<Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> Stability of the Boltzmann Equation for the Hard-Sphere Model

We study the L¹ stability of classical solutions to the Boltzmann equation for a hard-sphere model, when initial datum is a small perturbation of a vacuum, and tends to zero exponentially fast at infinity in the phase space. For this, we introduce nonlinear functionals measuring potential interactions between particles with different velocities and L¹ distance between classical solutions. We use pointwise estimates for a solution and the gain term of a collision operator to control the time-evolution of nonlinear functionals.Dedicated to Marshall Slemrod on the occasion of his 60th birthday     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> Formulation of Multidimensional Scalar Conservation Laws

We show that Kruzhkov’s theory of entropy solutions to multidimensional scalar conservation laws (Kruzhkov in Mat Sb (N.S.), 81(123), 228–255, 1970) can be entirely recast in L ² and fits into the general theory of maximal monotone operators in Hilbert spaces. Our approach is based on a combination of level-set, kinetic and transport-collapse approximations, in the spirit of previous works by Brenier (in C R Acad Sci Paris Ser I Math, 292, 563–566, 1981; in J Diff Equ, 50, 375–390, 1983; in SIAM J Numer Anal, 21, 1013–1037; in Methods Appl Anal, 11, 515–532, 2004), Giga and Miyakawa (in Duke Math J, 50, 505–515, 1983), and Tsai et al. (in Math Comp, 72, 159–181, 2003).     

Convergence Rates in <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> for Elliptic Homogenization Problems

We study rates of convergence of solutions in L ² and H ^1/2 for a family of elliptic systems {L_e}{\{\mathcal{L}_\varepsilon\}} with rapidly oscillating coefficients in Lipschitz domains with Dirichlet or Neumann boundary conditions. As a consequence, we obtain convergence rates for Dirichlet, Neumann, and Steklov eigenvalues of {L_e}{\{\mathcal{L}_\varepsilon\}} . Most of our results, which rely on the recently established uniform estimates for the L ² Dirichlet and Neumann problems in Kenig and Shen (Math Ann 350:867–917, 2011; Commun Pure Appl Math 64:1–44, 2011) are new even for smooth domains.     

A Remark on <Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript> Solutions to the 2-D Navier–Stokes Equations

A single-exponential growth estimate of the solutions to the 2-dimensional Navier–Stokes equations in the whole space for nondecaying initial velocity is established. The crucial idea is to decompose velocity into high and low frequency parts. Moreover, if the linear term and the vorticity decay exponentially, the velocity is bounded uniformly in time.      

<Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> Regularity for Infinity Harmonic Functions in Two Dimensions

A continuous function is said to be infinity harmonic if it satisfies the PDEin the viscosity sense. In this paper we prove that infinity harmonic functions are continuously differentiable when n=2.     

On the <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">s</Emphasis></Superscript> Theory of Hydrostatic Euler Equations

In this paper we study the two-dimensional hydrostatic Euler equations in a periodic channel. We prove the local existence and uniqueness of H ^s solutions under the local Rayleigh condition. This extends Brenier’s (Nonlinearity 12(3):495–512, 1999) existence result by removing an artificial condition and proving uniqueness. In addition, we prove weak–strong uniqueness, mathematical justification of the formal derivation and stability of the hydrostatic Euler equations. These results are based on weighted H ^s a priori estimates, which come from a new type of nonlinear cancellation between velocity and vorticity.     

A Theory of <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux

We propose a general framework for the study of L ¹ contractive semigroups of solutions to conservation laws with discontinuous flux:

Approximation of Flat <Emphasis Type="Italic">W</Emphasis><Superscript>2,2</Superscript> Isometric Immersions by Smooth Ones

Let \({S\subset\mathbb{R}^2}\) be a bounded Lipschitz domain and denote by \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)}\) the set of mappings \({u\in W^{2,2}(S;\mathbb{R}^3)}\) which satisfy \({(\nabla u)^T(\nabla u) = Id}\) almost everywhere. Under an additional regularity condition on the boundary \({\partial S}\) (which is satisfied if \({\partial S}\) is piecewise continuously differentiable), we prove that the strong W ^2,2 closure of \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)\cap C^{\infty}(\overline{S};\mathbb{R}^3)}\) agrees with \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)}\).     

