The incompressible Navier–Stokes limit of the Boltzmann equation for hard cutoff potentials

The present paper proves that all limit points of sequences of renormalized solutions of the Boltzmann equation in the limit of small, asymptotically equivalent Mach and Knudsen numbers are governed by Leray solutions of the Navier–Stokes equations. This convergence result holds for hard cutoff potentials in the sense of H. Grad, and therefore completes earlier results by the same authors [Invent. Math. 155 (2004) 81–161] for Maxwell molecules.     

Boundary Layers and Incompressible Navier‐Stokes‐Fourier Limit of the Boltzmann Equation in Bounded Domain I

We establish the incompressible Navier‐Stokes‐Fourier limit for solutions to the Boltzmann equation with a general cutoff collision kernel in a bounded domain. Appropriately scaled families of DiPerna‐Lions(‐Mischler) renormalized solutions with Maxwell reflection boundary conditions are shown to have fluctuations that converge as the Knudsen number goes to 0. Every limit point is a weak solution to the Navier‐Stokes‐Fourier system with different types of boundary conditions depending on the ratio between the accommodation coefficient and the Knudsen number. The main new result of the paper is that this convergence is strong in the case of the Dirichlet boundary condition. Indeed, we prove that the acoustic waves are damped immediately; namely, they are damped in a boundary layer in time. This damping is due to the presence of viscous and kinetic boundary layers in space. As a consequence, we also justify the first correction to the infinitesimal Maxwellian that one obtains from the Chapman‐Enskog expansion with Navier‐Stokes scaling. This extends the work of Golse and Saint‐Raymond [20,21] and Levermore and Masmoudi [28] to the case of a bounded domain. The case of a bounded domain was considered by Masmoudi and Saint‐Raymond [34] for the linear Stokes‐Fourier limit and Saint‐Raymond [41] for the Navier‐Stokes limit for hard potential kernels. Neither [34] nor [41] studied the damping of the acoustic waves. This paper extends the result of [34,41] to the nonlinear case and includes soft potential kernels. More importantly, for the Dirichlet boundary condition, this work strengthens the convergence so as to make the boundary layer visible. This answers an open problem proposed by Ukai [46]. © 2016 Wiley Periodicals, Inc.     

Hydrodynamic limits with shock waves of the Boltzmann equation

We show that piecewise smooth solutions with shocks of the Euler equations in gas dynamics can be obtained as the zero Knudsen number limit of solutions of the Boltzmann equation for hard sphere collision model. The construction of the Boltzmann solutions is done in two steps. First we introduce a generalized Hilbert expansion with shock layer correction to construct approximations to the solutions of the Boltzmann equations with small Knudsen numbers. We then apply the recently developed macro‐micro decomposition and energy method for Boltzmann shock layers to construct the exact Boltzmann solutions through the stability analysis. © 2004 Wiley Periodicals, Inc.     

A constructive approach to steady nonlinear kinetic equations

We study steady boundary value problems of nonlinear kinetic theory. Using a continuation argument based on the variation of the Knudsen number we derive a method for the construction of steady solutions of discrete velocity models in a slab. This method is readily transformed into a numerical code. In a preliminary numerical test case the numerical scheme turns out to yield solutions even for Knudsen numbers small enough to calculate with high precision the asymptotic flow field adjacent to a kinetic boundary layer. Thus, we are able to numerically simulate in a simplified situation the transition from a (mesoscopic) kinetic boundary layer to the (macroscopic) far field.     

An analysis of the different parameters influence on the microbearing load carrying capacity

An analytical solution for the two-dimensional non-isothermal compressible gas flow in a slider microbearing is presented in this paper. The solutions enable analysis of the relevant parameters influences on the load carrying capacity of the micro-bearing. Hence, the influences of the flow conditions expressed by Mach, Reynolds and Knudsen numbers, the ratio of the inlet to outlet heights, bearing number, as well as temperature field are investigated. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The applicability of continuum models in the transitional regime of hypersonic flow over blunt bodies

Hypersonic rarefied gas flow over blunt bodies in the transitional flow regime (from continuum to free-molecule) is investigated. Asymptotically correct boundary conditions on the body surface are derived for the full and thin viscous shock layer models. The effect of taking into account the slip velocity and the temperature jump in the boundary condition along the surface on the extension of the limits of applicability of continuum models to high free-stream Knudsen numbers is investigated. Analytic relations are obtained, by an asymptotic method, for the heat transfer coefficient, the skin friction coefficient and the pressure as functions of the free-stream parameters and the geometry of the body in the flow field at low Reynolds number; the values of these coefficients approach their values in free-molecule flow (for unit accommodation coefficient) as the Reynolds number approaches zero. Numerical solutions of the thin viscous shock layer and full viscous shock layer equations, both with the no-slip boundary conditions and with boundary conditions taking into account the effects slip on the surface are obtained by the implicit finite-difference marching method of high accuracy of approximation. The asymptotic and numerical solutions are compared with the results of calculations by the Direct Simulation Monte Carlo method for flow over bodies of different shape and for the free-stream conditions corresponding to altitudes of 75–150 km of the trajectory of the Space Shuttle, and also with the known solutions for the free-molecule flow regine. The areas of applicability of the thin and full viscous shock layer models for calculating the pressure, skin friction and heat transfer on blunt bodies, in the hypersonic gas flow are estimated for various free-stream Knudsen numbers.     

