a local error estimate of the method of multi scale asymptotic expansions for elliptic problems with rapidly oscillatory coefficients in this paper on basis of o a oleinik a s shamaev g a yosifian mathematical problems in elasticity and homogenization north holland amsterdam 1992 we present a local error estimate of the method of multi scale asymptotic expansions for second order elliptic problems with rapidly oscillatory coefficients

wavefront propagation for reaction diffusion systems of pde m i freidlin has introduced probabilistic techniques to study propagation for systems of reaction diffusion pde the motivating idea is that should a reaction diffusion system possess only a single unstable and a single stable equilibrium then the solution u of the system will presumably tend to switch for large times from near the former to near the latter state this paper brings to bear purely pde techniques to this problem especially the theory of viscosity solutions on hamilton jacobi equations due to crandall lions

homogenization of linear transport equations with oscillatory vector fields the l 2 weak limit as varepsilon downarrow 0 of the solutions of the linear transport equation is studied begin gathered u t bf a left frac x varepsilon right cdot nabla x u 0 u t 0 u 0 left x frac x varepsilon right end gathered where varepsilon 0 x in r 2 t in 0 t for any t 0 bf a is a smooth divergence free vector field on the two dimensional unit torus and has no equilibrium points the initial condition u 0 is assumed to be smooth compactly supported in x and periodic in x varepsilon using a weak l 2 formulation and the idea of rotation number for flows on the two dimensional torus the effective equations for the weak limit are found they depend sensitively on the underlying flow on the two dimensional torus and its ergodicity

finite difference heterogeneous multi scale method for homogenization problems in this paper we propose a numerical method the finite difference heterogeneous multi scale method fd hmm for solving multi scale parabolic problems based on the framework introduced in commun math sci 1 1 87 the numerical method relies on the use of two different schemes for the original equation at different grid level which allows to give numerical results at a much lower cost than solving the original equations we describe the strategy for constructing such a method discuss generalization for cases with time dependency random correlated coefficients nonconservative form and implementation issues finally the new method is illustrated with several test examples

the effect of turbulence on mixing in prototype reaction diffusion systems the effect of turbulence on mixing in prototype reaction diffusion systems is analyzed here in the special situation where the turbulence is modeled ideally with two separated scales consisting of a large scale mean flow plus a small scale spatiotemporal periodic flow in the limit of fast reaction and slow diffusion it is rigorously proved that the turbulence does not contribute to the location of the mixing zone in the limit and that this mixing zone location is determined solely by advection of the large scale velocity field this surprising result contrasts strongly with earlier work of the authors that always yields a large scale propagation speed enhanced by small scale turbulence for propagating fronts the mathematical reasons for these differences are pointed out this main theorem rigorously justifies the limit equilibrium approximations utilized in nonpremixed turbulent diffusion flames and condensation evaporation modeling in cloud physics in the fast reaction limit the subtle nature of this result is emphasized by explicit examples presented in the fast reaction and zero diffusion limit with a nontrivial effect of turbulence on mixing in the limit the situation with slow reaction and slow diffusion is also studied in the present work here the strong stirring by turbulence before significant reaction occurs necessarily leads to a homogenized limit with the strong mixing effects of turbulence expressed by a rigorous turbulent diffusivity modifying the reaction diffusion equations physical examples from non premixed turbulent combustion and cloud microphysics modeling are utilized throughout the paper to motivate and interpret the mathematical results c 2000 john wiley sons inc

convergence of a shock capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions convergence of a shock capturing streamline diffusion nite element method for a conservation law in two space dimensions