Solutions for Quasilinear Nonsmooth Evolution Systems in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>

We prove that nonsmooth quasilinear parabolic systems admit a local solution in L ^p strongly differentiable with respect to time over a bounded three-dimensional polyhedral space domain. The proof rests essentially on new elliptic regularity results for polyhedral Laplace interface problems for anisotropic materials. These results are based on sharp pointwise estimates for Greens function, which are also of independent interest. To treat the nonlinear problem, we then apply a classical theorem of Sobolevskii for abstract parabolic equations and recently obtained resolvent estimates for elliptic operators and interpolation results. As applications we have in mind primarily reaction-diffusion systems. The treatment of such equations in an L ^p context seems to be new and allows (by Gauss theorem) the proper definition of the normal component of currents across the boundary.     

A Numerical Study of the Recirculation Zones During Spin-Up and Spin-Down for Confined Rotating Flows<Superscript>*</Superscript>

This paper reports computational simulations of the Navier–Stokes equations for confined axisymmetric rotating flows induced by rotating the endwalls instantaneously at a different rate to the sidewall. The transient behavior of the recirculation zones in the meridional plane is investigated during the temporal evolution. The changes in the topological structure of the meridional-plane streamline pattern are significant and the temporal evolution from one pattern to another reveals similarities between spin-up and spin-down at the early stages but subsequently differs. As the onset bubble for the first recirculating period always sets out from a certain axial station, a recirculation factor, R_f, is suggested to predict the onset time and location for the first period of recirculation. Accordingly, a stagnation point is observed numerically from a central axial station for low Reynolds numbers around 70–80. The effect of changing the rotation of the sidewall is also discussed, but no substantial influence is observed on the characteristics of the recirculation zones if there is no appearance of the Taylor–Görtler vortices in the sidewall boundary layer.     

An experimental—Numerical evaluation of the<Emphasis Type="Italic">T</Emphasis><Stack><Subscript>ε</Subscript><Superscript>*</Superscript></Stack> integral for a three-dimensional crack front

TheT _ε ^* integral was calculated on the surface of single edge notched, three-point bend (SE(B)) specimens using experimentally obtained displacements. Comparison was made withT _ε ^* calculated with the measured surface displacements andT _ε ^* calculated at several points through the thickness of a finite element (FE) model of the SE(B) specimen. Good comparison was found between the surfaceT _ε ^* calculated from displacements extracted from the FE model and the surfaceT _ε ^* calculated from experimentally obtained displacements. The computedT _ε ^* integral was also observed to decrease as the crack front was traversed from the surface to the mid-plane of the specimen. Mid-planeT _ε ^* values tend to be approximately 10% of the surface values.     

The Stationary Navier-Stokes System in Nonsmooth Manifolds: The Poisson Problem in Lipschitz and <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> Domains

We consider the linearized version of the stationary Navier-Stokes equations on a subdomain of a smooth, compact Riemannian manifold M. The emphasis is on regularity: the boundary of is assumed to be only C¹ and even Lipschitz, and the data are selected from appropriate Sobolev-Besov scales. Our approach relies on the method of boundary integral equations, suitably adapted to the variable-coefficient setting we are considering here. Applications to the stationary, nonlinear Navier-Stokes equations in this context are also discussed.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> Stability Estimates for Shock Solutions of Scalar Conservation Laws Using the Relative Entropy Method

We consider scalar nonviscous conservation laws with strictly convex flux in one spatial dimension, and we investigate the behavior of bounded L ² perturbations of shock wave solutions to the Riemann problem using the relative entropy method. We show that up to a time-dependent translation of the shock, the L ² norm of a perturbed solution relative to the shock wave is bounded above by the L ² norm of the initial perturbation.     

