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.     

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.     

On <Emphasis Type="Italic">n</Emphasis>th-order nonlinear ordinary random differential equations

An existence result for a nonlinear nth-order ordinary random differential equation is proved under the Carathéodory condition. Two existence results for extremal random solutions are also proved in the Carathéodory case and the discontinuous case of the nonlinearity involved in the equations. Our investigations are carried out in the Banach space of continuous real-valued functions on closed bounded intervals of the real line together with the application of a random version of the Leray–Schauder principle.     

<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).     

Positive solutions to （n-1,1） m-point boundary value problems with dependence on the first order derivative

Using the extension of Krasnoselskii＇s fixed point theorem in a cone, we prove the existence of at least one positive solution to the nonlinear nth order m-point boundary value problem with dependence on the first order derivative. The associated Green＇s function for the nth order m-point boundary value problem is given, and growth conditions are imposed on the nonlinear term f which ensures the existence of at least one positive solution. A simple example is presented to illustrate applications of the obtained results.     

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.     

A simplified model system for <Emphasis Type="Italic">Toxoplasma gondii</Emphasis> spread within a heterogeneous environment

This study is dedicated to building and analyzing the spatial spread of Toxoplasma gondii through a heterogeneous predator–prey system. The spatial domain is made of N patches hosting various population species, some of them being prey, others being predators. Predators offer strong heterogeneities with respect to local sustainable resources yielding variable growth rates, from exponential decay to logistic regulation. T. gondii life cycle goes through several stages, starting in the environment where oocysts are released from cat feces, reaching prey within which asexual reproduction yields cysts and then predators wherein sexual reproduction takes place. The resulting model system is complex to handle. We consider some relevant toy models with three patches, two resident predator species and Lotka–Volterra functional responses to predation. We provide the existence and local stability of a persistent stationary state for the underlying predator–prey model systems. The reproduction number R ₀ is computed in the quasistationary case; it simplifies when slow–fast dynamics are considered. Numerical experiments illustrate our analysis.     

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.     

Minimizing Periodic Orbits with Regularizable Collisions in the <Emphasis Type="Italic">n</Emphasis>-Body Problem

In the Newtonian n-body problem, there are various subsystems with two degrees of freedom, such as the collinear three-body problem and the isosceles three-body problem. After we determine a normal form of the Lagrangians of these subsystems, we prove the existence of periodic solutions with regularizable collisions for these systems. Our result includes several examples, such as Schubart’s orbit with or without equal masses, among others.     

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.     

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

<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     

Differential Independence of <Emphasis Type="Italic">Γ</Emphasis> and <Emphasis Type="Italic">ζ</Emphasis>

In a celebrated theorem H?lder proved that the Euler Γ-function is differential transcendental, i.e. Γ(z) is not a solution of any (non-trivial) algebraic ordinary differential equation with coefficients that are complex numbers; and we extend his methods to the Riemann ζ-function. Moreover, we conjecture that Γ and ζ are differential independent, i.e. Γ(z) is not a solution of any such algebraic differential equation—even allowing coefficients that are differential polynomials in ζ(z). However, we are able to demonstrate only the partial result that Γ(z) and ζ(sin 2πz) are differential independent.     

Pullback attractor of 2D nonautonomous <Emphasis Type="Italic">g</Emphasis>-Navier-Stokes equations with linear dampness

The pullback attractors for the 2D nonautonomous g-Navier-Stokes equations with linear dampness are investigated on some unbounded domains. The existence of the pullback attractors is proved by verifying the existence of pullback D-absorbing sets with cocycle and obtaining the pullback D-asymptotic compactness. Furthermore, the estimation of the fractal dimensions for the 2D g-Navier-Stokes equations is given.     

Navier–Stokes Equations: Almost <Emphasis Type="Italic">L</Emphasis><Subscript>3,∞</Subscript>-Case

A sufficient condition of regularity for solutions to the Navier–Stokes equations is proved. It generalizes the so-called L _3,∞-case.     

<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.