Droplet spreading: Intermediate scaling law by PDE methods

We consider the spreading of a thin droplet of viscous liquid on a plane surface driven by capillarity. The standard lubrication approximation leads to an evolution equation for the film height h that is ill‐posed when the spreading is limited by the no‐slip boundary condition at the liquid‐solid interface due to a singularity at the moving contact line. The most common relaxation of the no‐slip boundary condition removes this singularity but introduces a new physical length scale: the slippage length b. It is believed that this microscopic‐length scale only enters logarithmically in the effective (that is, macroscopic) spreading behavior. In this paper, we rigorously show that the naively expected spreading rate is indeed only altered by a logarithmic term involving b. More precisely, we prove a scaling law for the diameter of the apparent (that is, macroscopic) support of the droplet in time. This is an intermediate scaling law: It takes an initial layer to “forget” the initial droplet shape, whereas after a long time, the droplet is so thin that its spreading is governed by the physics on the scale b. Our proof works by deriving suitable estimates for physically relevant integral quantities: the free energy, the length of the apparent support, and their respective rates of change. As opposed to matched asymptotic methods, this PDE approach closely mimics a simple heuristic argument based on the gradient flow structure. © 2002 John Wiley & Sons, Inc.     

The diffusive logistic model with a free boundary in heterogeneous environment

In this paper, we study the population dynamics of an invasive species in heterogeneous environment which is modeled by a diffusive logistic equation with free boundary condition. To understand the effect of the dispersal rate D and the parameter μ (the ratio of the expansion speed of the free boundary and the population gradient at the expanding front) on the dynamics of this model, we divide the heterogeneous environment into two cases: strong heterogeneous environment and weak heterogeneous environment. By choosing D and μ as variable parameters, we derive sufficient conditions for species spreading (resp. vanishing) in the strong heterogeneous environment; while in the weak heterogeneous environment, we obtain sharp criteria for the spreading and vanishing. Moreover, when spreading happens, we give an estimate for the asymptotic spreading speed of the free boundary. These theoretical results may have important implications for prediction and prevention of biological invasions.     

Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system

In this paper, we investigate the spatial dynamics of a nonlocal and time-delayed reaction-diffusion system, which is motivated by an age-structured population model with distributed maturation delay. The spreading speed c^*, the existence of traveling waves with the wave speed c?c^*, and the nonexistence of traveling waves with c<c^* are obtained. It turns out that the spreading speed coincides with the minimal wave speed for monotone traveling waves.     

Spreading speed and traveling waves for a nonmonotone reaction–diffusion model with distributed delay and nonlocal effect

In this paper, spreading speed and traveling waves for reaction–diffusion model with distributed delay and nonlocal effect without monotonicity are investigated. It is shown that there exists the spreading speed c^∗ which coincides with the minimal wave speed, and its limiting integral equation has an unique traveling wave with speed c > c^∗, and no traveling wave with c < c^∗. Moreover, the dependence of the spreading speed on the delay and the nonlocal effect is considered.     

On spaces admitting no ℓ p or c 0 spreading model

It is shown that for each separable Banach space X not admitting ? ₁ as a spreading model there is a space Y having X as a quotient and not admitting any ? _p for 1 ≤ p < ∞ or c ₀ as a spreading model. We also include the solution to a question of Johnson and Rosenthal (Studia Math 43:77–92, 1972) on the existence of a separable space not admitting as a quotient any space with separable dual.     

Cut-disks for level spheres in link and tangle complements

Wu has shown that if a link or a knot L in S³ in thin position has thin spheres, then the thin sphere of lowest width is an essential meridional surface in the link complement. In this paper we show that if we further assume that L⊂S³ is prime, then the thin sphere of lowest width also does not have any vertical cut-disks. We also prove the result for a specific kind of tangles in S²×[−1,1].     

Compressing thin spheres in the complement of a link

Let L be a link in S³ that is in thin position but not in bridge position and let P be a thin level sphere with compressing disk D. We introduce the idea of alternating level spheres for D and show that all such spheres are thin and their widths are monotone decreasing. This allows us to generalize a result of Wu by giving a bound on the number of disjoint irreducible compressing disks P can have in terms of the width of P, including identifying thin spheres with unique compressing disks. We also give conditions under which P must be incompressible on some side or be weakly incompressible. In particular we show that the thin level sphere of second lowest width is weakly incompressible. If P is strongly compressible we describe how a pair of compressing disks must lie relative to the link.     