Transport of Lead (Pb<Superscript>2+</Superscript>) Ions Through Silty-Clayey Soils Under Acidic Conditions

This study aimed to identify effects of pH on the transport of Pb²⁺ ions through a saturated silty-clayey soil layer by using advection–dispersion equation (ADE). The predictive accuracy of the solution of ADE depends on the proper determination of the retardation by adsorption and, therefore, the adsorption mechanism of lead onto silty-clayey soil was investigated first by performing batch equilibrium experiments. These results showed that the sorption mechanism of lead onto silty-clayey soil depended on pH and could be best described by the Langmuir isotherm. Based on the results of the sequential experiments, it was also concluded that the pH dependent charges in silty-clayey soil were mainly associated with the surfaces of carbonates and the specific adsorption of lead ions. The numerical solutions of the combined form of ADE with the Langmuir isotherm indicated that the migration profiles of lead in silty-clayey soil were a strong function of the parameters of the Langmuir isotherm rather than the infiltration velocity.     

On solutions continuously differentiable on ℝ<Superscript>+</Superscript> for systems of linear functional differential equations with linearly transformed argument

We obtain conditions for the existence of solutions continuous on ℝ⁺ for a system of linear functional differential equations with linearly transformed argument. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 3, pp. 322–327, July–September, 2007.     

Electrorheological response of SnO<Subscript>2</Subscript> and Y<Subscript>2</Subscript>O<Subscript>3</Subscript> nanoparticles in silicon oil

The electrorheological properties (ER) of some fluids containing particles change extensively under the external electrical field. This phenomenon is applicable in many industries and equipments, such as clutches and motor driven rotor, which would transfer the spin to a drive shaft through a thin layer of electrorheological fluid. In this investigation, the effects of external electrical field on ER properties of non-Newtonian fluids (silicon oil) with the addition of SnO₂ and Y₂O₃ nanoparticles were studied. The ER properties were measured for a wide range of SnO₂ and Y₂O₃ nanoparticle concentrations and DC electrical voltages using concentric cylinder rotary rheometer. Based on the results, ER properties of nanofluids, e.g., apparent viscosity, shear stress, and yield stress, were enhanced by applying electrical field and increasing SnO₂ and Y₂O₃ concentrations.

Assessment of <Emphasis Type="Italic">Σ−Y</Emphasis><Subscript>liq</Subscript> model predictions for air-assisted atomisation

In a series of recent works, R. Borghi and co-workers proposed a new Eulerian model of two-phase turbulent flows which introduced a transport equation for the average area of the liquid–gas interface. This work is devoted to the assessment of this model’s ability to predict the effects of liquid properties and injection regimes on the atomisation quality. Air-assisted atomisation, for which extensive experimental data are available, is chosen as a test case. It is shown that the model predictions are in good agreement with the observed trends for a wide range of variations of the liquid properties, such as density and surface tension, as well as the injection regimes, defined by the liquid and gas jet exit velocities.      

Nonequilibrium molecular radiation behind the front of a strong shock wave in the CO<Subscript>2</Subscript>-N<Subscript>2</Subscript>-O<Subscript>2</Subscript> mixture

Models of population of some radiating electron-vibrational states of CO, CN, and C₂ molecules are developed. The characteristics of radiation in a chemically nonequilibrium flow behind the front of a strong shock wave in a mixture of gases constituting the Martian atmosphere are calculated. The numerical data are compared with experimental results.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 13–22, March–April, 2005     

Numerical study of conjugate heat transfer in rectangular microchannel heat sink with Al<Subscript>2</Subscript>O<Subscript>3</Subscript>/H<Subscript>2</Subscript>O nanofluid

In the present paper, conjugate heat transfer approach has been used to numerically study laminar forced convective heat transfer characteristics of Al₂O₃/H₂O nanofluid flowing in a silicon microchannel heat sink (MCHS) of rectangular cross-section using thermal dispersion model. Results are presented in terms of thermal resistance that characterizes MCHS performance. It is observed that use of nanofluid improves MCHS performance by reducing fin (conductive) thermal resistance.