mathematics of granular materials this is a short and somewhat informal review on the most mathematical parts of the kinetic theory of granular media intended for physicists and for mathematicians outside the field

the discrete wavelet transform for a symmetric anti symmetric multiwavelet family on the interval the chui lian multiwavelet family of approximation order three is extended to serve as a basis for an orthogonal discrete multiwavelet transform on the interval this requires in addition to the original symmetric and antisymmetric pairs of scaling functions and wavelets four new functions of each type at each edge of the interval i e one more than required just to preserve the approximation order the criteria are given that such edge functions must satisfy in order to allow multiresolution analysis of data sequences without special constraints on endpoint behavior or use of periodic boundary conditions and a specific choice is examined in applications to both noiseless and noisy data these are accomplished through the development of interval basis extensions for both an earlier multiwavelet projection based prefilter postfilter combination and a multiwavelet vector thresholding algorithm

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 gotze about diffusions with periodic divergence free drift vector fields see bhagot1 and bhagot2

fast diffusion to self similarity complete spectrum long time asymptotics and numerology the complete spectrum is determined for the operator open image in new window on the sobolev space w1 2 rn formed by closing the smooth functions of compact support with respect to the norm open image in new window here the barenblatt profile is the stationary attractor of the rescaled diffusion equation open image in new window in the fast supercritical regime mopen image in new window the same diffusion dynamics represent the steepest descent down an entropy e u on probability measures with respect to the wasserstein distance d2 formally the operator h hess e is the hessian of this entropy at its minimum so the spectral gap h 2 n 1 m found below suggests the sharp rate of asymptotic convergence open image in new window from any centered initial data 0 u 0 x l1 rn with second moments this bound improves various results in the literature and suggests the conjecture that the self similar solution u t x r t n x r t is always slowest to converge the higher eigenfunctions which are polynomials with hypergeometric radial parts and the presence of continuous spectrum yield additional insight into the relations between symmetries of rn and the flow thus the rate of convergence can be improved if we are willing to replace the distance to with the distance to its nearest mass preserving dilation or still better affine image the strange numerology of the spectrum is explained in terms of the number of moments of

lagrangian numerical approximations to one dimensional convolution diffusion equations this work focuses on the numerical analysis of one dimensional nonlinear diffusion equations involving a convolution product first homogeneous friction equations are considered algorithms follow recent ideas on mass transportation methods and lead to simple schemes which can be proved to be stable to decrease entropy and to converge toward the unique solution of the continuous problem in particular for the first time homogeneous cooling states are displayed numerically further we present results on the more delicate fourth order thin film equation for which a nonnegativity preserving scheme is derived the dead core phenomenon is presented for the hele shaw cell

existence and uniqueness of monotone measure preserving maps introduction given a pair of borel probability measures and v on rd it is natural to ask whether v can be obtained from by redistributing its mass in a canonical way in the case of the line d 1 the answer is clear as long as both measures are free from atoms x v x 0mthere is a map y x of the line to itself for which uniquely determined almost everywhere this map may be taken to be nondecreasing by a suitable choice of y x e rw o at the remaining points interpreting and v as the initial and final distribution of a one dimensional fluid the transformation y x gives a rearrangement of fluid particles yielding final state v from the initial state this rearrangement is characterized by the fact that it preserves particle ordering obviating any need for two particles to cross although the generalization of this construction to higher dimensions is the focus of this paper the one dimensional case will be pursued slightly further when the measures are absolutely continuous with respect to lebesgue d x f x dx and dr y g y dymthen formally at least neglecting regularity issues the fundamental theorem of calculus yields

