Fluid-dynamic limit for the Ruijgrok-Wu model of the Boltzmann equation

In this paper we consider the fluid-dynamic limit for the Ruijgrok-Wu model derived from the Boltzmann equation. We use new technique developed in [S. Hwang, A.E. Tzavaras, Kinetic decomposition of approximate solutions to conservation laws: Applications to relaxation and diffusion-dispersion approximations, Comm. Partial Differential Equations 27 (2002) 1229-1254] in order to get the convergence. First, we obtain the approximate transport equation for the given kinetic model. Then using the averaging lemma, we obtain the convergence. This paper shows how to relate the given kinetic model with the averaging lemma to get the convergence.     

Kinetic decomposition for singularly perturbed higher order partial differential equations

In this paper, we consider singularly perturbed higher order partial differential equations. We establish the condition under which the approximate solutions converge in a strong topology to the entropy solution of a scalar conservation laws using methodology developed in Hwang and Tzavaras (Comm. Partial Differential Equations 27 (2002) 1229). First, we obtain the approximate transport equation for the given dispersive equations. Then using the averaging lemma, we obtain the convergence.     

Singular limit problem of the Camassa-Holm type equation

We consider a shallow water equation of Camassa-Holm type, containing nonlinear dispersive effects as well as fourth order dissipative effects. We prove the strong convergence and establish the condition under which, as diffusion and dispersion parameters tend to zero, smooth solutions of the shallow water equation converge to the entropy solution of a scalar conservation law using methodology developed by Hwang and Tzavaras [S. Hwang, A.E. Tzavaras, Kinetic decomposition of approximate solutions to conservation laws: Applications to relaxation and diffusion-dispersion approximations, Comm. Partial Differential Equations 27 (2002) 1229-1254]. The proof relies on the kinetic formulation of conservation laws and the averaging lemma.     

Well-posedness of weak and strong solutions to the kinetic Cucker–Smale model

In this paper, we develop a unified framework that can be used to establish the well-posedness of kinetic Cucker–Smale model with or without noise, for general initial data regardless of the supports; meanwhile we rigorously justify the vanishing noise limit. Our proof is based on weighted energy estimates and the velocity averaging lemma in kinetic theory.     

Existence,uniqueness and positivity of solutions for BGK models for mixtures

We consider kinetic models for a multi component gas mixture without chemical reactions. In the literature, one can find two types of BGK models in order to describe gas mixtures. One type has a sum of BGK type interaction terms in the relaxation operator, for example the model described by Klingenberg, Pirner and Puppo [20] which contains well-known models of physicists and engineers for example Hamel [16] and Gross and Krook [15] as special cases. The other type contains only one collision term on the right-hand side, for example the well-known model of Andries, Aoki and Perthame [1]. For each of these two models [20] and [1], we prove existence, uniqueness and positivity of solutions in the first part of the paper. In the second part, we use the first model [20] in order to determine an unknown function in the energy exchange of the macroscopic equations for gas mixtures described by Dellacherie [11].     

Cauchy problem for the ellipsoidal BGK model for polyatomic particles

We establish the existence and uniqueness of mild solutions for the polyatomic ellipsoidal BGK model, which is a relaxation type kinetic model describing the evolution of polyatomic gaseous system at the mesoscopic level.     

Embedding open Riemann surfaces in Riemannian manifolds

For the Riemann surface of the topological type, we can get a conformai model in orientable Riemannian manifolds. We will prove that there is a conformally equivalent model in orientable Riemannian manifolds for a given open Riemann surface. To end up we utilize Garsia 's Continuity lemma and Brouwer's Fixed Point lemma along with the Teichmüller theory.     

Initial–boundary value problems for conservation laws with source terms and the Degasperis–Procesi equation

We consider conservation laws with source terms in a bounded domain with Dirichlet boundary conditions. We first prove the existence of a strong trace at the boundary in order to provide a simple formulation of the entropy boundary condition. Equipped with this formulation, we go on to establish the well-posedness of entropy solutions to the initial–boundary value problem. The proof utilizes the kinetic formulation and the averaging lemma. Finally, we make use of these results to demonstrate the well-posedness in a class of discontinuous solutions to the initial–boundary value problem for the Degasperis–Procesi shallow water equation, which is a third order nonlinear dispersive equation that can be rewritten in the form of a nonlinear conservation law with a nonlocal source term.     

Recovery of the scale-ε~2 pattern by lattice BGK model

IntroductionThe chemical oscillation and chemical waves are the chemical system of the order structuresdecided by the nonlinear features far from the state of equilibrium. If we consider the effecto f the diffusion and nonlinear reaction, then we can find two types of chemical phenomena:Turing pattern and nonlinear chemical wavesll--3]. Turing pattern is the periodic structure inthe spatial distribution. In 1952, Turing pointed out that the structure ealsts. The character ofthe TUring patter…     

Weak Entropy Boundary Conditions for Isentropic Gas Dynamics via Kinetic Relaxation

We consider a kinetic BGK model relaxing to isentropic gas dynamics previously introduced by the authors, but with Dirichlet boundary condition on the incoming velocities. We pass to the limit as the relaxation parameter tends to zero by compensated compactness inside the domain, and obtain that the limit satisfies entropy inequalities on the boundary involving weak traces of entropy fluxes. Our method is very general and could be applied to any entropy satisfying BGK model as soon as we have strong compactness of the macroscopic variables inside the domain.     