Thue inequalities with a small number of primitive solutions

Several upper bounds are known for the numbers of primitive solutions (x; y) of the Thue equation (1) j F(x; y) j = m and the more general Thue inequality (3) 0 < j F(x; y) j m. A usual way to derive such an upper bound is to make a distinction between "small" and "large" solutions, according as max( j x j ; j y j ) is smaller or larger than an appropriate explicit constant Y depending on F and m; see e.g. [1], [11], [6] and [2]. As an improvement and generalization of some earlier results we give in Section 1 an upper bound of the form cn for the number of primitive solutions (x; y) of (3) with max( j x j ; j y j )Y0 , wherec 25 is a constant and n denotes the degree of the binary form F involved (cf. Theorem 1). It is important for applications that our lower bound Y0 for the large solutions is much smaller than those in [1], [11], [6] and [4], and is already close to the best possible in terms of m. ByusingTheorem1 we establish in Section 2 similar upper bounds for the total number of primitive solutions of (3), provided that the height or discriminant of F is suficiently large with respect to m (cf. Theorem 2 and its corollaries). These results assert in a quantitative form that, in a certain sense, almost all inequalities of the form (3) have only few primitive solutions. Theorem 2 and its consequences are considerable improvements of the results obtained in this direction in [3], [6], [13] and [4]. The proofs of Theorems 1 and 2 are given in Section 3. In the proofs we use among other things appropriate modifications and refenements of some arguments of [1] and [6].     

ON THE FUNCTIONAL-DIFFERENTIAL EQUATION OF THE DISTRIBUTION OF PRIME NUMBERS

1IntroductionsandStatementsInt-,iloIf)4-0'sCllorw'3I](see(ieVisme[4],Wright[:l]alldDriver[1])al,t}(}Inr)1,cdforil](lal[(lllrisl-,i(ital]yarlalyti(:itlI)rooftilt)prim(?number1,if(3or'3m5whi相似文献   

任意体上的双矩阵分解与矩阵方程

本文给出了任意体上具有相同行数或相同列数的双矩阵分解定理；利用此定理，给出了任意体上的矩阵方程ＡＸＢ＋ＣＹＤ＝Ｅ及［Ａ１ＸＢ１，Ａ２ＸＢ２］＝［Ｅ１；Ｅ２］有解的充要条件及其一般解的表达式．     

Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions

The classical existence-and-uniqueness theorem of the solution to a stochastic differential delay equation (SDDE) requires the local Lipschitz condition and the linear growth condition (see e.g. [11], [12] and [20]). The numerical solutions under these conditions have also been discussed intensively (see e.g. [4], [10], [13], [16], [17], [18], [21], [22] and [24]). Recently, Mao and Rassias [14] and [15] established the generalized Khasminskii-type existence-and-uniqueness theorems for SDDEs, where the linear growth condition is no longer imposed. These generalized Khasminskii-type theorems cover a wide class of highly nonlinear SDDEs but these nonlinear SDDEs do not have explicit solutions, whence numerical solutions are required in practice. However, there is so far little numerical theory on SDDEs under these generalized Khasminskii-type conditions. The key aim of this paper is to close this gap.     

Stability analysis of the plane Couette flow for a model kinetic equation

The stability of the plane Couette flow is studied using the simplified Boltzmann equation (the BGK equation) in which the high modes in the space of velocities and coordinates are truncated. The solution to the Navier-Stokes equation with small additional terms depending on the Knudsen number is used as the stationary solution. We assume that the perturbations depend only on the coordinate that is orthogonal to the flow. The density perturbations are assumed to be nonzero. In this approximation, the problem is found to be unstable in the case of small Knudsen numbers.     

Fuzzy system reliability analysis by the use of Tω (the weakest t-norm) on fuzzy number arithmetic operations

In general, the sup-min convolution has been used for fuzzy arithmetic to analyze fuzzy system reliability, where the reliability of each system component is represented by fuzzy numbers. It is well known that T_ω-based addition preserves the shape of L-R type fuzzy numbers. In this paper, we show T_ω-based multiplication also preserves the shape of L-R type fuzzy numbers. We then apply T_ω-based arithmetic operations to fuzzy system reliability analysis. In fact, we show that we can simplify fuzzy arithmetic operations and even get the exact solutions for L-R type fuzzy system reliability, while others [Singer, Fuzzy Sets Syst. 34 (1990) 145; Cheng and Mon, Fuzzy Sets Syst. 56 (1993) 29; Chen, Fuzzy Sets Syst. 64 (1994) 31] have got the approximate solutions using sup-min convolution for evaluating fuzzy system reliability.     