Spreading speeds and traveling waves for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity

This paper is concerned with the traveling wave solutions and the spreading speeds for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity, which is motivated by an age-structured population model with time delay. We first prove the existence of traveling wave solution with critical wave speed c = c*. By introducing two auxiliary monotone birth functions and using a fluctuation method, we further show that the number c = c* is also the spreading speed of the corresponding initial value problem with compact support. Then, the nonexistence of traveling wave solutions for c < c* is established. Finally, by means of the (technical) weighted energy method, we prove that the traveling wave with large speed is exponentially stable, when the initial perturbation around the wave is relatively small in a weighted norm.     

Spreading speed and traveling waves for a multi-type SIS epidemic model

The theory of asymptotic speeds of spread and monotone traveling waves for monotone semiflows is applied to a multi-type SIS epidemic model to obtain the spreading speed c^∗, and the nonexistence of traveling waves with wave speed c<c^∗. Then the method of upper and lower solutions is used to establish the existence of monotone traveling waves connecting the disease-free and endemic equilibria for c?c^∗. This shows that the spreading speed coincides with the minimum wave speed for monotone traveling waves. We also give an affirmative answer to an open problem presented by Rass and Radcliffe [L. Rass, J. Radcliffe, Spatial Deterministic Epidemics, Math. Surveys Monogr. 102, Amer. Math. Soc., Providence, RI, 2003].     

The time-periodic diffusive competition models with a free boundary and sign-changing growth rates

To understand the spreading of invasive and native species, in this paper we consider the diffusive competition models with a free boundary in the heterogeneous time-periodic environments, in which the variable intrinsic growth rates of these two species change signs and may be very negative in some large regions. We study the spreading–vanishing dichotomy, long-time dynamical behavior of solution, sharp criteria for spreading and vanishing, and estimates of the asymptotic spreading speed of the free boundary. Moreover, we establish the existence of positive solutions to a T-periodic boundary value problem of the diffusive competition system with sign-changing growth rates in the half line.     

Simulation of turbulent dispersion using a simple random model of the flow field

A method is devised to simulate the movement and spreading of a patch of contaminant in two-dimensional turbulent flow. The turbulent motion is exponentially divided into components of differing wave number, adjacent components being made to have correlation times differing by a factor of two. The turbulent motion is then reconstructed by replacing each component with a sinusoidal advection field having a randomly directed wave number. Contaminant particles are advected by each of the reconstructed components, the smallest scale components being applied first. A computer simulation was performed, using a Kolmogorov k^{-$\text{5}\text{3}$} turbulent energy spectrum. Batchelor's σ ∝ t^{$\text{3}\text{2}$} law for the spreading of a contaminant patch was reproduced, approximately, as was Richardson's non-Gaussian asymptotic form of the distance-neighbour function.     

Symmetries and similarity solutions for the axisymmetric spreading under gravity of a thin power-law liquid drop on a horizontal plane

We perform a complete analysis of all the Lie point symmetries admitted by the equation describing the axisymmetric spreading under gravity of a thin power-law liquid drop on a horizontal plane. We then investigate the existence of group-invariant solutions for particular values of the power-law parameter β.     

Completely simple codes

A code X is said to be completely simple if the kernel of the syntactic monoid of X ^? is a completely simple semigroup. This class of codes is a common generalization of two important classes of codes, that is, thin codes and group codes, and has remarkable algebraic and combinatorial properties. In this paper, we give systematic characterizations of completely simple codes and, in particular, show that many fundamental properties of thin codes are preserved in this generalization.     

Information spreading in Delay Tolerant Networks based on nodes’ behaviors

Information spreading in DTNs (Delay Tolerant Networks) adopts a store–carry–forward method, and nodes receive the message from others directly. However, it is hard to judge whether the information is safe in this communication mode. In this case, a node may observe other nodes’ behaviors. At present, there is no theoretical model to describe the varying rule of the nodes’ trusting level. In addition, due to the uncertainty of the connectivity in DTN, a node is hard to get the global state of the network. Therefore, a rational model about the node’s trusting level should be a function of the node’s own observing result. For example, if a node finds k nodes carrying a message, it may trust the information with probability p(k). This paper does not explore the real distribution of p(k), but instead presents a unifying theoretical framework to evaluate the performance of the information spreading in above case. This framework is an extension of the traditional SI (susceptible-infected) model, and is useful when p(k) conforms to any distribution. Simulations based on both synthetic and real motion traces show the accuracy of the framework. Finally, we explore the impact of the nodes’ behaviors based on certain special distributions through numerical results.     