stochastic superparameterization in quasigeostrophic turbulence in this article we expand and develop the authors recent proposed methodology for efficient stochastic superparameterization algorithms for geophysical turbulence geophysical turbulence is characterized by significant intermittent cascades of energy from the unresolved to the resolved scales resulting in complex patterns of waves jets and vortices conventional superparameterization simulates large scale dynamics on a coarse grid in a physical domain and couples these dynamics to high resolution simulations on periodic domains embedded in the coarse grid stochastic superparameterization replaces the nonlinear deterministic eddy equations on periodic embedded domains by quasilinear stochastic approximations on formally infinite embedded domains the result is a seamless algorithm which never uses a small scale grid and is far cheaper than conventional sp but with significant success in difficult test problems various design choices in the algorithm are investigated in detail here including decoupling the timescale of evolution on the embedded domains from the length of the time step used on the coarse grid and sensitivity to certain assumed properties of the eddies e g the shape of the assumed eddy energy spectrum we present four closures based on stochastic superparameterization which elucidate the properties of the underlying framework a null hypothesis stochastic closure that uncouples the eddies from the mean a stochastic closure with nonlinearly coupled eddies and mean a nonlinear deterministic closure and a stochastic closure based on energy conservation the different algorithms are compared and contrasted on a stringent test suite for quasigeostrophic turbulence involving two layer dynamics on a plane forced by an imposed background shear the success of the algorithms developed here suggests that they may be fruitfully applied to more realistic situations they are expected to be particularly useful in providing accurate and efficient stochastic parameterizations for use in ensemble based state estimation and prediction

taylor dispersion in curved channels the taylor dispersion problem is to determine the long time behavior of the concentration density of a passive scalar contaminant diffusing in an incompressible channel flow given an initial injection of contaminant which is slowly varying order epsilon in the longitudinal direction taylor showed that the cross channel mean of the concentration experiences an enhanced diffusion relative to the mean flow this is due to mechanical rather than molecular events mercer and roberts have treated this problem by a formal application of infinite dimensional center manifold theory and extended the analysis to the case of a curved channel which is slowly varying in the longitudinal direction it is shown that after the introduction of appropriate multiple scales their center manifold technique can be used to carry out the taylor dispersion calculations for a channel whose cross section has large order 1 variation in the longitudinal directions also more general underlying flows can be allowed than was heretofore possible a heuristic argument is given to show the existence of some channel shape and flow field for which effective diffusion is reduced rather than enhanced a similar advection diffusion problem but with no channel boundary has been rigorously analyzed by homogenization theory and discussed in a recent paper of majda andmore mclaughlin the two problems are briefly compared less

the support of the equilibrium measure in the presence of a monomial external field on 1 1 the support of the equilibrium measure associated with an external field of the form q x cx2m 1 x c 1 1 with c 0 and m a positive integer is investigated it is shown that the support consists of at most two intervals this resolves a question of deift kriecherbauer and mclaughlin

large deviations estimates for some non local equations general bounds and applications large deviation estimates for the following linear parabolic equation are studied frac partial u partial t tr big a x d 2u big b x cdot d u int r n big u x y u x d u x cdot y ind y 1 y big d mu y where mu is a levy measure which may be singular at the origin assuming only that some negative exponential integrates with respect to the tail of mu it is shown that given an initial data solutions defined in a bounded domain converge exponentially fast to the solution of the problem defined in the whole space the exact rate which depends strongly on the decay of mu at infinity is also estimated

on the slowness of phase boundary motion in one space dimension we study the limiting behavior of the solution of u t e 2 u xx u 3 u 0 a x b with a neumann boundary condition or an appropriate dirichlet condition the analysis is based on energy methods we assume that the initial data has a transition layer structure i e ue 1 except near finitely many transition points we show that in the limit as e 0 the solution maintains its transition layer structure and the transition points move slower than any power of e

slow motion in higher order systems and g convergence in one space dimension the slow motion in higher order systems and convergence in one space dimension was studied some of the theorems and computations that support and proves the stability of the equations are presented

numerical analysis of the m p p model of turbulence abstract two different numerical schemes are proposed for the study of the behavior of a viscous incompressible fluid in a turbulent regime using the m p p model they differ in their treatment of both velocity and turbulent kinetic energy at the time discretization stage the first a gear scheme is fully implicit while the second is of the alternate direction type in both the lagrangian coordinate is calculated by applying an explicit lax wendroff scheme various numerical experiments showed good agreement between the two schemes