An integral representation of functions in gas-kinetic models

Motivated by the theory of kinetic models in gas dynamics, we obtain an integral representation of lower semicontinuous functions on \({{\mathbb{R}}^d,}\)\({d\geq1}\). We use the representation to study the problem of compactness of a family of the solutions of the discrete time BGK model for the compressible Euler equations. We determine sufficient conditions for strong compactness of moments of kinetic densities, in terms of the measures from their integral representations.     

Homogenization and Hydrodynamic Limit for Fermi‐Dirac Statistics Coupled to a Poisson Equation

This paper deals with the diffusion approximation of a Boltzmann‐Poisson system modeling Fermi‐Dirac statistics in the presence of an extra external oscillating electrostatic potential. Here we extend the analysis done in [19] to the case of a nonlinear collision operator. In addition to the averaging lemma and control from entropy dissipation used in [19], here we use two‐scale Young measures and renormalization techniques to prove the convergence. This result rigorously justifies the formal analysis of [3]. © 2015 Wiley Periodicals, Inc.     

Properties of Convergence for a Class of Generalized $q$-Gamma Operators.

In this paper, a generalization of $q$-Gamma operators based on the concept of $q$-integer is introduced. We investigate the moments and central moments of the operators by computation, obtain a local approximation theorem and get the pointwise convergence rate theorem and also obtain a weighted approximation theorem. Finally, a Voronovskaya type asymptotic formula was given.     

Convergence of a continuous BGK model for conservation laws on the quarter plane

We consider a scalar conservation law in the quarter plan. This equation is approximated in a kinetic BGK model with infinite set of velocities. The convergence is established in the general BV framework, without special restrictions on the flux nor on the equilibrium problem's data.     

变双障碍问题解的抽象稳定性

本文考虑了主部为非线性变双障碍问题解的抽象稳定性 (连续依赖性 ) .由于采用了弱收敛原理和文 [2 ]中取检验函数的技巧 ,我们的证明无需像 [1 ]那样应用Minty引理 .     

Kinetic approximation of a boundary value problem for conservation laws

Summary. We design numerical schemes for systems of conservation laws with boundary conditions. These schemes are based on relaxation approximations taking the form of discrete BGK models with kinetic boundary conditions. The resulting schemes are Riemann solver free and easily extendable to higher order in time or in space. For scalar equations convergence is proved. We show numerical examples, including solutions of Euler equations.Mathematics Subject Classification (2000): 65M06, 65M12, 76M20Correspondence to: D. Aregba-Driollet     

Stability and consistency of kinetic upwinding for advection-diffusion equations

^** Email: matorril{at}ust.hk^*** Email: makxu{at}ust.hk Numerical methods based on kinetic models of fluid flows, likethe so-called BGK scheme, are becoming increasingly popularfor the solution of convection-dominated viscous fluid equationsin a finite-volume approach due to their accuracy and robustness.Based on kinetic-gas theory, the BGK scheme approxi-mately solvesthe BGK kinetic model of the Boltzmann equation at each cellinterface and obtains a numerical flux from integration of thedistribution function. This paper provides the first analyticalinvestigations of the BGK-scheme and its stability and consistencyapplied to a linear advection–diffusion equation. Thestructure of the method and its limiting cases are discussed.The stability results concern explicit time marching and demonstratethe upwinding ability of the kinetic method. Furthermore, itsstability domain is larger than that of common finite-volumemethods in the under-resolved case, i.e. where the grid Reynoldsnumber is large. In this regime, the BGK scheme is shown toallow the time step to be controlled from the advection alone.We show the existence of a third-order ‘super-convergence’on coarse grids independent of the initial condition. We alsoprove a limiting order for the local consist-ency error andshow the error of the BGK scheme to be asymptotically firstorder on very fine grids. However, in advection-dominated regimessuper-convergence is responsible for the high accuracy of themethod.     

Generalized Baskakov–Durrmeyer type operators

In the present paper, we study some approximation properties of the Durrmeyer type modification of generalized Baskakov operators introduced by Erencin (Appl Math Comput 218(3):4384–4390, 2011). First, we establish a Lorentz-type lemma for the derivatives of the kernel of the generalized Baskakov operators and then obtain a recurrence relation for the moments of their Durrmeyer type modification. Next, we discuss some direct results in simultaneous approximation by these operators e.g. pointwise convergence theorem, Voronovskaja-type theorem and an estimate of error in terms of the modulus of continuity. Finally, we estimate the error in the approximation of functions having derivatives of bounded variation.     

Nonlinear diffusive-dispersive limits for scalar multidimensional conservation laws

We consider a class of multidimensional conservation laws with vanishing nonlinear diffusion and dispersion terms. Under a condition on the relative size of the diffusion and dispersion coefficients, we show that the approximate solutions converge in a strong topology to the entropy solution of a scalar conservation law. Our proof is based on methodology developed in [S. Hwang, A.E. Tzavaras, Kinetic decomposition of approximate solutions to conservation laws: Applications to relaxation and diffusion-dispersion approximations, Comm. Partial Differential Equations 27 (2002) 1229-1254] which uses the averaging lemma.     

Discussion on the convergence of Rosen's gradient projection method

Since Rosen's gradient projection method was published in 1960, a rigorous convergence proof of the method has remained to be an open question. A convergence proof for the three dimensional case is given in this paper. The whole proof, except one lemma which we failed to prove for the general case, is applicable to the general case. For the general case a convergence condition is given in the main theorem.This research was supported in part by the National Science Foundation under the research grant MCS 81-01214.