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

a finite volume scheme for the patlak keller segel chemotaxis model a finite volume method is presented to discretize the patlak keller segel pks modeling chemosensitive movements first we prove existence and uniqueness of a numerical solution to the proposed scheme then we give a priori estimates and establish a threshold on the initial mass for which we show that the numerical approximation converges to the solution to the pks system when the initial mass is lower than this threshold numerical simulations are performed to verify accuracy and the properties of the scheme finally in the last section we investigate blow up of the solution for large mass

on explosions of solutions to a system of partial differential equations modelling chemotaxis a system of partial differential equations modelling chemotactic aggregation is analysed keller segel model conditions on the system of paramaters are given implying global existence of smooth solutions in two space dimensions and radially symmetric situations explosion of the bacteria concentration in finite time is shown for a class of initial values

identification of asymptotic decay to self similarity for one dimensional filtration equations the objective of this paper is the derivation and the analysis of a simple explicit numerical scheme for general one dimensional filtration equations it is based on an alternative formulation of the problem using the pseudoinverse of the density s repartition function in particular the numerical approximations can be proven to satisfy a contraction property for a wasserstein metric various numerical results illustrate the ability of this numerical process to capture the time asymptotic decay towards self similar solutions even for fast diffusion equations

point dynamics in a singular limit of the keller segel model 2 formation of the concentration regions this paper continues the analysis started in the first part of this article cf j j l velazquez siam j appl math 64 2004 pp 1198 1223 it was seen there using the method of matched asymptotics that a regularized version of the keller segel system admits for a suitable asymptotic limit solutions with some regions of high concentrations for the cell density this paper considers the relation between the phenomenon of blow up for the limit problem and the dynamics of the concentration regions described in j j l velazquez siam j appl math 64 2004 pp 1198 1223 in particular this paper analyzes the precise way in which the regularization introduced in the keller segel system stops the aggregation process and yields the formation of concentration regions

a special class of stationary flows for two dimensional euler equations a statistical mechanics description we consider the canonical gibbs measure associated to an vortex system in a bounded domain at inverse temperature widetilde beta and prove that in the limitn widetilde beta n n 1 where 8 here denotes the vorticity intensity of each vortex the one particle distribution function n nx x converges to a superposition of solutions of the following mean field equation left begin array 20 c varrho beta x frac e beta psi mathop smallint limits lambda e beta psi delta psi varrho beta in lambda psi partial lambda 0 end array right

weak solutions to a parabolic elliptic system of chemotaxis abstract we study a parabolic elliptic system of partial differential equations which describes the chemotactic feature of slime molds it is known that the blowup solution forms singularities such as delta functions referred to as the collapses here we study the case that the domain is a flat torus and show that the post blowup continuation of the solution is possible only when those collapses are quantized with the mass 8

biorthogonal multiresolution analysis on a triangle and applications we present in this paper new constructions of biorthogonal multiresolution analysis on the triangle we use direct method based on the tensor product to construct dual scaling spaces on next we construct the associated wavelet spaces and we prove that the associated wavelets have compact support and preserve the original regularity finally we describe some regular results which are very useful to establish the norm equivalences as applications we prove that the wavelet bases constructed in this paper are adapted for the study of the sobolev spaces h 0 s and h s s n and are easy to implement wavelets have unfolded their full computation efficiently in numerical and applied analysis the properties of wavelet bases provide a rigorous analysis for dynamical systems we construct in this work biorthogonal wavelet bases on a triangle these bases are adapted to the study of the sobolev spaces the bases allow many concrete numerical examples as numerical simulation for elliptic problems or image processing

diffusion and mixing in fluid flow we study enhancement of diffusive mixing on a compact riemannian manifold by a fast incompressible flow our main result is a sharp description of the class of flows that make the deviation of the solution from its average arbitrarily small in an arbitrarily short time provided that the flow amplitude is large enough the necessary and suficient condition on such flows is expressed naturally in terms of the spectral properties of the dynamical system associated with the flow in particular we find that weakly mixing flows always enhance dissipation in this sense the proofs are based on a general criterion for the decay of the semigroup generated by an operator of the form g ial with a negative unbounded self adjoint operator g a self adjoint operator l and parameter a 1 in particular they employ the rage theorem describing evolution of a quantum state belonging to the continuous spectral subspace of the hamiltonian related to a classical theorem of wiener on fourier transforms of measures applications to quenching in reaction diffusion equations are also considered

stability of some mechanisms of chemotactic aggregation in this paper the stability of a blow up mechanism for a particular keller segel model that yields chemotactic aggregation is considered it is shown using matched asymptotic methods that the considered blow up mechanism is stable more precisely small perturbations of the initial data produce a small shifting on the blow up time and the blow up point but not modification in the blow up mechanism

