Heat Kernel Estimates and Homogenization for Asymptotically Lower Dimensional Processes on Some Nested Fractals

We consider the class of diffusions on fractals first constructed in [12] on the Sierpinski and abc gaskets. We give an alternative construction of the diffusion process using Dirichlet forms and extend the class of fractals considered to some nested fractals. We use the Dirichlet form to deduce Nash inequalities which give upper bounds on the short and long time behaviour of the transition density of the diffusion process. For short times, even though the diffusion lives mainly on a lower dimensional subset of the fractal, the heat flows slowly. For the long time scales the diffusion has a homogenization property in that rescalings converge to the Brownian motion on the fractal.     

Random point fields with Markovian refinements and the geometry of fractally disordered media

We give a general construction of the probability measure for describing stochastic fractals that model fractally disordered media. For these stochastic fractals, we introduce the notion of a metrically homogeneous fractal Hansdorff-Karathéodory measure of a nonrandom type. We select a classF[q] of random point fields with Markovian refinements for which we explicitly construct the probability distribution. We prove that under rather weak conditions, the fractal dimension D for random fields of this class is a self-averaging quantity and a fractal measure of a nonrandom type (the Hausdorff D-measure) can be defined on these fractals with probability 1. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 490–505, September, 2000.     

Elementary density bounds for self-similar sets and application

Falconer[1] used the relationship between upper convex density and upper spherical density to obtain elementary density bounds for s-sets at H S-almost all points of the sets. In this paper, following Falconer[1], we first provide a basic method to estimate the lower bounds of these two classes of set densities for the self-similar s-sets satisfying the open set condition （OSC）, and then obtain elementary density bounds for such fractals at all of their points. In addition, we apply the main results to the famous classical fractals and get some new density bounds.     

Diffusion processes on fractal fields: heat kernel estimates and large deviations

A fractal field is a collection of fractals with, in general, different Hausdorff dimensions, embedded in ². We will construct diffusion processes on such fields which behave as Brownian motion in ² outside the fractals and as the appropriate fractal diffusion within each fractal component of the field. We will discuss the properties of the diffusion process in the case where the fractal components tile ². By working in a suitable shortest path metric we will establish heat kernel bounds and large deviation estimates which determine the trajectories followed by the diffusion over short times.     

A spatial-fractional thermal transport model for nanofluid in porous media

In this paper, a spatial fractional-order thermal transport equation with the Caputo derivative is proposed to describe convective heat transfer of nanofluids within disordered porous media in boundary layer flow. This equation arises naturally when the effect of anomalous migration of nanoparticles on heat transfer is considered. The numerical results show that local Nusselt numbers of four different kinds of nanofluids are all inversely proportional to the fractional derivative exponent β. Based on this finding, it is concluded that the anomalous diffusion of nanoparticles improves the convective heat transfer of nanofluids and the space fractional thermal transport equation may serve as a candidate model for studying nanofluids. Additionally, the effects of other involved physical parameters on temperature distribution and Nusselt number are presented and analyzed.     

Schwarz Problem for Multiply Connected Domains and its Application to Diffusion Around Fractals

Stationary diffusion perpendicular to unidirectional circular cylinders arranged in two-dimensional fractals set is considered. Complex potentials and the method of functional equations are used to determine the concentration field. This result is applied to the calculation of the effective diffusion tensor of fractals randomly diluted on the plane. In particular, we discuss a special fractal derived by a group of Möbius transformations and deduce a generalization of the Clausius-Mossotti approximation.     

A Diffusion Equation on Fractals in Random Media

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Heat kernel estimates for FIN processes associated with resistance forms

Quenched and annealed heat kernel estimates are established for Fontes–Isopi–Newman (FIN) processes on spaces equipped with a resistance form. These results are new even in the case of the one-dimensional FIN diffusion, and also apply to fractals such as the Sierpinski gasket and carpet.     

Integral inequalities for the fundamental solutions of diffusions on manifolds with divergence-free drift

The main purpose of this article is to present uniform integral inequalities for the fundamental solutions of diffusions on compact manifolds with divergence free drift vector fields. The method relies on the fact that the heat flow depends on the isoperimetric function. The isoperimetric function is used to construct a suitable comparison manifold. The heat kernel of this comparison manifold gives uniform bounds for the fundamental solutions of the original diffusion problem. The results presented here can be used to solve some open problem of Bhattacharya and Götze about diffusions with periodic, divergence free drift vector fields (see [Bhagöt1] and [Bhagöt2]). Mathematics Subject Classification (2000): 58J35, 47D07This research was supported by the German Exchange Service DAAD     

Dimension of hyperfractals

It is shown that multivalued fractals have the same address structure as the associated hyperfractals. Hyperfractals may be used to model self-similar diffusion limited aggregations, structure of urban settlements, and clusters of nanoparticles. We establish that the Hausdorff dimensions of a particular class of hyperfractals can be calculated by means of the Moran–Hutchinson formula.     

The distributional dimension of fractals

In the book [1] H.Triebel introduces the distributional dimension of fractals in an analytical form and proves that： for Г as a non-empty set in R^n with Lebesgue measure ｜Г｜ = 0, one has dimH Г = dimD Г, where dimD Г and dimH Г are the Hausdorff dimension and distributional dimension, respectively. Thus we might say that the distributional dimension is an analytical definition for Hausdorff dimension. Therefore we can study Hausdorff dimension through the distributional dimension analytically. By discussing the distributional dimension, this paper intends to set up a criterion for estimating the upper and lower bounds of Hausdorff dimension analytically. Examples illustrating the criterion are included in the end.     