A symmetry-breaking problem in the annulus

We study questions of degeneracy and bifurcation for radial solutions of the semilinear elliptic equation ?u(x) + f(u(x)) = 0, x isin; [math001], [math001]an annulus in Rⁿ, with homogeneous Dirichlet boundary conditions. For certain nonlinearities f(u), we prove existence of degenerate radial solutions u (for which the kernel of the linearized operator Lz = ?z + [math001](u)z, z isin; C²We study questions of degeneracy and bifurcation for radial solutions of the semilinear elliptic equation ?u(x) + f(u(x)) = 0, x isin; [math001], [math001]an annulus in Rⁿ, with homogeneous Dirichlet boundary conditions. For certain nonlinearities f(u), we prove existence of degenerate radial solutions u (for which the kernel of the linearized operator Lz = ?z + [math001](u)z, z isin; C²$⁰([math001]), is non-trivial) and existence of nonradial solutions for the semi-linear equation. These nonradial (asymmetric) solutions are obtained via a bifurcation procedure from the radial (symmetric) ones. This phenomena is called symmetry-breaking. The bifurcation results are proved by a Conley index argument     

A comparative analysis of approaches for investigating hypersonic flow over blunt bodies in a transitional regime

Hypersonic flows of a viscous perfect rarefied gas over blunt bodies in a transitional flow regime from continuum to free molecular, characteristic when spacecraft re-enter Earth's atmosphere at altitudes above 90-100 km, are considered. The two-dimensional problem of hypersonic flow is investigated over a wide range of free stream Knudsen numbers using both continuum and kinetic approaches: by numerical and analytical solutions of the continuum equations, by numerical solution of the Boltzmann kinetic equation with a model collision integral in the form of the S-model, and also by the direct simulation Monte Carlo method. The continuum approach is based on the use of asymptotically correct models of a thin viscous shock layer and a viscous shock layer. A refinement of the condition for a temperature jump on the body surface is proposed for the viscous shock layer model. The continuum and kinetic solutions, and also the solutions obtained by the Monte Carlo method are compared. The effectiveness, range of application, advantages and disadvantages of the different approaches are estimated.     

群体多目标规划αβ-较多联合有效解类的几何特性

对于群体多目标规划问题，文[1]和[2]分别引进了它的联合有效解类和带参数α的α-较多联合有效解类，并且建立了这些解类的最优性条件．文[3]则研究了联合有效解类的几何特性．本文借助供选方案集的带两个参数α和β的αβ-较多有效数，定义了群体多目标规划问题的更一般的αβ-较多联合有效解类，并且研究了这些解的几何特性，得到了若干必要条件和充分条件．     

Periodicity of Balancing Numbers

The balancing numbers originally introduced by Behera and Panda [2] as solutions of a Diophantine equation on triangular numbers possess many interesting properties. Many of these properties are comparable to certain properties of Fibonacci numbers, while some others are more interesting. Wall [14] studied the periodicity of Fibonacci numbers modulo arbitrary natural numbers. The periodicity of balancing numbers modulo primes and modulo terms of certain sequences exhibits beautiful results, again, some of them are identical with corresponding results of Fibonacci numbers, while some others are more fascinating. An important observation concerning the periodicity of balancing numbers is that, the period of this sequence coincides with the modulus of congruence if the modulus is any power of 2. There are three known primes for which the period of the sequence of balancing numbers modulo each prime is equal to the period modulo its square, while for the Fibonacci sequence, till date no such prime is available.     

Taylor-Galerkin algorithms for viscoelastic flow: Application to a model problem

Coupled and decoupled Taylor-Galerkin algorithms are considered for viscoelastic flow and a model problem—transient startup Poiseuille flow in a channel under a fixed pressure gradient. All algorithms reproduce the steady-state solutions and are stable at high elasticity numbers (E). For a fixed mesh, the coupled and decoupled versions (TGC and TGD) give exceptional time-accuracy at low elasticity numbers [to within O(1%) at E = 1] and reasonable accuracy at high elasticity numbers [to within O(10%) at E = 10, 100]. By definition, the decoupled false-transient scheme (TGF), which uses different time scales for velocity and stress time stepping, provides a poor transient history. Where the main requirement is to compute a steady-state algorithm efficiency is crucial. The TGF scheme attains a steady state between six to eight times faster than does the TGC scheme, and the latter is over twice as fast as the TGD form. © 1994 John Wiley & Sons, Inc.     

Calibrating microfluidic pressure correction functions in Laval nozzle flow by eliminating production accuracy influences

The investigated case is a micronozzle flow with Knudsen numbers in the slip-flow regime near the nozzle throat in vacuum environment. Compared gases are neon, argon, krypton and xenon. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)