Incompressible navier-stokes and euler limits of the boltzmann equation

We consider solutions of the Boltzmann equation, in a d-dimensional torus, d = 2, 3, For macroscopic times τ = t/?^N, ? « 1, t ≧ 0, when the space variations are on a macroscopic scale x = ?^N?1r, N ≧ 2, x in the unit torus. Let u(x, t) be, for t ≦ t₀, a smooth solution of the incompressible Navier Stokes equations (INS) for N = 2 and of the Incompressible Euler equation (IE) for N > 2. We prove that (*) has solutions for t ≦ t₀ which are close, to O(?²) in a suitable norm, to the local Maxwellian [p/(2πT)^d/2]exp{?[v ? ?u(x,t)]²/2T } with constant density p and temperature T . This is a particular case, defined by the choice of initial values of the macroscopic variables, of a class of such solutions in which the macroscopic variables satisfy more general hydrodynamical equations. For N ≧ 3 these equations correspond to variable density IE while for N = 2 they involve higher-order derivatives of the density.     

Oscillation of Systems of Neutral Equations with Variable Coefficients

Consider the system of the neutral delay differential equations (1) $ \frac{d}{{dt}}\left[{Y\left(t \right) - R\left(T \right)Y\left({T - \varrho } \right)} \right] + P\left(t \right)Y\left({t - \tau } \right) - Q\left(t \right)Y\left({t - \sigma } \right) = 0 $ where P(t) = (p_ij(t)), Q(t) = (q_ij(t)) and R(t) = (r_ij(t)) are n x n matrices for t ≧ 0 and the delays τ, σ and ? are nonnegative numbers. We obtain sufficient conditions for the oscillation of all solutions of (1) under the following hypotheses:      

A power law for connectedness of some random graphs at the critical point

We consider P(G is connected) when G is a graph with vertex set Z⁺ = {1,2, …}, and the edge between i and j is present with probability p(i, j) = min(λ h(i, j), 1) for certain functions h(i, j) homogeneous of degree -1. It is known that there is a critical value λ_c of λ such that . We show that the probability, at the critical point λ_c, that n₁, and n₂ are connected satisfies a power law, in the sense that for n₂ ≧ n_t ≧ 1 for any δ > 0 and certain constants c₁ and c₂.     

Gevrey Asymptotic Representation of the Solutions of Equations with One Turning Point

An ordinary differential equation of the type with parameterξ ? IRⁿ and smooth coefficients a_j,a ? C^∞([-T,T]) is studied. It is assumed that all the characteristic roots of the equation vanish at t = 0 while for t ≠ 0 they are real and distinct. The constructions of real-valued phase functions ?pH_kl (k,l = 1., m) and of amplitude functions A_jkl such that for a given s ? [-T, T] every solution u(t, ξ) of the equation can be represented as where Ψ_j(s, ξ)= D^j_tu(s,ξ), j = 0,m-1 are given.     

The decay of solutions of the Carleman model

We prove that for positive initial data u₀, v₀ ? C¹ ( R ) ∩ L¹ ( R ) vanishing at infinity, the solution u(x, t) v(x, t) of the Carleman model satisfies the estimate The constant C depends only on the initial mass m.     

Sojourns above a high level for a gaussian process with a point of maximum variance

Let X(t), 0≦t≦ 1, be a measurable Gaussian process with mean 0 and variance σ²(t) = EX²(t). Suppose that σ²(t) assumes a unique maximum value σ² at a point τ [0,1]. Define L_u = mes(t: 0≦t≦1, X(t)>)Under appropriate conditions, there exists a function fσ(u) such that fσ(u) → ∞ for u ∞, and The function fσ and the constant δ > 0 are determined by the behavior of the function R(x) = mes(t: 0 ≦ t ≦ 1, σ² - σ²(t) ≦ x) for x > 0.     

Longtime behavior for a nonlinear wave equation arising in elasto‐plastic flow

The paper studies the longtime behavior of solutions to the initial boundary value problem (IBVP) for a nonlinear wave equation arising in elasto‐plastic flow u_tt?div{|?u|^m?1?u}?λΔu_t+Δ²u+g(u)=f(x). It proves that under rather mild conditions, the dynamical system associated with above‐mentioned IBVP possesses a global attractor, which is connected and has finite Hausdorff and fractal dimension in the phase spaces X₁=H(Ω) × L²(Ω) and X=(H³(Ω)∩H(Ω)) × H(Ω), respectively. Copyright © 2008 John Wiley & Sons, Ltd.     