Comparison inequalities for heat semigroups and heat kernels on metric measure spaces

We prove a certain inequality for a subsolution of the heat equation associated with a regular Dirichlet form. As a consequence of this inequality, we obtain various interesting comparison inequalities for heat semigroups and heat kernels, which can be used for obtaining pointwise estimates of heat kernels. As an example of application, we present a new method of deducing sub-Gaussian upper bounds of the heat kernel from on-diagonal bounds and tail estimates.     

Effective Conductivity of Nonlinear Two-Phase Media: Homogenization and Two-Point Padé Approximants

The aim of this paper is a study of the quasilinear transport equation, for instance the stationary heat equation. For periodically microheterogeneous media asymptotic homogenization has been performed with the local problem formulated as a minimization problem. The Golden–Papanicolaou integral representation theorem and some bounds developed for the linear equation have been extended. Two-point Padé approximants have been used to calculate bounds. Examples are also provided.     

THE NAGUMO EQUATION ON SELF－SIMILAR FRACTAL SETS

The Nagumo equationut ut=△u+bu(u-a)(1-u),t＞0 is investigated with initial data and zero Neumann boundary conditions on post-critically finite (p.c.f.) self-similar fractals that have regular harmonic structures and satisfy the separation condition. Such a nonlinear diffusion equation has no travelling wave solutions because of the“pathological” property of the fractal. However, it is shown that a global Hoelder continuous solution in spatial variables exists on the fractal considered. The Sobolev-type inequality plays a crucial role, which holds on such a class of p.c.f self-similar fractals. The heat kernel has an eigenfunction expansion and is well-defined due to a Weyl‘s formula. The large time asymptotic behavior of the solution is discussed, and the solution tends exponentially to the equilibrium state of the Nagumo equation as time tends to infinity if b is small.     

On gradient bounds for the heat kernel on the Heisenberg group

It is known that the couple formed by the two-dimensional Brownian motion and its Lévy area leads to the heat kernel on the Heisenberg group, which is one of the simplest sub-Riemannian space. The associated diffusion operator is hypoelliptic but not elliptic, which makes difficult the derivation of functional inequalities for the heat kernel. However, Driver and Melcher and more recently H.-Q. Li have obtained useful gradient bounds for the heat kernel on the Heisenberg group. We provide in this paper simple proofs of these bounds, and explore their consequences in terms of functional inequalities, including Cheeger and Bobkov type isoperimetric inequalities for the heat kernel.     

Fractional Generalized Random Fields on Bounded Domains

Using the theory of generalized random fields on fractional Sobolev spaces on bounded domains, and the concept of dual generalized random field, this paper introduces a class of random fields with fractional-order pure point spectra. The covariance factorization of an α-generalized random field having a dual is established, leading to a white-noise linear-filter representation, which reduces to the usual Markov representation in the ordinary case when α∈N and the covariance operator of the dual random field is local. Fractional-order differential models commonly arising from anomalous diffusion in disordered media can be studied within this framework.     

Transition Density Estimates for Diffusion Processes on Post Critically Finite Self-Similar Fractals

The framework of post critically finite (p.c.f) self-similarfractals was introduced to capture the idea of a finitely ramifiedfractal, that is, a connected fractal set where any componentcan be disconnected by the removal of a finite number of points.These ramification points provide a sequence of graphs whichapproximate the fractal and allow a Laplace operator to be constructedas a suitable limit of discrete graph Laplacians. In this paperwe obtain estimates on the heat kernel associated with the Laplacianon the fractal which are best possible up to constants. Theseare short time estimates for the Laplacian with respect to anatural measure and expressed in terms of an effective resistancemetric. Previous results on fractals with spatial symmetry haveobtained heat kernel estimates of a non-Gaussian form but whichare of Aronson type. By considering a range of examples whichare not spatially symmetric, we show that uniform Aronson typeestimates do not hold in general on fractals. 1991 MathematicsSubject Classification: 60J60, 60J25, 28A80, 31C25.     

Growth of heat trace and heat content asymptotic coefficients

We show in the smooth category that the heat trace asymptotics and the heat content asymptotics can be made to grow arbitrarily rapidly. In the real analytic context, however, this is not true and we establish universal bounds on their growth.     

Intrinsic ultracontractivity on Riemannian manifolds with infinite volume measures

By establishing the intrinsic super-Poincar'e inequality,some explicit conditions are presented for diffusion semigroups on a non-compact complete Riemannian manifold to be intrinsically ultracontractive.These conditions,as well as the resulting uniform upper bounds on the intrinsic heat kernels,are sharp for some concrete examples.     

A note on Smoluchowski’s equations with diffusion

We consider an infinite system of reaction–diffusion equations that models aggregation of particles. Under suitable assumptions on the diffusion coefficients and aggregation rates, we show that this system can be reduced to a scalar equation, for which an explicit self-similar solution is obtained. In addition, pointwise bounds for the solutions of associated initial and initial-boundary value problems are provided.