left definite multiparameter eigenvalue problems we study the problem k tmxm anvmnxm o xmehm m i k n 1 where tm and vmn are selfadjoint linear operators on separable hilbert spaces hm with tm positive r i compact and vmn bounded we assume left definiteness which involves positivity of certain linear combinations of cofactors in the determi nant with m n th entry xm vmnxm we establish a spectral theory for that is in some way simpler and more complete than those hitherto available for this case in particular we make use of operators bn a iao where the an are determinantal operators on i hm this complements an established approach to the alternative right definite problem where ao is positive via the operators fn aoian 1 introduction in this section we shall discuss how the left and right definiteness conditions arose and what methods and results are currently available for the left definite case then in 1 3 we shall summarise our approach here throughout we shall be considering the system of the abstract 1 1 definiteness conditions early work on was motivated by separation of variables in certain classical boundary value problems the resulting ordinary differential equations in most cases do satisfy the conditions we impose on the operators tm and vmn the hm being l2 spaces the bulk of the literature which also covers finite dimensional equations including difference approximations involves sign definiteness of the determinant 80 x whose m n th entry is xm vmnx m atkinson 1 has treated such problems systematically and his work together with various recent extensions now provides an analogue of much of the basic spectral theory for a single equation tx avx x 1 0 with x vx sign definite such problems are nowadays called right definite actually various right definiteness conditions have been used and two which are relevant to our discussion are the strong 80 condition

universality of the double scaling limit in random matrix models we study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support we establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit the limiting kernels are constructed out of functions associated with the second painleve equation this extends a result of bleher and its for the special case of a critical quartic potential the two main tools we use are equilibrium measures and riemann hilbert problems in our treatment of equilibrium measures we allow a negative density near the critical point which enables us to treat all cases simultaneously the asymptotic analysis of the riemann hilbert problem is done with the deift zhou steepest descent analysis for the construction of a local parametrix at the critical point we introduce a modification of the approach of baik deift and johansson so that we are able to satisfy the required jump properties exactly c 2005 wiley periodicals inc

generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields the equilibrium measure in the presence of an external field plays a role in a number of areas in analysis for example in random matrix theory the limiting mean density of eigenvalues is precisely the density of the equilibrium measure typical behavior for the equilibrium measure is 1 it is positive on the interior of a finite number of intervals 2 it vanishes like a square root at endpoints and 3 outside the support there is strict inequality in the euler lagrange variational conditions if these conditions hold then the limiting local eigenvalue statistics is loosely described by a bulk in which there is universal behavior involving the sine kernel and edge effects in which there is a universal behavior involving the airy kernel through techniques from potential theory and integrable systems we show that this regular behavior is generic for equilibrium measures associated with real analytic external fields in particular we show that for any one parameter family of external fields v c the equilibrium measure exhibits this regular behavior except for an at most countable number of values of c we discuss applications of our results to random matrices orthogonal polynomials and integrable systems 2000 john wiley sons inc

multiparameter definiteness conditions a previous attempt to systematize various conditions from multiparameter spectral theory is extended to weaker forms of definiteness the latter not only occur in the established literature but are also under active investigation at present several algebraic and geometrical formulations exist and questions concerning their equivalence are approached in a unified fashion where possible

pattern formation in a flux limited reaction diffusion equation of porous media type a non linear pde featuring flux limitation effects together with those of the porous media equation non linear fokker planck is presented in this paper we analyze the balance of such diverse effects through the study of the existence and qualitative behavior of some admissible patterns namely traveling wave solutions to this singular reaction diffusion equation we show the existence and qualitative behavior of different types of traveling waves classical profiles for wave speeds high enough and discontinuous waves that are reminiscent of hyperbolic shock waves when the wave speed lowers below a certain threshold some of these solutions are of particular relevance as they provide models by which the whole solution and not just the bulk of it as it is the case with classical traveling waves spreads through the medium with finite speed