Global solutions for operator Riccati equations with unbounded coefficients: A non-linear semigroup approach

Let X be a Banach space of real-valued functions on [0, 1] and let ?(X) be the space of bounded linear operators on X. We are interested in solutions R:(0, ∞) → ?(X) for the operator Riccati equation where T is an unbounded multiplication operator in X and the B_i(t)'s are bounded linear integral operators on X. This equation arises in transport theory as the result of an invariant embedding of the Boltzmann equation. Solutions which are of physical interest are those that take on values in the space of bounded linear operators on L¹(0, 1). Conditions on X, R(0), T, and the coefficients are found such that the theory of non-linear semigroups may be used to prove global existence of strong solutions in ?(X) that also satisfy R(t) ? ?(L¹(0,1)) for all t ≥ 0.     

Decay estimates for the wave and Dirac equations with a magnetic potential

We study the electromagnetic wave equation and the perturbed massless Dirac equation on ℝ_t × ℝ³: where the potentials A(x), B(x), and V(x) are assumed to be small but may be rough. For both equations, we prove the expected time decay rate of the solution where the norm ‖f‖_X can be expressed as the weighted L²-norm of a few derivatives of the data f. © 2006 Wiley Periodicals, Inc.     

Complex Powers of the Neumann Laplacian in Lipschitz Domains

Let L^q_r(Ω) be the usual scale of Sobolev spaces and let Δ_N be the Neumann Laplacian in an arbitrary Lipschitz domain Ω. We present an interpolation based approach to the following question: for what range of indices does map isomorphically onto L^q_r(Ω)/ℝ?     

Numerical solution of a time-like Cauchy problem for the wave equation

Let D ? ?ⁿ be a bounded domain with piecewise-smooth boundary, and q(x,t) a smooth function on D × [0, T]. Consider the time-like Cauchy problem Given g, h for which the equation has a solution, we show how to approximate u(x,t) by solving a well posed fourth-order elliptic partial differential equation (PDE). We use the method of quasi-reversibility to construct the approximating PDE. We derive error estimates and present numerical results.     

Entropy Numbers in Weighted Function Spaces and Eigenvalue Distributions of Some Degenerate Pseudodifferential Operators II

This paper is the continuation of [17]. We investigate mapping and spectral properties of pseudodifferential operators of type Ψ with χ χ ? ? and 0 ≤ γ ≤ 1 in the weighted function spaces B (?ⁿ, w(x)) and F (?ⁿ, w(x)) treated in [17]. Furthermore, we study the distribution of eigenvalues and the behaviour of corresponding root spaces for degenerate pseudodifferential operators preferably of type b₂(x) b(x, D) b₁(x), where b₁(x) and b₂(x) are appropriate functions and b(x, D) ? Ψ. Finally, on the basis of the Birman-Schwinger principle, we deal with the “negative spectrum” (bound states) of related symmetric operators in L₂.     

On the Nonoscillatory Behavior of Solutions of a Second Order Linear Differential Equation

In this paper, wo improve the Sturm comparison theorem and two nonoscillation criteria of Leighton and Wintner, and establish two variants of a Wintner' s nonoscillatory criterion of the second order linear differential equation where r, c : t₀,∞) →, R > 0 a. e. on t₀,∞) and 1/r, c ε L^l(t₀,b) for each b ∞ (t₀,>) for some t₀ > 0. Using these two criteria, we improve some nonoscillation criteria of Hartman. Hille. Moore. Potter. WintnEr, and Willett. These proofs are more elegant and concise than those of theirs.     

Estimating the Diffusion Coefficient for Diffusions Driven by fBm

When the Hurst coefficient of a fBm B _t ^H is greater than 1/2, it is possible to define a stochastic integral with respect to B _t ^H as the pathwise limit of Riemann sums. In this article we consider diffusion equations of the type X_t = x₀ + ₀ ^T (X_s) dB_s ^H. We then construct a simple-to-use estimator of the diffusion coefficient (x), based on the number of crossings of level x of the process X _t. We then study consistency in probability of this estimator and calculate convergence rates in probability.     

