On domains of regularity of the solutions of some special classes of Monge-Ampère-type equations

We present several results on estimates from above of the regularity domains for special classes of solutions to Monge-Ampère-type equations (in particular, periodic at least in one variable). Also, we present some geometrical and geophysical applications. Namely, we are concerned with the problem on dimensions of a single-valued projection on a plane of a surface with separated-from-zero negative Gaussian curvature and discuss the existence or nonexistence of a solution to the pressure-wind balance equation on a torus. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 1, pp. 237–246, 2006.     

Controllability for a Nonlinear Abstract Evolution Equation

We prove a theorem on the local controllability of a system described by a nonlinear evolution equation in Banach space when the control is a multiplier on the right-hand side. We obtain sufficient conditions on the size of the neighborhood from which we can take the function from the overdetermination condition so that the inverse problem is uniquely solvable.     

Nonexistence results for a compressible non-Newtonian fluid with magnetic effects in the whole space

We consider a generalization of the compressible barotropic Navier-Stokes equations to the case of non-Newtonian fluid in the whole space. The viscosity tensor is assumed to be coercive with an exponent q>1. We prove that if the total mass and momentum of the system are conserved, then one can find a constant qγ>1 depending on the dimension of space n and the heat ratio γ such that for q∈[qγ,n) there exists no global in time smooth solution to the Cauchy problem. We prove also an analogous result for solutions to equations of magnetohydrodynamic non-Newtonian fluid in 3D space.     

Thermo-Sensitivity of poly-N-isopropylacrylamide with Statistically Introduced D,L-Allylglycine Betainic Units

Abstract

Statistical copolymers of N-isopropylacrylamide containing 2.2 – 2.7?mol % D,L-allylglycine were synthesized. The sample with 2.2?mol % allylglycine and molar mass 320000?g/mol was studied using light scattering and turbidimetry while heated in water and water–salt solutions. Temperature dependencies of light scattering intensity and optical transmission as well as hydrodynamic size and composition of the scattering entities were obtained in a wide range of pH. The onset temperatures of phase separation were independent of pH; however, in the basic pH region the phase transition was widened as compared to the neutral and acid media. It was shown that a small content of allylglycine results in a different thermoresponsive behavior of the copolymer as compared with N-isopropylacrylamide homopolymer. The addition of a physiological amount of NaCl shifts the copolymer thermo-sensitivity toward lower temperatures.     

On the position of a vortex in a two-dimensional model of atmosphere

As a continuation of our work, Rozanova et al. (2010) [1] we study possible trajectories of a long time existing vortex in a model of the atmosphere dynamics, where the vortex can be interpreted as a tropical cyclone. The model can be obtained from the system of primitive equations governing the motion of air over the Earth’s surface after averaging over the height. We consider approximations of l-plane and β-plane used in geophysics for modeling of middle scale processes and equations on the whole sphere as well. We associate with a cyclone a special class of smooth solutions having a form of a localized steady non-singular vortex moving with a bearing field. We show that the solutions satisfy the equations of the model either exactly or with a discrepancy which is small in a neighborhood of the trajectory of the center of vortex. We show both analytically and numerically that the trajectory of a localized vortex keeps the features of trajectory of vortex with a linear profile of velocity, where the exact solution can be obtained.     

Frozen and almost frozen structures in the compressible rotating fluid

We study a possibilityof existence of localized two-dimensional structures, both smooth and non-smooth, that can move without significant change of their shape in a leading stream of compressible barotropic fluid on a rotating plane.     

A stationary solution to the air motion equations in the lower part of a typhoon

The existence of a stationary axisymmetric solution is proved to the model equations describing the air motion in the lower part of a typhoon. These are the equations (averaged over the vertical component) of motion of a barotropic viscous gas with constant coefficients of the dynamic and volume viscosities, under the conditions of a constant external friction, the Coriolis force, and the mass outflow from the central part. It is required that the components of the solution at infinity be continuous and decrease to zero at infinity. It is shown that, for the existence of a nontrivial solution of this type, it is necessary that the Coriolis parameter be nonzero and that there be a mass outflow small as compared to the coefficients of the external friction and viscosity.     

Photothermal ablation therapy for cancer based on metal nanostructures

Besides conventional surgery, radiation therapy, and chemotherapy, which all tend to have side-effects and damage normal tissues, new medical strategies, such as photothermal sensitization and photo-thermal ablation therapy (PTA) with near-IR laser light, have been explored for treating cancer. Much of the current excitement surrounding nanoscience is directly connected to the promise of new nanotechnology for cancer diagnosis and therapy. The basic principle behind PTA is that heat generated from light can be used to destroy cancer cells. Strong optical absorption and high efficiency of photothermal conversion at the cancer sites are critical to the success of PTA. Because of their unique optical properties, e.g., strong surface plasmon resonance (SPR) absorption, noble metal nanomaterials, such as gold and silver, have been found to significantly enhance photothermal conversion for PTA applications. Substantial effort has been made to develop metal nanostructures with optimal structural and photothermal properties. Ideal metal nanostructures should have strong and tunable SPR, be easy to deliver, have low toxicity, and be convenient for bioconjugation for actively targeting specific cancer cells. This review would highlight some gold nanostructures with various shapes and properties, including nanoparticles (NPs), nanorods (NRs), nanoshells, nanocages, and hollow nanospheres, which have been studied for PTA applications. Among these structures, hollow gold nanospheres (HGNs) exhibit arguably the best combined properties because of their small size (30―50 nm), spherical shape, and strong, narrow, and tunable SPR absorption.     

Hydrodynamic approach to constructing solutions of the nonlinear Schrödinger equation in the critical case

Proceeding from the hydrodynamic approach, we construct exact solutions to the nonlinear Schrödinger equation with special properties. The solutions describe collapse, in finite time, and scattering, over infinite time, of wave packets. They generalize known blow-up solutions based on the ``ground state'.