Maxwell equations and the direction of the electromagnetic field

Let M₀ be the Minkowski space, let Λ₂(M₀) be the space of bivectors in M₀, and let G¹ ⊂ Λ₂(M₀) be the manifold of directions of the physical space, consisting of simple bivectors with square −1. A mapping F: U → Λ₂(M₀), U ⊂ ℝ⁴, satisfying the Maxwell equations is regarded as the tensor of an electromagnetic field in vacuum. The field is described on the basis of a special decomposition F = eω + h(*ω), where the mapping ω: U → G¹ is called the direction of the field, and e: U → (0, +∞) and h: U → ℝ are the electric and magnetic coefficients of the field. The Maxwell equations are reformulated in terms of ω, e, and h. Electromagnetic fields whose set of directions is a point or a one-dimensional subset of G¹ are considered. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 118–146.     

Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones

The boundary value problem for the similar stream function f = f(η;λ) of the Cheng–Minkowycz free convection flow over a vertical plate with a power law temperature distribution T_w(x) = T_∞ + Ax^λ in a porous medium is revisited. It is shown that in the λ-range − 1/2 < λ < 0 , the well known exponentially decaying “first branch” solutions for the velocity and temperature fields are not some isolated solutions as one has believed until now, but limiting cases of families of algebraically decaying multiple solutions. For these multiple solutions well converging analytical series expressions are given. This result yields a bridging to the historical quarreling concerning the feasibility of exponentially and algebraically decaying boundary layers. Owing to a mathematical analogy, our results also hold for the similar boundary layer flows induced by continuous surfaces stretched in viscous fluids with power-law velocities u_w(x)∼ x^λ. (Received: June 7, 2005)     

Reduction of PDEs on Unbounded Domains. Application: Unsteady Water-Waves Problem

Summary. For a certain class of partial differential equations in cylindrical domains, we show that all small time-dependent solutions are described by a reduced system of equations on the real line, which contains nonlocal terms. As an application, we investigate the system describing nonlinear water waves travelling on the free surface of an inviscid fluid. Two-dimensional gravity waves are characterized by the parameter λ , the inverse square of the Froude number. For λ close to the critical value λ ₀ =1 , we obtain a reduced system of four nonlocal equations. We show that the terms of lowest order in μ=λ-1 lead to the Korteweg—de Vries equation for the lowest-order approximation of the free surface. Received February 23, 1994; final revision received October 13, 1997; accepted for publication October 16, 1997.     

Small prime solutions of a pair of linear equations in five prime variables

Assume an additional congruent condition on the coefficients. We prove that the pair 5 of linear equations ∑j=1^5 αλjpj = bλ （λ= 1, 2） has solutions in primes pj satisfying pj 〈〈 （｜b1｜＋｜b2｜＋1） maxλ,j ｜αλj｜^2318＋ε. This improves the exponent 79680 without assuming the additional condition of the second author＇s.     

On a class of semilinear weakly hyperbolic equations

Abstract  In this paper, we deal with some global existence results for the large data smooth solutions of the Cauchy Problem associated with the semilinear weakly hyperbolic equations

Homogenization of eigenvalue problems in perforated domains

In this paper, we treat some eigenvalue problems in periodically perforated domains and study the asymptotic behaviour of the eigenvalues and the eigenvectors when the number of holes in the domain increases to infinity Using the method of asymptotic expansion, we give explicit formula for the homogenized coefficients and expansion for eigenvalues and eigenvectors. If we denote by ε the size of each hole in the domain, then we obtain the following aysmptotic expansion for the eigenvalues: Dirichlet: λ_ε = ε⁻² λ + λ₀ +O (ε), Stekloff: λ_ε = ελ₁ +O (ε²), Neumann: λ_ε = λ₀ + ελ₁ +O (ε²). Using the method of energy, we prove a theorem of convergence in each case considered here. We briefly study correctors in the case of Neumann eigenvalue problem.     

The solvability of the initial boundary-value problem for equations of motion of a viscous compressible barotropic liquid in the spacesW 2 l+1,l/2+1 (Q T )

A new method of estimating the solutions of the Navier-Stokes equations for a viscous compressible barotropic fluid in a bounded domain Ω⊂ℝ³ is suggested, which makes it possible to investigate the problem for the whole scale of anisotropic spaces W ₂ ^l+2,l/2+1 (Q_T), Q_T=Ω×(0,T), for arbitrary l>1/2. Bibliography: 10 titles. To dear Olga Alexandrovna Ladyzenskaya on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992, pp. 177–186. Translated by V. A. Solonnikov.     