Growth of a PSL2R Manifold Group

The purpose of this paper is to present characterizations of the inhomogeneous Hardy-Bessel potential spaces F^p_α (K) over the 2-series field K defined by Littlewood-Paley type function, where Δ₀(x) = 1(|x|≥1), = 0(other), Δ_j(x) = 2^j ≥, = 0(other) (j ≥ 1). These characterizations are given by difference of functions, ball means of difference and atoms. As applications of these results we shall determine when F^p_α(K) is a multiplication algebra, and prove the lower majorant property, the uniform localization property and the equivalence of Fourier multipliers.     

Polynomials Annihilating the Witt Ring

Let F be a non-formally real field of characteristic not 2 and let W(F) be the Witt ring of F. In certain cases generators for the annihilator ideal are determined. Aim the primary decomposition of A(F) is given. For formally d fields F, as an analogue the primary decomposition of A_t(F) = {f(X) ∈ Z[X]| f(ω) = 0 for all ω ∈ W_t(F)}, where W_t(F) is the torsion part of the Witt group, is obtained.     

Turánnical hypergraphs

This paper is motivated by the question of how global and dense restriction sets in results from extremal combinatorics can be replaced by less global and sparser ones. The result we consider here as an example is Turán's theorem, which deals with graphs G = ([n],E) such that no member of the restriction set \begin{align*}\mathcal {R}\end{align*} = \begin{align*}\left( {\begin{array}{*{20}c} {[n]} \\ r \\ \end{array} } \right)\end{align*} induces a copy of K_r. Firstly, we examine what happens when this restriction set is replaced by \begin{align*}\mathcal {R}\end{align*} = {X∈ \begin{align*}\left( {\begin{array}{*{20}c} {[n]} \\ r \\ \end{array} } \right)\end{align*}: X ∩ [m]≠??}. That is, we determine the maximal number of edges in an n ‐vertex such that no K_r hits a given vertex set. Secondly, we consider sparse random restriction sets. An r ‐uniform hypergraph \begin{align*}\mathcal R\end{align*} on vertex set [n] is called Turánnical (respectively ε ‐Turánnical), if for any graph G on [n] with more edges than the Turán number t_r(n) (respectively (1 + ε)t_r(n) ), no hyperedge of \begin{align*}\mathcal {R}\end{align*} induces a copy of K_r in G. We determine the thresholds for random r ‐uniform hypergraphs to be Turánnical and to be ε ‐Turánnical. Thirdly, we transfer this result to sparse random graphs, using techniques recently developed by Schacht [Extremal results for random discrete structures] to prove the Kohayakawa‐?uczak‐Rödl Conjecture on Turán's theorem in random graphs.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Zolotarevω-Polynomials inWH[0, 1]

The main result of this paper characterizes generalizationsof Zolotarev polynomials as extremal functions in the Kolmogorov–Landauproblemwhereω(t) is a concave modulus of continuity,r, m: 1?m?r,are integers, andB?B₀(r, m, ω). We show that theextremal functionsZ_Bhaver+1 points of alternance andthe full modulus of continuity ofZ^(r)_B: ω(Z^(r)_B; t)=ω(t) for allt∈[0, 1]. This generalizesthe Karlin's result on the extremality of classical Zolotarevpolynomials in the problem () forω(t)=tand allB?B_r.     

On the riemann problem for a prototype of a mixed type conservation law

We study the system of conservation laws given by With initial value The system is elliptic when u² + v² < ρ² and hyperbolic when u² + v² ≧ ρ². Following Liu's construction it is found that the system always has a weak solution which however is not necessarily unique.     

Generalized herglotz domains

Let F(θ k, α) be the far field pattern arising from the scattering of a time harmonic plane acoustic wave of wave number k and direction a by a sound-soft cylinder of cross section D. Suppose F has the Fourier expansion where a_n = a_n(k, . Then if ?² is a Dirichlet eigenvalue for D, sufficient conditions are given on D for the existence of a nontrivial sequence |b_n| where the b_n are independent of such that for all directions Domains for which this is true are called generalized Herglotz domains. The conditions for a domain to be a generalized Herglotz domain are given either in terms of the Schwarz function for the analytic boundary ?D or in terms of the Rayleigh hypothesis in acoustic scattering theory and examples are given showing the applicability of these conditions.