nonuniform right definiteness abstract the multiparameter eigenvalue problem wm x m xm w m t m n 1 k n v mn m 1 k where gl ge c k xm is a nonzero element of the separable hilbert space hm and tm and vmn are compact symmetric is studied various properties including existence and uniqueness of i c k for which the imth greatest eigenvalue of wm i equals one are proved right definiteness is assumed which means positivity of the determinant with m n th entry ym vmnym for all nonzero ym hm m 1 k this gives a klein oscillation theorem for systems of an o d e satisfying a definiteness condition that is usefully weaker than in previous such results an expansion theorem in terms of the corresponding eigenvectors xmi is also given thereby connecting the abstract oscillation theory with a result of atkinson

homogenization of two dimensional linear flows with integral invariance we study the homogenization of two dimensional linear transport equations u t av xv ep cdot nabla xv u 0 where av is a nonvanishing vector field with integral invariance on the torus t 2 when the underlying flow on t 2 is ergodic we derive the efficient equation which is a linear transport equation with constant coefficients and quantify the pointwise convergence rate this result unifies and illuminates the previously known results in the special cases of incompressible flows and shear flows when the flow on t 2 is nonergodic the homogenized limit is an average over t 1 of solutions of linear transport equations with constant coefficients the convergence here is in the weak sense of w 1 infty loc rra 1 and the sharp convergence rate is oe one of the main ingredients in our analysis is a classical theorem due to kolmogorov regarding flows with integral invariance on t 2 to which we present here an elementary and constructive proof

homogenization of degenerate porous medium type equations in ergodic algebras abstract we consider the homogenization problem for general porous medium type equations of the form u t f x x e u the pressure function f x y may be of two different types in the type 1 case f x y is a general strictly increasing function this is a mildly degenerate case in the type 2 case f x y has the form h x y f u s x y where f u is just a nondecreasing function this is a strongly degenerate case we address the initial boundary value problem for a general bounded or unbounded domain with null or more generally steady pressure condition on the boundary the homogenization is carried out in the general context of ergodic algebras as far as the authors know homogenization of such degenerate quasilinear parabolic equations is addressed here for the first time we also review the existence and stability theory for such equations and establish new results needed for the homogenization analysis further we include some new results on algebras with mean value specially a new criterion establishing the null measure of level sets of elements of the algebra which is useful in connection with the homogenization of porous medium type equations in the type 2 case

homogenization of linear and nonlinear transport equations we study the homogenization of the linear and nonlinear transport equations with oscillatory velocity fields two types of homogenized equations are derived for general n dimensional linear and nonlinear problems we derive homogenized equations by introducing additional independent variables to represent the small scales for the two dimensional linear transport equations we derive effective equations for the averaged quantities such equations take the form of either a degenerate non local diffusion equation with memory or a higher order hyperbolic equation to study the nonlinear transport equations we introduce the concept of two scale young measure and extend diperna s method to prove that it reduces to a family of dirac measures

on point mass riesz external fields on the real axis abstract the purpose of this work is twofold first we aim to extend for 0 s 1 the results of one of the authors about equilibrium measures in the real axis in external fields created by point mass charges for the case of logarithmic potentials s 0 our second motivation comes from the work of the other two authors on riesz s equilibrium problems on finitely many intervals on the real line in the presence of external fields they have shown that when the signed equilibrium measure has concave positive part on every interval then the s equilibrium support is also a union of finitely many intervals with one of them at most included in each of the initial intervals as the positive part of the signed equilibrium for point mass external fields is not necessarily concave the investigation of the corresponding s equilibrium support is of comparative interest moreover we provide simple examples of compactly supported equilibrium measures in external fields q not satisfying the usual requirements about the growth at infinity that is lim x q x our main tools are signed equilibrium measures and iterated balayage algorithm in the context of riesz s equilibrium problems on the real line as these techniques are not mainstream work in the field and can be applied in other contexts we highlight their use here

