Continuous solutions of systems of nonlinear difference equations with continuous argument and their properties

We obtain sufficient conditions for the existence and uniqueness of continuous N-periodic solutions (N is a positive integer number) for a certain class of systems of nonlinear difference equations with continuous argument and study their properties. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 2, pp. 177–183, April–June, 2007.     

Viscoelastic properties of ultra-high viscosity alginates

Ultra-high viscosity alginates were extracted from the brown seaweeds Lessonia nigrescens (UHV_N, containing 61% mannuronate (M) and 2% guluronate (G)) and Lessonia trabeculata (UHV_T, containing 22% M and 78% G). The viscoelastic behavior of the aqueous solutions of these alginates was determined in shear flow in terms of the shear stress σ ₂₁, the first normal stress difference N ₁, and the shear viscosity η in isotonic NaCl solutions (0.154 mol/L) at T = 298 K in dependence of the shear rate [(g)\dot]\dot{\gamma} for solutions of varying concentrations and molar masses (3–10 × 10⁵ g/mol, homologous series was prepared by ultrasonic degradation). Data obtained in small-amplitude oscillatory shear (SAOS) experiments obey the Cox–Merz rule. For comparison, a commercial alginate with intermediate chemical composition was additionally characterized. Particulate substances which are omnipresent in most alginates influenced the determination of the material functions at low shear rates. We have calculated structure–property relationships for the prediction of the viscosity yield, e.g., η–M _w–c–[(g)\dot]\dot{\gamma} for the Newtonian and non-Newtonian region. For the highest molar masses and concentrations, the elasticity yield in terms of N ₁ could be determined. In addition, the extensional flow behavior of the alginates was measured using capillary breakup extensional rheometry. The results demonstrate that even samples with the same average molar mass but different molar mass distributions can be differentiated in contrast to shear flow or SAOS experiments.     

Sufficient Conditions for the Existence of Bounded Solutions of Systems of Nonlinear Difference Equations

We obtain sufficient conditions for systems of nonlinear difference equations x(n + 1) = A(x(n))x(n) + f(n), n ∈ ℤ, where A(x) is a matrix function continuous on ℝ ^m , to have solutions in the space of bilateral number sequences. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 165–173, April–June, 2005.     

Asymptotic approximation for the solution of a boundary-value problem with varying type of boundary conditions in a thick two-level junction

We consider a mixed boundary-value problem for a Poisson equation in a plane two-level junction Ω_ε that is the union of a domain Ω₀ and a large number 3N of thin rods with thickness of order . The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are ε-periodically alternated. The homogeneous Dirichlet conditions and inhomogeneous Neumann conditions are given on the sides of the thin rods from the first level and the second level, respectively. Using the method of matched asymptotic expansions and special junction-layer solutions, we construct an asymptotic approximation for the solution and prove the corresponding estimates in the Sobolev space H ¹(Ω_ε) as ε → 0 (N → +∞). Published in Neliniini Kolyvannya, Vol. 9, No. 3, pp. 336–355, July–September, 2006.     

Unique Solutions to Hartree–Fock Equations for Closed Shell Atoms

In this paper we study the problem of uniqueness of solutions to the Hartree and Hartree–Fock equations of atoms. We show, for example, that the Hartree–Fock ground state of a closed shell atom is unique provided the atomic number Z is sufficiently large compared to the number N of electrons. More specifically, a two-electron atom with atomic number Z\geqq 35{Z\geqq 35} has a unique Hartree–Fock ground state given by two orbitals with opposite spins and identical spatial wave functions. This statement is wrong for some Z > 1, which exhibits a phase segregation.     

Optimal Gradient Estimates and Asymptotic Behaviour for the Vlasov–Poisson System with Small Initial Data

The Vlasov–Poisson system describes interacting systems of collisionless particles. For solutions with small initial data in three dimensions it is known that the spatial density of particles decays as t ⁻³ at late times. In this paper this statement is refined to show that each derivative of the density which is taken leads to an extra power of decay, so that in N dimensions for N \geqq 3{N \geqq 3} the derivative of the density of order k decays as t ^−N-k . An asymptotic formula for the solution at late times is also obtained.     

Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics

The Bogoliubov–Dirac–Fock (BDF) model is the mean-field approximation of no-photon quantum electrodynamics. The present paper is devoted to the study of the minimization of the BDF energy functional under a charge constraint. An associated minimizer, if it exists, will usually represent the ground state of a system of N electrons interacting with the Dirac sea, in an external electrostatic field generated by one or several fixed nuclei. We prove that such a minimizer exists when a binding (HVZ-type) condition holds. We also derive, study and interpret the equation satisfied by such a minimizer. Finally, we provide two regimes in which the binding condition is fulfilled, obtaining the existence of a minimizer in these cases. The first is the weak coupling regime for which the coupling constant α is small whereas αZ and the particle number N are fixed. The second is the non-relativistic regime in which the speed of light tends to infinity (or equivalently α tends to zero) and Z, N are fixed. We also prove that the electronic solution converges in the non-relativistic limit towards a Hartree–Fock ground state.     