contractions in the 2 wasserstein length space and thermalization of granular media an algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions this rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite dimensional manifold an abstract theory is developed for gradient flows in length spaces which shows how degenerate convexity or even non convexity if uniformly controlled will quantify contractivity limit expansivity of the flow

the behavior of the spectral gap under growing drift we analyze the behavior of the spectral gap of the laplace beltrami operator on a compact riemannian manifold when a divergence free drift vector field is added we increase the drift by multiplication with a large constant c and ask the question how the spectral gap behaves as c goes to infinity it turns out that the spectral gap stays bounded if and only if the drift vector field has eigenfunctions in h 1 in that case the spectral gaps converge and we determine the limit

infinite time aggregation for the critical patlak keller segel model in ℝ2 we analyze the two dimensional parabolic elliptic patlak keller segel model in the whole euclidean space r2 under the hypotheses of integrable initial data with finite second moment and entropy we first show local in time existence for any mass of free energy solutions namely weak solutions with some free energy estimates we also prove that the solution exists as long as the entropy is controlled from above the main result of the paper is to show the global existence of free energy solutions with initial data as before for the critical mass 8 actually we prove that solutions blow up as a delta dirac at the center of mass when t keeping constant their second moment at any time furthermore all moments larger than 2 blow up as t if initially bounded

convergence of the mass transport steepest descent scheme for the subcritical patlak keller segel model variational steepest descent approximation schemes for the modified patlak keller segel equation with a logarithmic interaction kernel in any dimension are considered we prove the convergence of the suitably interpolated in time implicit euler scheme defined in terms of the euclidean wasserstein distance associated with this equation for subcritical masses as a consequence we recover the recent result about the global in time existence of weak solutions to the modified patlak keller segel equation for the logarithmic interaction kernel in any dimension in the subcritical case moreover we show how this method performs numerically in dimension one in this particular case this numerical scheme corresponds to a standard implicit euler method for the pseudoinverse of the cumulative distribution function we demonstrate its capabilities to reproduce the blow up of solutions for supercritical masses easily without the need of mesh refinement

long time asymptotics of kinetic models of granular flows we analyze the long time asymptotics of certain one dimensional kinetic models of granular flows which have been recently introduced in 22 in connection with the quasi elastic limit of a model boltzmann equation with dissipative collisions and variable coefficient of restitution these nonlinear equations classified as nonlinear friction equations split naturally into two classes depending on whether or not the temperature of their similarity solutions homogeneous cooling states reduce to zero in finite time for both classes we show uniqueness of the solution by proving decay to zero in the wasserstein metric of any two solutions with the same mass and mean velocity furthermore if the temperature of the similarity solution decays to zero in finite time we prove by computing explicitly upper bounds for the lifetime of the solution in terms of the length of the support that the temperature of any other solution with initially bounded support must also decay to zero in finite time

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

meshfree methods and boundary conditions summary we perform here some meshfree methods to inhomogeneous laplace equations we prove the efficiency of those methods compared with classical ones for one or two dimensional case for numerics and for one dimensional for theoretical results copyright 2006 john wiley sons ltd

stirring up trouble multi scale mixing measures for steady scalar sources abstract the mixing efficiency of a flow advecting a passive scalar sustained by steady sources and sinks is naturally defined in terms of the suppression of bulk scalar variance in the presence of stirring relative to the variance in the absence of stirring these variances can be weighted at various spatial scales leading to a family of multi scale mixing measures and efficiencies we derive a priori estimates on these efficiencies from the advection diffusion partial differential equation focusing on a broad class of statistically homogeneous and isotropic incompressible flows the analysis produces bounds on the mixing efficiencies in terms of the peclet number a measure of the strength of the stirring relative to molecular diffusion we show by example that the estimates are sharp for particular source sink and flow combinations in general the high peclet number behavior of the bounds scaling exponents as well as prefactors depends on the structure and smoothness properties of and length scales in the scalar source and sink distribution the fundamental model of the stirring of a monochromatic source sink combination by the random sine flow is investigated in detail via direct numerical simulation and analysis the large scale mixing efficiency follows the upper bound scaling within a logarithm at high peclet number but the intermediate and small scale efficiencies are qualitatively less than optimal the peclet number scaling exponents of the efficiencies observed in the simulations are deduced theoretically from the asymptotic solution of an internal layer problem arising in a quasi static model