trilinear resonant interactions of semilinear hyperbolic waves an important question in the study of nonlinear equations is the understanding of the asymptotic behavior of families of solutions or approximate solutions one of the difficulty is known to be the nonlinear interaction of oscillations this phenomenon has been studied for special classes of solutions such as oscillatory functions with phases and amplitudes this is the domain of geometric optics in some cases general families of bounded solutions have also been considered the present paper enters the later category but is highly inspired by the results of nonlinear geometric optics geometric optics provides a precise description of the asymptotic structure of solutions to both linear e g wh and nonlinear j hk mr hmr jmr 1 jmr 2 s hyperbolic pdes whose initial data u 0 x u0 x eu1 x 0 x e have rapid oscillations with prescribed phases 0 and a single prescribed scale 1e such solutions are important in both theory and applications l kk in the lowest approximation oscillations having different spatial scales interact only when m equals zero since otherwise the amplitudes of higher frequency oscillations become negligible relative to those of lower frequencies taking m 0 requires l estimates systems that are well posed in l occur mostly in one spatial dimension for such systems arbitrary uniformly bounded sequences of initial data may be considered but comparatively little is known about the asymptotic structure of the resulting solutions since the oscillations in such solutions may be extremely complex a natural problem is to determine their non oscillatory part which may be identified with the weak limit of the sequence of solutions the highly oscillatory nature of the solutions manifests itself in the absence of strong convergence in general of course some information about those oscillations will generally be needed in order to determine the weak limit the asymptotic structure of uniformly bounded solutions depends strongly on the nature of the nonlinearity for entropy solutions of a genuinely nonlinear conservation law t 1 or of a pair of such equations in one spatial dimension dp oscillations of amplitude o 1 do not persist for positive times so a sequence of solutions converges strongly to the solution whose initial data is the weak limit of the initial data although oscillations do persist in linear systems the weak limit of a sequence of solutions to such

multiscale young measures in homogenization of continuous stationary processes in compact spaces and applications abstract we introduce a framework for the study of nonlinear homogenization problems in the setting of stationary continuous processes in compact spaces the latter are functions f t r n q q with f t x f t x where q is a compact hausdorff topological space f c q and t x q q x r n is an n dimensional continuous dynamical system endowed with an invariant radon probability measure it can be easily shown that for almost all q the realization f t x belongs to an algebra with mean value that is an algebra of functions in buc r n containing all translates of its elements and such that each of its elements possesses a mean value this notion was introduced by zhikov and krivenko v v zhikov e v krivenko homogenization of singularly perturbed elliptic operators mat zametki 33 1983 571 582 english transl in math notes 33 1983 294 300 we then establish the existence of multiscale young measures in the setting of algebras with mean value where the compactifications of r n provided by such algebras plays an important role these parametrized measures are useful in connection with the existence of correctors in homogenization problems we apply this framework to the homogenization of a porous medium type equation in r n with a stationary continuous process as a stiff oscillatory external source this application seems to be new even in the classical context of periodic homogenization

on the fkpp equation with gaussian shear advection abstract a boundary value problem for the fisher kolmogorov petrovskii piskunov fkpp equation with shear random advection is investigated it is demonstrated that the upper bounds for front propagation in the fkpp equation with gaussian advection obtained by an analysis of its linear ensemble averaged upper solution can give too rough predictions it is shown that the unboundedness of the gaussian advection affects ensemble averaged solutions of linear and nonlinear problems in a different way this analytical prediction is confirmed by direct numerical simulations the phenomenon of propagation failure due to zero boundary conditions is studied and the critical conditions are found numerical procedure for the analysis of random wave propagation is suggested by introduction of mean front position and its variance some numerical experiments are presented

probability density functions of decaying passive scalars in periodic domains an application of sinai yakhot theory employing the formalism introduced by phys rev lett 63 1962 1989 we study the probability density functions pdf s of decaying passive scalars in periodic domains under the influence of smooth large scale velocity fields the particular regime we focus on is one where the normalized scalar pdf s attain a self similar profile in finite time i e the so called strange or statistical eigenmode regime in accordance with the work of sinai and yakhot the central regions of the pdf s are power laws however the details of the pdf profiles are dependent on the physical parameters in the problem interestingly for small peclet numbers the pdf s resemble stretched or pure exponential functions whereas in the limit of large peclet numbers there emerges a universal gaussian form for the pdf numerical simulations are used to verify these predictions

