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>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     

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.     

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.     

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">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.     

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

Levitan Almost Periodic and Almost Automorphic Solutions of <Emphasis Type="Italic">V</Emphasis>-monotone Differential Equations

In the present article we consider a special class of equations

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.     

<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).     

Bifurcation Structure of a Class of S<Subscript><Emphasis Type="Italic">N</Emphasis></Subscript>-invariant Constrained Optimization Problems

In this paper we investigate the bifurcations of solutions to a class of constrained optimization problems. This study was motivated by annealing problems which have been used to successfully cluster data in many different applications. Solving these problems numerically is challenging due to the size of the space being optimized over, which depends on the size and the complexity of the data being analyzed. The type of constraints and the form of the cost functions make them invariant to the action of the symmetric group on N symbols, S_N, and we capitalize on this symmetry to describe the bifurcation structure. We ascertain the existence of bifurcating branches, address their stability, and compare the stability to optimality in the constrained problem. These theoretical results are used to explain numerical results obtained from an annealing problem used to cluster data.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.Albert E. Parker-This research was partially supported by IGERT Grant NSF-DGE 9972824Tomá Gedeon-This research was partially supported by NSF EIA-BITS Grant 426411     

Globally Minimizing Parabolic Motions in the Newtonian <Emphasis Type="Italic">N</Emphasis>-body Problem

We consider the N-body problem in with the Newtonian potential 1/r. We prove that for every initial configuration x _i and for every minimizing normalized central configuration x ₀, there exists a collision-free parabolic solution starting from x _i and asymptotic to x ₀. This solution is a minimizer in every time interval. The proof exploits the variational structure of the problem, and it consists in finding a convergent subsequence in a family of minimizing trajectories. The hardest part is to show that this solution is parabolic and asymptotic to x ₀.     

Some Results on the <Emphasis Type="BoldItalic">m</Emphasis>-Laplace Equations with Two Growth Terms

We prove the existence of positive radial solutions of the following equation:

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.     

Rheological and micro structural studies of methylcellulose solutions in <Emphasis Type="Italic">N</Emphasis>,<Emphasis Type="Italic">N</Emphasis>-dimethylformamide for preparation of extruded beads: controlled urea release

The rheological properties of methylcellulose in N,N-dimethylformamide (MC-DMF) gel are investigated to prepare extruded beads. The temperature scan under dynamic compression for various concentrations of MC in DMF is performed to investigate the rapture of MC gel at a constant frequency of 1 Hz. The morphological studies are performed using a scanning electron microscope (SEM) to analyze the size and shape of dried bead. However, during swelling studies, the MC beads have the capability to swell and retain a large amount of water >?9150% by weight and 9192.6% by volume. The mechanism of swelling is thermodynamically verified, where the enthalpy of hydration of initial layer of MC bead is negative. The newly defined electrostatic penta-pole model explains the anomalous behavior of urea release, where urea is assumed to be electrostatically bounded with the MC molecules.

Open image in new window

Grapichal abstract ?

     

Stationary Solutions and Connecting Orbits for <Emphasis Type="Italic">p</Emphasis>-Laplace Equation

We deal with one dimensional p-Laplace equation of the form

$$\begin{aligned} u_t = (|u_x|^{p-2} u_x )_x + f(x,u), \ x\in (0,l), \ t>0, \end{aligned}$$

under Dirichlet boundary condition, where \(p>2\) and \(f:[0,l]\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous function with \(f(x,0)=0\). We will prove that if there is at least one eigenvalue of the p-Laplace operator between \(\lim _{u\rightarrow 0} f(x,u)/|u|^{p-2}u\) and \(\lim _{|u|\rightarrow +\infty } f(x,u)/|u|^{p-2}u\), then there exists a nontrivial stationary solution. Moreover we show the existence of a connecting orbit between stationary solutions. The results are based on Conley index and detect stationary states even when those based on fixed point theory do not apply. In order to compute the Conley index for nonlinear semiflows deformation along p is used.

     

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.