the Boltzmann theory. Based on a systematic derivation and study of the viscous layer equations and the $L^2$ to $L^\infty$ framework, we establish the validity of the Hilbert expansion for the Boltzmann equation with specular reflection boundary conditions, which leads to derivations of compressible Euler equations and acoustic equations.

Correction to: The Landau Equation with the Specular Reflection Boundary Condition

Correction to: The Landau Equation with the Specular Reflection Boundary Condition

An Investigation of Transition Flow in Porous Media by Event Driven Molecular Dynamics Simulation

Aim of this study is to investigate the properties of mono-atomic gas flow through the porous medium by using Event-Driven Molecular Dynamics (EDMD) simulation in the transition regime. The molecules and the solid particles forming the porous structure were modelled as hard spheres hence molecule trajectories, collision partners, interaction times and post-collision velocities were calculated deterministically. The porous medium is formed of spherical particles suspended in the middle of the channel and these particles are distributed into the channel in a regular cubic array. Collisions of gas molecules with porous medium were provided by means of the specular reflection boundary condition. A negative pressure boundary condition was applied to the inlet and outlet of the porous media to ensure gas flow. Porosity, solid sphere diameter and Knudsen number (Kn) were initially input to the simulation for different Cases. Thus, the effects of these parameters on mass flow rate, dynamic viscosity, tortuosity and permeability were calculated by EDMD simulation. The results were compared with the literature and were found to be consistent.

Global Mild Solutions of the Landau and Non‐Cutoff Boltzmann Equations

This paper proves the existence of small‐amplitude global‐in‐time unique mild solutions to both the Landau equation including the Coulomb potential and the Boltzmann equation without angular cutoff. Since the well‐known works [45] and [3, 43] on the construction of classical solutions in smooth Sobolev spaces which in particular are regular in the spatial variables, it still remains an open problem to obtain global solutions in an framework, similar to that in [49], for the Boltzmann equation with the cutoff assumption in general bounded domains. One main difficulty arises from the interaction between the transport operator and the velocity‐diffusion‐type collision operator in the non‐cutoff Boltzmann and Landau equations; another major difficulty is the potential formation of singularities for solutions to the boundary value problem.In the present work we introduce a new function space with low regularity in the spatial variable to treat the problem in cases when the spatial domain is either a torus or a finite channel with boundary. For the latter case, either the inflow boundary condition or the specular reflection boundary condition is considered. An important property of the function space is that the norm, in velocity and time, of the distribution function is in the Wiener algebra A(Ω) in the spatial variables. Besides the construction of global solutions in these function spaces, we additionally study the large‐time behavior of solutions for both hard and soft potentials, and we further justify the property of propagation of regularity of solutions in the spatial variables. © 2019 Wiley Periodicals, Inc.

Propagation of ion-acoustic surface waves in a turbulent quantum plasma with quantum recoil: transverse truncation method approach

The effect of quantum recoil on the low-frequency electrostatic surface waves on a semi-bounded quantum turbulent plasma is investigated based on the kinetic dielectric permittivity of a plasma where the wave frequency is in the range of ion-acoustic wave. The dispersion relation and the damping rate of the surface wave are derived by employing the specular reflection boundary condition and the transverse truncation method. It is shown that the frequency of surface wave is enhanced by the quantum recoil for both non-degenerate and degenerate cases. The magnitude of damping rate is also increased by the quantum effect. The quantum recoil effect on the wave propagation is significantly reduced as the wave number becomes small. In the range of such small wave numbers, the diffusional dissipation becomes more important for the wave propagation.