long time behavior of the continuum limit of the toda lattice and the generation of infinitely many gaps from c initial data we analyze a continuum limit of the finite non periodic toda lattice through an associated constrained maximization problem over spectral density functions the maximization problem was derived by deift and mclaughlin using the lax levermore approach initially developed for the zero dispersion limit of the kdv equation it encodes the evolution of the continuum limit for all times including evolution through shocks the formation of gaps in the support of the maximizer is indicative of oscillations in the toda lattice and the lack of strong convergence of the continuum limit for large times the maximizer tends to have zero gaps which is the continuum analogue of the sorting property of the finite lattice using methods from logarithmic potential theory we show that this behavior depends crucially on the initial data we exhibit initial data for which the zero gap ansatz holds uniformly in the spatial parameter at large times and other initial data for which this uniformity fails at all times we then construct an example of c smooth initial data generating at a later time infinitely many gaps in the support of the maximizer while for even larger times all gaps have closed

convergence analysis of krylov subspace iterations with methods from potential theory krylov subspace iterations are among the best known and most widely used numerical methods for solving linear systems of equations and for computing eigenvalues of large matrices these methods are polynomial methods whose convergence behavior is related to the behavior of polynomials on the spectrum of the matrix this leads to an extremal problem in polynomial approximation theory how small can a monic polynomial of a given degree be on the spectrum this survey gives an introduction to a recently developed technique to analyze this extremal problem in the case of symmetric matrices it is based on global information on the spectrum in the sense that the eigenvalues are assumed to be distributed according to a certain measure then depending on the number of iterations the lanczos method for the calculation of eigenvalues finds those eigenvalues that lie in a certain region which is characterized by means of a constrained equilibrium problem from potential theory the same constrained equilibrium problem also describes the superlinear convergence of conjugate gradients and other iterative methods for solving linear systems

on multiparameter theory abstract in this paper we point out a mistake in the paper abstract multiparameter theory i of p j browne we give new proofs for some of the results of p j browne

homogenization and convergence of the vortex method for 2 d euler equations with oscillatory vorticity fields the 2 d incompressible euler equations with oscillatory vorticity fields are studied a homogenization result for 2 d euler equations in velocity vorticity formulation is obtained and weak continuity of the equations is proved convergence of the vortex method is analyzed in the case when the continuous vorticity is not well resolved by the computational particles numerical results are given comparisons are made with the corresponding finite difference approximation

periodic homogenization for inertial particles we study the problem of homogenization for inertial particles moving in a periodic velocity field and subject to molecular diffusion we show that under appropriate assumptions on the velocity field the large scale long time behavior of the inertial particles is governed by an effective diffusion equation for the position variable alone to achieve this we use a formal multiple scale expansion in the scale parameter this expansion relies on the hypo ellipticity of the underlying diffusion an expression for the diffusivity tensor is found and various of its properties studied in particular an expansion in terms of the non dimensional particle relaxation time the stokes number is shown to co incide with the known result for passive non inertial tracers in the singular limit 0 this requires the solution of a singular perturbation problem achieved by means of a formal multiple scales expansion in incompressible and potential fields are studied as well as fields which are neither and theoretical findings are supported by numerical simulations

perturbation theory of multiparameter eigenvalue problems abstract we investigate the multiparameter eigenvalue problem in the case where some of the operators depend holomorphically on a perturbation parameter the behaviour of eigenvalues of finite multiplicity together with the associated eigenvectors is considered as the perturbation parameter is varied in addition we discuss the effect of the perturbation on a complete orthonormal system of eigenvectors the theory allows fairly general holomorphic perturbations and only requires a weak multiparameter definiteness condition