Scherk-Type Capillary Graphs

This paper concerns the regularity of a capillary graph (the meniscus profile of liquid in a cylindrical tube) over a corner domain of angle α. By giving an explicit construction of minimal surface solutions previously shown to exist (Indiana Univ. Math. J. 50 (2001), no. 1, 411–441) we clarify two outstanding questions. Solutions are constructed in the case α = π/2 for contact angle data (γ₁, γ₂) = (γ, π − γ) with 0 < γ < π. The solutions given with |γ − π/2| < π/4 are the first known solutions that are not C² up to the corner. This shows that the best known regularity (C^{1, ∈}) is the best possible in some cases. Specific dependence of the H?lder exponent on the contact angle for our examples is given. Solutions with γ = π/4 have continuous, but horizontal, normal vector at the corners in accordance with results of Tam (Pacific J. Math. 124 (1986), 469–482). It is shown that our examples are C^{0, β} up to and including the corner for any β < 1. Solutions with |γ − π/2| > π/4 have a jump discontinuity at the corner. This kind of behavior was suggested by numerical work of Concus and Finn (Microgravity sci. technol. VII/2 (1994), 152–155) and Mittelmann and Zhu (Microgravity sci. technol. IX/1 (1996), 22–27). Our explicit construction, however, allows us to investigate the solutions quantitatively. For example, the trace of these solutions, excluding the jump discontinuity, is C^2/3.     

Isolated Singularities of the 1D Complex Viscous Burgers Equation

The Cauchy problem for the 1D real-valued viscous Burgers equation u _t +uu _x  = u _xx is globally well posed (Hopf in Commun Pure Appl Math 3:201–230, 1950). For complex-valued solutions finite time blow-up is possible from smooth compactly supported initial data, see Poláčik and Šverák (J Reine Angew Math 616:205–217, 2008). It is also proved in Poláčik and Šverák (J Reine Angew Math 616:205–217, 2008) that the singularities for the complex-valued solutions are isolated if they are not present in the initial data. In this paper we study the singularities in more detail. In particular, we classify the possible blow-up rates and blow-up profiles. It turns out that all singularities are of type II and that the blow-up profiles are regular steady state solutions of the equation.     

Continuous Dependence on Initial Data in Fluid–Structure Motions

We prove a continuous dependence theorem for weak solutions of equations governing a fluid–structure interaction problem in two spatial dimensions. The proof is based on a priori estimates which, in particular, convey uniqueness of weak solutions. The estimates are obtained using Eulerian coordinates, without remapping the problem into a fixed domain.     

A Revised Analytic Solution to the Linear Displacement Problem Including Capillary Pressure Effects

Fokas and Yortsos (SIAM J. Appl. Math. 42(2), 318–332, 1982) and Yortsos and Fokas (SPEJ 23(1), 115–124, 1983) presented the only published exact solution of the linear waterflood problem that includes capillary effects. Despite the importance of this breakthrough, their approach has largely been disregarded due to the perceived limitations that it presented in modeling real physical situations. In this article, we show that by appropriately normalizing relevant parameters of the governing equation involved, a substantial level of the limitations is taken care of. The resultant governing equation obtained is one in terms of a parameter N _M related to the mobility ratio and another parameter N _V, representing a ratio of the viscous to capillary forces. The results of the explicit solutions obtained indicate that these two parameters are indeed the controlling parameters of the flow, and that the capillary effects are practically non-existent even when N _V = 100. These analytical results serve a very useful utility in validating numerical simulators.     

Prevalent Behavior of Strongly Order Preserving Semiflows

Classical results in the theory of monotone semiflows give sufficient conditions for the generic solution to converge toward an equilibrium or toward the set of equilibria (quasiconvergence). In this paper, we provide new formulations of these results in terms of the measure-theoretic notion of prevalence, developed in Christensen (Israel J. Math., 13, 255–260, 1972) and Hunt et al. (Bull. Am. Math. Soc., 27, 217–238, 1992). For monotone reaction–diffusion systems with Neumann boundary conditions on convex domains, we show the prevalence of the set of continuous initial conditions corresponding to solutions that converge to a spatially homogeneous equilibrium. We also extend a previous generic convergence result to allow its use on Sobolev spaces. Careful attention is given to the measurability of the various sets involved.     

