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><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.     

<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).     

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.     

Generalized elastic curves in the Lorentz flat space <Emphasis Type="Italic">L</Emphasis><Superscript>4</Superscript>

In this paper, we study the extremals of the curvature energy actions on non-null curves in the four-dimensional Lorentz-Minkowski space. We derive the motion equations and find three Killing fields along the generalized elastic curves. We also construct a cylindrical coordinate system using these Killing fields and express the generalized elastic curves by means of quadratures.     

<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">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Solvability of Higher Order Parabolic and Elliptic Systems with BMO Coefficients

We prove the solvability in Sobolev spaces for both divergence and non-divergence form higher order parabolic and elliptic systems in the whole space, on a half space, and on a bounded domain. The leading coefficients are assumed to be merely measurable only in the time variable and have small mean oscillations with respect to the spatial variables in small balls or cylinders. For the proof, we develop a set of new techniques to produce mean oscillation estimates for systems on a half space.     

<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.     

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:

<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.     

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 \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)}\).     

Regularity and finite dimensionality of attractor for plate equation on R<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

This paper addresses the regularity and finite dimensionality of the global attractor for the plate equation on the unbounded domain. The existence of the attractor in the phase space has been established in an earlier work of the author. It is shown that the attractor is actually a bounded set of the phase space and has finite fractal dimensionality.     

Three solutions to inequalities of Dirichlet problem driven by p（x）-Laplacian

A class of nonlinear elliptic problems driven by p（x）-Laplacian with a nonsmooth locally Lipschitz potential is considered.By applying the version of the nonsmooth three-critical-point theorem,the existence of three solutions to the problems is proved.     

Torus generated by <Emphasis Type="Italic">Escherichia coli</Emphasis>

The bioluminescence images of unstirred cultures show that lux reporter E. coli (0.10 mg biomass per ml of the broth medium) in 6.4–10 mm diameter circular containers induce center-fluid-rising toroidal convection of ≤1 mm/min. The bioconvective torus is stable in a Teflon vessel and is deformed by 3.2–4.4 mm wavelength azimuthal waves in polystyrene or glass vessels.     

Adhesion Energy of <Emphasis Type="Italic">C. elegans</Emphasis>

The surface adhesion between C. elegans and the agar plates on which they are commonly grown has yet to be accurately quantified. C. elegans is a scientifically important species of nematode whose simple structure allowed the first mapping of the complete nervous system in a multicellular organism. One of the current topics of research in the C. elegans community is the investigation of neuronal function in locomotion. Models of locomotion are used in these studies to aid in determination of the functions of specific neurons involved in locomotion. The adhesion force plays a critical role in developing these models. This paper presents the experimental determination of the adhesion energy of a representative sample of C. elegans. Adhesion energy was determined by a direct pull-off technique. In this approach, nematodes are anesthetized to prevent movement and secured to a small load cell before an agar plate is slowly brought into contact with the specimen and then removed. The maximum tensile force is then fit to a JKR-type adhesion model, which assumes that the nematode is a cylinder in order to determine the adhesion energy. Repeated adhesions are also investigated to determine the importance of drying on the measured adhesion force. From these experiments, the adhesion energy was found to be W =?4.94 ± 1.19 mJ/m². Limited experiments on the rol-6 cuticle mutant found a lower adhesion energy W =?2.65 ± 1.16 mJ/m² for these animals.     

The effect of 1,3:2,4-<Emphasis Type="Italic">bis</Emphasis>-<Emphasis Type="Italic">O</Emphasis>-(<Emphasis Type="Italic">p</Emphasis>-methylbenzylidene)-<Emphasis Type="SmallCaps">d</Emphasis>-sorbitol (PDTS) on uniaxial elongational viscosity of polypropylene

We investigated the dynamic viscoelasticity and elongational viscosity of polypropylene (PP) containing 0.5 wt% of 1,3:2,4-bis-O-(p-methylbenzylidene)-d-sorbitol (PDTS). The PP/PDTS system exhibited a sol–gel transition (T _gel) at 193 °C. The critical exponent n was nearly equal to 2/3, in agreement with the value predicted by a percolation theory. This critical gel is due to a three-dimensional network structure of PDTS crystals. The elongational viscosity behavior of neat PP followed the linear viscosity growth function 3η ⁺ (t), where η ⁺ (t) is the shear stress growth function in the linear viscoelastic region. The elongational viscosity of the PP/PDTS system also followed the 3η ⁺ (t) above T _gel but did not follow the 3η ⁺ (t) and exhibited strong strain-softening behavior below T _gel. This strain softening can be attributed to breakage of the network structure of PDTS with a critical stress (σ _c) of about 10⁴ Pa.     

Mechanical properties and structure of Haliotis discus hannai Ino and Hemifusus tuba conch shells： a comparative study

Haliotis discus hannai Ino （abalone shell） and Hemifusus tuba conch shell have been studied for the purpose to comparatively investigate the mechanisms by which nature designs composites. It is shown that both shells are composed of aragonite and a small amount of proteins while the conch shell shows finer microstructure but lower strength than aba- lone shell. It is also shown that the fresh shells exhibits better property than those after heat-treatments. It is therefore sup- posed that the size of inorganic substance is not a dominant factor to improve strength, while both proteins in shells and the microstructure of inorganic matter also play important roles.