spectral analysis and computation for homogenization of advection diffusion processes in steady flows advective diffusion plays a key role in the transport of salt heat buoys and markers in geophysical flows in the dispersion of pollutants and trace gases in the atmosphere and even in the dynamics of sea ice floes influenced by winds and ocean currents the long time large scale behavior of such systems is equivalent to an enhanced diffusion process with an effective diffusivity matrix d three decades ago a stieltjes integral representation for the homogenized matrix involving a spectral measure of a self adjoint operator was developed however analytical calculations of d have been obtained for only a few simple flows and numerical computations of the effective behavior based on this spectral representation have apparently not been attempted we overcome these limitations by providing a mathematical foundation for the computation of stieltjes integral representations of d we explore two different approaches and for each approach we derive new stieltjes integral representations and rigorous bounds for the symmetric and antisymmetric parts of d involving the molecular diffusivity and a spectral measure of a self adjoint operator that depends on the characteristics of a randomly perturbed periodic flow in discrete formulations of each approach we express explicitly in terms of standard or generalized eigenvalues and eigenvectors of hermitian matrices we develop and implement an efficient numerical algorithm that combines beneficial numerical attributes of each approach we use this method to compute the effective behavior for model flows and relate spectral characteristics to flow geometry and transport properties advective diffusion plays a key role in the transport of salt heat buoys and markers in geophysical flows in the dispersion of pollutants and trace gases in the atmosphere and even in the dynamics of sea ice floes influenced by winds and ocean currents the long time large scale behavior of such systems is equivalent to an enhanced diffusion process with an effective diffusivity matrix d three decades ago a stieltjes integral representation for the homogenized matrix involving a spectral measure of a self adjoint operator was developed however analytical calculations of d have been obtained for only a few simple flows and numerical computations of the effective behavior based on this spectral representation have apparently not been attempted we overcome these limitations by providing a mathematical foundation for the computation of stieltjes integral representations of d we explore two different approaches and for each approach we derive new stieltjes integral representations and rigorou

metastable patterns in solutions of ut ϵ2uxx f u we consider the equation in question on the interval 0 x 1 having neumann boundary conditions with f u f u where f is a double well energy density with equal minima at u 1 the only stable states of the system are patternless constant solutions but given two phase initial data a pattern of interfacial layers typically forms far out of equilibrium the ensuing nonlinear relaxation process is extremely slow patterns persist for exponentially long times proportional to exp a l where a f 1 and l is the minimum distance between layers physically a tiny potential jump across a layer drives its motion we prove the existence and persistence of these metastable patterns and characterise accurately the equations governing their motion the point of view is reminiscent of center manifold theory a manifold parametrising slowly evolving states is introduced a neighbourhood is shown to be normally attracting and the parallel flow is characterised to high relative accuracy proofs involve a detailed study of the dirichlet problem spectral gap analysis and energy estimates

upper bounds for the reaction front in d dimensional turbulent flow we develop an asymptotic method that yields analytic results for the upper bounds for the ensemble averaged reaction front position and speed in a d dimensional high reynolds number turbulent flow the chemical reaction is assumed to be of kolmogorov petrovskii piskunov type and the velocity is an incompressible gaussian random field in addition to the general formalism some examples are worked out in detail

a variational principle for kpp front speeds in temporally random shear flows we establish the variational principle of kolmogorov petrovsky piskunov kpp front speeds in temporally random shear flows with sufficiently decaying correlations a key quantity in the variational principle is the almost sure lyapunov exponent of a heat operator with random potential to prove the variational principle we use the comparison principle of solutions the path integral representation of solutions and large deviation estimates of the associated stochastic flows the variational principle then allows us to analytically bound the front speeds the speed bounds imply the linear growth law in the regime of large root mean square shear amplitude at any fixed temporal correlation length and the sublinear growth law if the temporal decorrelation is also large enough the so called bending phenomenon