global in time weak measure solutions and finite time aggregation for nonlocal interaction equations in this paper we provide a well posedness theory for weak measure solutions of the cauchy problem for a family of nonlocal interaction equations these equations are continuum models for interacting particle systems with attractive repulsive pairwise interaction potentials the main phenomenon of interest is that even with smooth initial data the solutions can concentrate mass in finite time we develop an existence theory that enables one to go beyond the blow up time in classical norms and allows for solutions to form atomic parts of the measure in finite time the weak measure solutions are shown to be unique and exist globally in time moreover in the case of sufficiently attractive potentials we show the finite time total collapse of the solution onto a single point for compactly supported initial measures our approach is based on the theory of gradient flows in the space of probability measures endowed with the wasserstein metric in addition to classical tools we exploit the stability of the flow with respect to the transportation distance to greatly simplify many problems by reducing them to questions about particle approximations

an integro differential equation arising as a limit of individual cell based models in this paper we study mathematical properties of an integro differential equation that arises as a particular limit case in the study of individual cell based model we obtain global well posedness for some classes of interaction potentials and finite time blow up for others the existence of space homogeneous steady states as well as long time asymptotics for the solutions of the problem is also discussed

solution of cartesian and curvilinear quantum equations via multiwavelets on the interval abstract it is shown how orthogonal compact support multiwavelets may be used for the solution of quantum mechanical eigenvalue problems subject to specific boundary conditions special scaling functions and wavelets with convenient limiting behaviors at the edges of an interval are constructed in analogy to earlier work on single wavelet families all of the integrals required for hamiltonian matrix elements involving both regular and edge functions are calculated efficiently through use of recursion and quadrature methods it is demonstrated through accurate eigenvalue determination that both cartesian and curvilinear degrees of freedom are readily accommodated with such a basis using as examples the particle in a box and the hydrogen atom in spherical polar coordinates

compactly supported wavelets and boundary conditions abstract wavelet bases subject to various homogeneous boundary conditions on an interval are constructed they are obtained from daubechies compactly supported wavelets which are restricted to the interval following an idea by y meyer and then suitably modified the sobolev spaces arising from homogeneous boundary problems are characterized in terms of wavelet coefficients in these bases

exponential speed of uniform convergence of the cell density toward equilibrium for subcritical mass in a patlak keller segel model this paper is concerned with a chemotaxis aggregation model for cells more precisely with a parabolic elliptic semilinear patlak keller segel system in a ball of rn exhibiting a critical mass phenomenon for n 2 the main result of this paper is the exponential speed of uniform convergence of radial solutions toward the unique steady state in the subcritical case we stress that this covers in particular the classical keller segel system with n 2 well known for its critical mass 8 and that the result improves on the known results even for this most studied problem a key tool is an associated one dimensional degenerate parabolic problem the proof exploits its formal gradient flow structure ut f u t on an infinite dimensional riemannian manifold in particular we show a new hardy type inequality equivalent to the strict convexity of f at any steady state of subcritical mass which heuristically explains the exponential speed of convergence

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

averaging of incompressible flows on two dimensional surfaces we consider a process on a compact two dimensional surface which consists of the fast motion along the stream lines of an incompressible periodic vector field perturbed by white noise it gives rise to a process on the graph naturally associated to the structure of the stream lines of the unperturbed flow it has been shown by freidlin and wentzell that if the unperturbed motion is periodic for almost all the initial points then the corresponding process on the graph weakly converges to a markov process we consider the situation when the unperturbed motion is not periodic and the flow has ergodic components of positive measure we show that the process on the graph still converges to a markov process which spends a positive proportion of time in the vertices corresponding to the ergodic components of the flow as shown in a companion paper 5 these results allow one to describe the viscosity regularization of dissipative deterministic perturbations of incompressible flows in the case of surfaces of higher genus the limiting process may exhibit intermittent behavior in the sense that the time axis can be divided into the intervals of random lengths where the particle stays inside one ergodic component separated by the intevals where the particle moves deterministically between the components