Uniqueness of weak solutions of time-dependent 3-D Ginzburg-Landau model for superconductivity

We prove the uniqueness of weak solutions of the time-dependent 3-D Ginzburg-Landau model for superconductivity with (Ψ ₀, A ₀) ∈ L ²(Ω) initial data under the hypothesis that (Ψ, A) ∈ C([0, T]; L ³(Ω)) using the Lorentz gauge.      

Cubic Diophantine Inequalities

Let λ₁, λ₂,..., λ₇ be real numbers satisfying λ _i ≥ 1. In this paper, we prove there are integers x ₁,..., x ₇ such that the inequalities |λ₁ x ³ ₁ + λ₂ x ³ ₂ + ⋯ + λ₇ x ³ ₇| < 1 and hold simultaneously. Received November 18, 1997, Accepted October 23, 1998     

Four types of bounded wave solutions of CH-γ equation

Recently, many authors have studied the following CH-γ equationut c0ux 3uux - α2(uxxt uuxxx 2uxuxx) γuxxx =0,where α2, c0 and γ are paramters. Its bounded wave solutions have been investigated mainly for the case α2 ＞ 0. For the case α2 ＜ 0, the existence of three bounded waves (regular solitary waves,compactons, periodic peakons) was pointed out by Dullin et al. But the proof has not been given.In this paper, not only the existence of four types of bounded waves periodic waves, compacton-like waves, kink-like waves, regular solitary waves, is shown, but also their explicit expressions or implicit expressions are given for the case α2 ＜ 0. Some planar graphs of the bounded wave solutions and their numerical simulations are given to show the correctness of our results.     

Stability results for a class of non-linear parabolic equations

Summary We study the asymptotic behaviour of the solutions of the equation u_t=Au+λu−|u|^αu. Denoting by λ₀ the principal eigenvalue of the second-order differential operator A, we shall prove that if λ ⩽ λ₀ the only equilibrium solution, namely zero, is asymptotically stable, whereas, if λ>λ₀, the nontrivial equilibrium solutions without internal zeros are asymptotically stable. Attractivity and stability are proved both in the L²-norm and in the H ₀ ¹ -norm. Entrata in Redazione il 15 ottobre 1976.     

Discrete spectrum in the gaps of a perturbed pseudorelativistic hamiltonian

The pseudorelativistic Hamiltonian is considered under wide conditions on potentials A(x), W(x). It is assumed that a real point λ is regular for G_1/2. Let G_1/2(α)=G_1/2−αV, where α>0, V(x)≥0, and V ∈L _d(ℝ^d). Denote by N(λ, α) the number of eigenvalues of G_1/2(t) that cross the point λ as t increases from 0 to α. A Weyl-type asymptotics is obtained for N(λ, α) as α→∞. Bibliography: 5 titles. To O. A. Ladyzhenskaya Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 249, 1997. pp. 102–117. Translated by A. B. Pushnitskii.     

Life span of solutions of a weakly coupled parabolic system

We consider the Cauchy problem for the weakly coupled parabolic system ∂ _t w _λ−Δ w _λ = F(w _λ) in R ^N , where λ > 0, w _λ = (u _λ, v _λ), F(w _λ) = (v _λ ^p , u _λ ^q ) for some p, q ≥ 1, pq > 1, and , for some nonnegative functions φ₁, φ₂ C ₀(R ^N ). If (p, q) is sub-critical or either φ₁ or φ₂ has slow decay at ∞, w _λ blows up for all λ > 0. Under these conditions, we study the blowup of w _λ for λ small.      

Symmetry breaking for a class of semilinear elliptic problems and the bifurcation diagram for a 1-dimensional problem

We study the behaviour of the positive solutions to the Dirichlet problem IR ⁿ in the unit ball in IR ^R wherep<(N+2)/(N−2) ifN≥3 and λ varies over IR. For a special class of functionsg viz.,g(x)=u ₀ ^p (x) whereu ₀ is the unique positive solution at λ=0, we prove that for certain λ’s nonradial solutions bifurcate from radially symmetric positive solutions. WhenN=1, we obtain the complete bifurcation diagram for the positive solution curve.     

Life span of solutions of a weakly coupled parabolic system

We consider the Cauchy problem for the weakly coupled parabolic system ∂ _t w _λ−Δ w _λ = F(w _λ) in R ^N , where λ > 0, w _λ = (u _λ, v _λ), F(w _λ) = (v _λ ^p , u _λ ^q ) for some p, q ≥ 1, pq > 1, and w_l(0) = (lj₁, l^\fracq+1p+1j₂)w_{\lambda}(0) = ({\lambda}{\varphi}_1, {\lambda}^{\frac{q+1}{p+1}}{\varphi}_2), for some nonnegative functions φ₁, φ₂ ?\in C ₀(R ^N ). If (p, q) is sub-critical or either φ₁ or φ₂ has slow decay at ∞, w _λ blows up for all λ > 0. Under these conditions, we study the blowup of w _λ for λ small.     

