<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">1</Emphasis></Superscript> Stability of Multi-Dimensional Discrete-Velocity Boltzmann Equations

We study the L ¹ stability of multi-dimensional discrete-velocity Boltzmann equations. Under suitable smallness assumption on initial data, we show that bounded mild solutions are L ¹ stable. For a stability estimate, we employ Bonys multi-dimensional analysis for total interactions over characteristic planes.     

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.     

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.     

<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     

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.     

<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.     

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.     

<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).     

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.     

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:

A new zig-zag theory and C<Superscript>0</Superscript> plate bending element for composite and sandwich plates

A higher-order zig-zag theory for laminated composite and sandwich structures is proposed. The proposed theory satisfies the interlaminar continuity conditions and free surface conditions of transverse shear stresses. Moreover, the number of unknown variables involved in present model is independent of the number of layers. Compared to the zig-zag theory available in literature, the merit of present theory is that the first derivatives of transverse displacement have been taken out from the in-plane displacement fields, so that the C⁰ interpolation functions is only required during its finite element implementation. To obtain accurately transverse shear stresses by integrating three-dimensional equilibrium equations within one element, a six-node triangular element is employed to model the present zig-zag theory. Numerical results show that the present zig-zag theory can predict more accurate in-plane displacements and stresses in comparison with other zig-zag theories. Moreover, it is convenient to obtain transverse shear stresses by integration of equilibrium equations, as the C⁰ shape functions is only used when implemented in a finite element.     

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.     

Motion of a Graph by <Emphasis Type="Italic">R</Emphasis>-Curvature

We show the existence of weak solutions to the partial differential equation which describes the motion by R-curvature in R ^d , by the continuum limit of a class of infinite particle systems. We also show that weak solutions of the partial differential equation are viscosity solutions and give the uniqueness result on both weak and viscosity solutions.     

<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.     

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.     

Measurement of velocity profiles in a rectangular microchannel with aspect ratio <Emphasis Type="Italic">α</Emphasis> = 0.35

In this work, we measured 14 horizontal velocity profiles along the vertical direction of a rectangular microchannel with aspect ratio α = h/w = 0.35 (h is the height of the channel and w is the width of the channel) using microPIV at Re = 1.8 and 3.6. The experimental velocity profiles are compared with the full 3D theoretical solution, and also with a Poiseuille parabolic profile. It is shown that the experimental velocity profiles in the horizontal and vertical planes are in agreement with the theoretical profiles, except for the planes close to the wall. The discrepancies between the experimental data and 3D theoretical results in the center vertical plane are less than 3.6%. But the deviations between experimental data and Poiseuille’s results approaches 5%. It indicates that 2D Poiseuille profile is no longer a perfect theoretical approximation since α = 0.35. The experiments also reveal that, very near the hydrophilic wall (z = 0.5–1 μm), the measured velocities are significantly larger than the theoretical velocity based on the no-slip assumption. A proper discussion on some physical effects influencing the near wall velocity measurement is given.     

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.      

Low-speed streak and internal shear layer motions in a turbulent boundary layer

Some features of the inner region of a flat plate turbulent boundary layer are investigated by a Digital Particle Image Velocimetry technique. Measurements in planes parallel to the wall are examined. The energetic spanwise modes of the streaky motions are analysed by spatial Fourier analysis at different distances from the wall. Internal shear layers are deduced by applying VISA technique at y⁺=20 and detected events are ensemble averaged. The deduced flow structure highlights the dominant spatial relationship between low-speed streak and internal shear layer motions.     

Numerical investigation of natural convection of Al<Subscript>2</Subscript>O<Subscript>3</Subscript> nanofluid in vertical annuli

This paper presents the numerical study of internal free convection of Al₂O₃ water nanofluid in vertical annuli. Vertical walls are maintained at constant temperatures and horizontal walls are adiabatic. Results are validated by experimental data. Effect of nanofluids on natural convection is investigated as a function of geometrical and physical parameters and particle fractions for aspect ratio of 1 ≤ H/L ≤ 5, Grashof number of 10³ ≤ Gr ≤ 10⁵ and concentration of 0 ≤ ϕ ≤ 0.06. More than 330 different numerical cases are investigated to develop a new correlation for the Nusselt number. This correlation is presented as a function of Nusselt number of base fluid and particle fraction which is a linear decreasing function of particle fraction. The developed correlation for annuli is also valid for the natural convection of Al₂O₃ water nanofluid in a square cavity. Furthermore, the effect of the viscosity and conductivity models on the Nusselt number of nanofluids in cylindrical cavities are discussed.