Removing Collision Singularities from Action Minimizers for the N-Body Problem with Free Boundaries

For the planar and spatial N-body problems, it has been proved by Marchal and Chenciner that solutions for the minimizing problem with fixed ends are free from interior collisions. This important result has been extended by Ferrario & Terracini to Newtonian-type problems and equivariant problems. It has also been used to construct many symmetric solutions for the N-body problem. In this paper we are interested in action minimizing solutions in function spaces with free boundaries. The function spaces are imposed with boundary conditions, such that every mass point starts and ends on two transversal proper subspaces of ℝ^d, d≥2. We will prove that solutions for this minimizing problem with free boundaries are always free from collisions, including boundary collisions. This result can be used to construct certain classes of relative periodic solutions of the N-body problem.     

A Navier–Stokes Approximation of the 3D Euler Equation with the Zero Flux on the Boundary

Under assumptions on smoothness of the initial velocity and the external body force, we prove that there exists T ₀ > 0, ν ₀ > 0 and a unique continuous family of strong solutions u ^ν (0 ≤ ν < ν ₀) of the Euler or Navier–Stokes initial-boundary value problem on the time interval (0, T ₀). In addition to the condition of the zero flux, the solutions of the Navier–Stokes equation satisfy certain natural boundary conditions imposed on curl u ^ν and curl ² u ^ν .      

On the structure of the set of continuously differentiable solutions of one boundary-value problem for a system of functional differential equations of neutral type

We study the structure of the set of solutions, continuously differentiable for t ∈ R ⁺ = [0; + ∞), of one limit problem for systems of nonlinear functional differential equations of neutral type with nonlinear deviations of argument that depend on an unknown function. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 2, pp. 277–289, April–June, 2007.     

Dynamics of a chain system of rigid bodies with gravity-friction seismic dampers: fixed supports

A design model for a chain system of N elastically linked rigid bodies with a spheroidal gravity-friction damper is proposed. The Lagrange–Painlevé equations of the first kind are used to construct nonlinear dynamical models of a mechanical system undergoing translational vibrations about the equilibrium position. The conditions under which the system moves in one plane are established. The double nonstationary phase–frequency resonance of a system with N = 2 is analyze. After the numerical integration of the systems of differential equations, the phase–frequency surfaces are plotted and examined for several combinations of system parameters under two-frequency loading     

Weak in Space, Log in Time Improvement of the Lady?enskaja�CProdi�CSerrin Criteria

In this article we present a Ladyženskaja–Prodi–Serrin Criteria for regularity of solutions for the Navier–Stokes equation in three dimensions which incorporates weak L ^p norms in the space variables and log improvement in the time variable.     

Continuous Dependence of Entropy Solutions to the Euler Equations on the Adiabatic Exponent and Mach Number

We establish the L ¹-estimates for continuous dependence of entropy solutions to the full Euler equations away from the vacuum on two physical parameters: the adiabatic exponent γ → 1 that passes from the non-isentropic to isothermal Euler equations and the Mach number that passes from the compressible to incompressible Euler equations. Our analysis involves the effective approach developed in our earlier work and additional new techniques that generalize this approach to the setting of the full Euler equations.     

A procedure to determine the viscosity function from experimental data of capillary flow

The accurate calculation of the viscosity η as function of the shear rate &γdot; from capillary viscometry is still a matter of debate in the literature. In fact, this problem involves the inversion of an integral equation, which leads to multiple solutions due to the unavoidable noise present in the experimental data. The purpose of this work is to develop an efficient procedure to determine the viscosity function from experimental data of capillary flow without presenting the difficulties inherent in other methods discussed previously in the literature. The system identification procedure is used here to estimate the parameters of a viscosity model, which is appropriately selected for the fluid under study through preliminary calculations involving the apparent shear rate – shear stress data. Once the model is chosen by satisfying criteria for the fit goodness and its parameters are evaluated, a smooth and continuous function η(γdot;) is obtained in the range of experimental shear rates. The procedure proposed is also applicable to fluids in shear flow that present two Newtonian plateaus, as it is typically found in macromolecular dilute solutions. The mean value theorem of continuous functions is used to reduce significantly the computational time. Received: 15 November 1999 Accepted: 7 November 2000     

Lack of Uniqueness for Weak Solutions of the Incompressible Porous Media Equation

In this work we consider weak solutions of the incompressible two-dimensional porous media (IPM) equation. By using the approach of De Lellis–Székelyhidi, we prove non-uniqueness for solutions in L ^∞ in space and time.