The asymptotic behavior of Chern-Simons Higgs model on a compact Riemann surface with boundary

We study the self-dual Chern-Simons Higgs equation on a compact Riemann surface with the Neumann boundary condition.In the previous paper,we show that the Chern-Simons Higgs equation with parameter λ0 has at least two solutions(uλ1,uλ2) for λ sufficiently large,which satisfy that uλ1→u0 almost everywhere as λ→∞,and that uλ2→∞ almost everywhere as λ→∞,where u 0 is a(negative) Green function on M.In this paper,we study the asymptotic behavior of the solutions as λ→∞,and prove that uλ2-uλ2 converges to a solution of the Kazdan-Warner equation if the geodesic curvature of the boundary M is negative,or the geodesic curvature is nonpositive and the Gauss curvature is negative where the geodesic curvature is zero.     

A weighted version of a result of Hamada on minihypers and on linear codes meeting the Griesmer bound

Minihypers were introduced by Hamada to investigate linear codes meeting the Griesmer bound. Hamada (Bull Osaka Women’s Univ 24:1–47, 1985; Discrete Math 116:229–268, 1993) characterized the non-weighted minihypers having parameters , with k−1 > λ₁ > λ₂ > ... > λ _h ≥ 0, as the union of a λ₁-dimensional space, λ₂-dimensional space, ..., λ _h -dimensional space, which all are pairwise disjoint. We present in this article a weighted version of this result. We prove that a weighted -minihyper , with k−1 > λ₁ > λ₂ > ... > λ _h ≥ 0, is a sum of a λ₁-dimensional space, λ₂-dimensional space, ..., and λ _h -dimensional space. This research was supported by the Project Combined algorithmic and theoretical study of combinatorial structures between the Fund for Scientific Research Flanders-Belgium (FWO-Flanders) and the Bulgarian Academy of Sciences. This research is also part of the FWO-Flanders project nr. G.0317.06 Linear codes and cryptography.     

Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones

The boundary value problem for the similar stream function f  =  f(η;λ) of the Cheng–Minkowycz free convection flow over a vertical plate with a power law temperature distribution T_w(x)  =  T_∞ + Ax^λ in a porous medium is revisited. It is shown that in the λ-range  − 1/2  < λ  <  0 , the well known exponentially decaying “first branch” solutions for the velocity and temperature fields are not some isolated solutions as one has believed until now, but limiting cases of families of algebraically decaying multiple solutions. For these multiple solutions well converging analytical series expressions are given. This result yields a bridging to the historical quarreling concerning the feasibility of exponentially and algebraically decaying boundary layers. Owing to a mathematical analogy, our results also hold for the similar boundary layer flows induced by continuous surfaces stretched in viscous fluids with power-law velocities u_w(x)∼ x^λ.     

A coefficient inequality for Bloch functions with applications to uniformly locally univalent functions

We give a Fekete-Szeg? type inequality for an analytic function on the unit disk with Bloch seminorm ≤1. As an application of it, we derive a sharp inequality for the third coefficient of a uniformly locally univalent function f(z) = z + a ₂ z ² + a ₃ z ³ + ⋯ on the unit disk with pre-Schwarzian norm ≤λ for a given λ > 0.     

Critical thresholds in the semiclassical limit of 2-D rotational Schrödinger equations

We consider a two-dimensional convection model augmented with the rotational Coriolis forcing, centrifugal forcing as well as the quadratic potential with a fixed Ω > 0 being the rotational frequency. This model arises in the semiclassical limit of the Gross–Pitaevskii equation for Bose–Einstein condensates in a rotational frame. We investigate whether the action of dispersive rotational forcing complemented with the underlying potential prevents the generic finite time breakdown of the free nonlinear convection. We show that the rotating equations admit global smooth solutions for and only for a subset of generic initial configurations. Thus, the global regularity depends on whether the initial configuration crosses an intrinsic critical threshold, which is quantified in terms of the initial spectral gap associated with the 2 × 2 initial velocity gradient, λ ₂ (0) − λ ₁ (0), λ _j (0)=λ _j (∇ _x U₀) as well as the initial divergence, div_x (U₀). We also prove that for the case of isotropic trapping potential the smooth velocity field is periodic if and only if the ratio of the rotational frequency and the potential frequency is a rational number. The critical thresholds are also established for the case of repulsive potential. Finally the position density and the velocity field are explicitly recorded along the deformed flow map. Received: November 12, 2003; revised: May 4, 2004