Boundary Effects on Exact Solutions of the Lagrangian-Averaged Navier–Stokes-α Equations

In order to clarify the behavior of solutions of the Lagrangian-averaged Navier–Stokes- (LANS-) equations in the presence of solid walls, we identify a variety of exact solutions of the full equations and their boundary layer approximations. The solutions demonstrate that boundary conditions suggested for the LANS- equations in the literature⁽¹⁾ for a bounded domain do not apply in a semi-infinite domain. The convergence to solutions of the Navier–Stokes equations as 0 is elucidated for infinite-energy solutions in a semi-infinite domain, and non-uniqueness of these solutions is discussed. We also study the boundary layer approximation of LANS- equations, denoted the Prandtl- equations, and report solutions for turbulent jets and wakes. Our version of the Prandtl- equations includes an extra term necessary to conserve linear momentum and corrects an earlier result of Cheskidov.⁽²⁾     

A note on interconnections

The regular interconnection problem is considered in the context of Fliess models defined over an arbitrary noetherian ring. It is shown that the problem always has a solution provided that the plant is strongly controllable.     

Photo-Isomerization Kinetics of Azobenzene Containing Surfactant Conjugated with Polyelectrolyte

Ionic complexation of azobenzene-containing surfactants with any type of oppositely charged soft objects allows for making them photo-responsive in terms of their size, shape and surface energy. Investigation of the photo-isomerization kinetic and isomer composition at a photo-stationary state of the photo-sensitive surfactant conjugated with charged objects is a necessary prerequisite for understanding the structural response of photo-sensitive complexes. Here, we report on photo-isomerization kinetics of a photo-sensitive surfactant in the presence of poly(acrylic acid, sodium salt). We show that the photo-isomerization of the azobenzene-containing cationic surfactant is slower in a polymer complex compared to being purely dissolved in aqueous solution. In a photo-stationary state, the ratio between the trans and cis isomers is shifted to a higher trans-isomer concentration for all irradiation wavelengths. This is explained by the formation of surfactant aggregates near the polyelectrolyte chains at concentrations much lower than the bulk critical micelle concentration and inhibition of the photo-isomerization kinetics due to steric hindrance within the densely packed aggregates.     

Extrapolation results in grand Lebesgue spaces defined on product sets

Extrapolation results in weighted grand Lebesgue spaces defined with respect to product measure \(\mu \times \nu \) on \(X\times Y\), where \((X, d, \mu )\) and \((Y, \rho , \nu )\) are spaces of homogeneous type, are obtained. As applications of the derived results we prove new one-weight estimates for multiple integral operators such as strong maximal, Calderón–Zygmund and fractional integral operators with product kernels in these spaces.     

On Noisy Extensions of Nonholonomic Constraints

We propose several stochastic extensions of nonholonomic constraints for mechanical systems and study the effects on the dynamics and on the conservation laws. Our approach relies on a stochastic extension of the Lagrange–d’Alembert framework. The mechanical system we focus on is the example of a Routh sphere, i.e., a rolling unbalanced ball on the plane. We interpret the noise in the constraint as either a stochastic motion of the plane, random slip or roughness of the surface. Without the noise, this system possesses three integrals of motion: energy, Jellet and Routh. Depending on the nature of noise in the constraint, we show that either energy, or Jellet, or both integrals can be conserved, with probability 1. We also present some exact solutions for particular types of motion in terms of stochastic integrals. Next, for an arbitrary nonholonomic system, we consider two different ways of including stochasticity in the constraints. We show that when the noise preserves the linearity of the constraints, then energy is preserved. For other types of noise in the constraint, e.g., in the case of an affine noise, the energy is not conserved. We study in detail a class of Lagrangian mechanical systems on semidirect products of Lie groups, with “rolling ball type” constraints. We conclude with numerical simulations illustrating our theories, and some pedagogical examples of noise in constraints for other nonholonomic systems popular in the literature, such as the nonholonomic particle, the rolling disk and the Chaplygin sleigh.     

Rubio de Francía's weighted extrapolation in mixed-norm spaces and applications

Weighted extrapolation for pairs of functions in mixed-norm Banach function spaces defined on the product of quasi-metric measure spaces $(X,d,\mu )$ and $(Y,\rho ,\nu )$ are derived. As special cases, we have appropriate results for mixed-norm Lebesgue, Lorentz, and Orlicz spaces. Some of the derived results are applied to get weighted extrapolation in mixed-norm grand Lebesgue spaces.