Existence and uniqueness of traveling waves for non-monotone integral equations with applications

A class of integral equations without monotonicity is investigated. It is shown that there is a spreading speed c^∗>0 for such an integral equation, and that its limiting integral equation admits a unique traveling wave (up to translation) with speed c?c^∗ and no traveling wave with c<c^∗. These results are also applied to some nonlocal reaction-diffusion population models.     

Thin-type dense sets and related properties

We build on Gruenhage, Natkaniec, and Piotrowski?s study of thin, very thin, and slim dense sets in products, and the related notions of (NC) and (GC) which they introduced. We find examples of separable spaces X such that X² has a thin or slim dense set but no countable one. We characterize ordered spaces that satisfy (GC) and (NC), and we give an example of a separable space which satisfies (GC) but not witnessed by a collection of finite sets. We show that the question of when the topological sum of two countable strongly irresolvable spaces satisfies (NC) is related to the Rudin-Keisler order on βω. We also introduce and study the concepts of <κ-thin and superslim dense sets.     

Navier-Stokes equations in 3D thin domains with Navier friction boundary condition

In this article we study the 3D Navier-Stokes equations with Navier friction boundary condition in thin domains. We prove the global existence of strong solutions to the 3D Navier-Stokes equations when the initial data and external forces are in large sets as the thickness of the domain is small. We generalize the techniques developed to study the 3D Navier-Stokes equations in thin domains, see [G. Raugel, G. Sell, Navier-Stokes equations on thin 3D domains I: Global attractors and global regularity of solutions, J. Amer. Math. Soc. 6 (1993) 503-568; G. Raugel, G. Sell, Navier-Stokes equations on thin 3D domains II: Global regularity of spatially periodic conditions, in: Nonlinear Partial Differential Equations and Their Application, College de France Seminar, vol. XI, Longman, Harlow, 1994, pp. 205-247; R. Temam, M. Ziane, Navier-Stokes equations in three-dimensional thin domains with various boundary conditions, Adv. Differential Equations 1 (1996) 499-546; R. Temam, M. Ziane, Navier-Stokes equations in thin spherical shells, in: Optimization Methods in Partial Differential Equations, in: Contemp. Math., vol. 209, Amer. Math. Soc., Providence, RI, 1996, pp. 281-314], to the Navier friction boundary condition by introducing a new average operator Mε in the thin direction according to the spectral decomposition of the Stokes operator Aε. Our analysis hinges on the refined investigation of the eigenvalue problem corresponding to the Stokes operator Aε with Navier friction boundary condition.     

Evaluation of mechanical parameters of an elastic thin film system by modeling and numerical simulation of telephone cord buckles

The morphology of telephone cord buckles of elastic thin films can be used to evaluate the initial residual stress and interface toughness of the film-substrate system. The maximum out-of-plane displacement δ, the wavelength λ and amplitude A of the wave buckles can be measured in physical experiments. Through δ, λ, and A, the buckle morphology is obtained by an annular sector model established using the von Karman plate equations in polar coordinates for the elastic thin film. The mode-mix fracture criterion is applied to determine the shape and scale parameters. A numerical algorithm combining the Newmark-β scheme and the Chebyshev collocation method is adopted to numerically solve the problem in a quasi-dynamic process. Numerical experiments show that the numerical results agree well with physical experiments.     

Inverse heat conduction problem in a thin circular plate and its thermal deflection

An inverse problem of transient heat conduction in a thin finite circular plate with the given temperature distribution on the interior surface of a thin circular plate being a function of both time and position has been solved with the help of integral transform technique and also determine the thermal deflection on the outer curved surface of a thin circular plate defined as 0 ? r ? a, 0 ? z ? h. The results, obtained in the series form in terms of Bessel’s functions, are illustrated numerically.     

Studying designs via multisets

A new method to study families of finite sets, in particular t-designs, by studying families of multisets (also called lists) and their relationships with families of sets, is developed. Notion of the tag for a subset defined earlier by one of the authors is extended to a submultiset. A new concept t-(v, k, λ) list design is defined and studied. Basic existence theory for designs is extended to a new set up of list designs. In particular tags are used to prove that signed t-list designs exist whenever necessary conditions are satisfied. The concepts of homomorphisms and block spreading are extended to this new set up.