calculating effective diffusivities in the limit of vanishing molecular diffusion in this paper we study the problem of the numerical calculation by monte carlo methods of the effective diffusivity for a particle moving in a periodic divergent free velocity field in the limit of vanishing molecular diffusion in this limit traditional numerical methods typically fail since they do not represent accurately the geometry of the underlying deterministic dynamics we propose a stochastic splitting method that takes into account the volume preserving property of the equations of motion in the absence of noise and when inertial effects can be neglected an extension of the method is then proposed for the cases where the noise has a non trivial time correlation structure and when inertial effects cannot be neglected the method of modified equations is used to explain failings of euler based methods the new stochastic geometric integrators are shown to outperform standard euler based integrators various asymptotic limits of physical interest are investigated by means of numerical experiments using the new integrators

some notes on periodic beltrami fields in cartesian geometry some mathematical facts about beltrami fields which are time independent solutions of the three dimensional incompressible euler equations with nontrivial helicity are assembled the linearization about an arbitrary fixed beltrami field is studied in a hamiltonian framework a factorization of this linearization is introduced and used to characterize null spaces for beltrami fields with ergodic streamlines an interesting property of unstable modes is noticed and its consequences concerning the nature of potential instabilities are discussed

convection of microstructure and related problems we study how two flows that vary spatially in two widely separated scales evolve under the dynamics of euler s equations we use a multiple scale approach which we motivate by studying some simpler model problems

multiscale young measures in almost periodic homogenization and applications we prove the existence of multiscale young measures associated with almost periodic homogenization we give applications of this tool in the homogenization of nonlinear partial differential equations with an almost periodic structure such as scalar conservation laws nonlinear transport equations hamilton jacobi equations and fully nonlinear elliptic equations motivated by the application in nonlinear transport equations we also prove basic results on flows generated by lipschitz almost periodic vector fields which are of interest in their own in our analysis an important role is played by the so called bohr compactification mathbb g n of mathbb r n this is a natural parameter space for the young measures our homogenization results provide also the asymptotic behavior for the whole set of mathbb g n translates of the solutions which is in the spirit of recent studies on the homogenization of stationary ergodic processes

limit theorems for large deviations and reaction diffusion equations the equation u t u xx u 1 u is the simplest reaction diffusion equation introduction of a small parameter allows construction of geometric optics approximations for the solutions of such equations these solutions are approximated by step functions with the values 0 and 1 the region where the solution is close to 1 propagates according to the huygens principle for the corresponding velocity field v x e which is calculated via the equation new effects may emerge such as stops and jumps of the wave front the feynman kac formula implies that the solutions of certain cauchy problems obey some integral equations in the space of trajectories of the corresponding markov processes examination of this equation requires the study of laplace type asymptotics for functional integrals these asymptotics are defined by large deviations for the corresponding family of processes and are expressed through action functionals

an integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection diffusion by a steady incompressible velocity field are developed here under these conditions a rigorous stieltjes integral representation for effective diffusivity in turbulent transport is derived this representation is valid for all peclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the peclet number one consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all peclet numbers a nonlocal variational principle for effective diffusivity is developed along with applications to advection diffusion by random arrays of vortices a new class of rigorous examples is introduced these examples have an explicit stieltjes measure for the effective diffusivity furthermore the effective diffusivity behaves likek0 pe 1 2 in the limit of large peclet numbers wherek0 is the molecular diffusivity formal analogies with the theory of composite materials are exploited systematically

kpp equation and supercritical branching brownian motion in the subcritical speed area application to spatial trees if r t is the position of the rightmost particle at time t in a one dimensional branching brownian motion u t x p r t x is a solution of kpp equation frac partial u partial t frac 1 2 frac partial 2 u partial x 2 f u where f u 1 u g 1 u g is the generating function of the reproduction law and the inverse of the mean lifetime if m g 1 1 and g 0 0 it is known that frac r t t xrightarrow p c 0 sqrt 2a m 1 text when t to infty for the general kpp equation we show limit theorems for u t ct c c 0 t large deviations for r t and probabilities of presence of particles for the branching process are deduced where z t denotes the random point measure of particles living at time t and a yaglom type theorem is proved the conditional distribution of the spatial tree given z t ct 0 is studied in the limit as t

interface dynamics of the fisher equation with degenerate diffusion abstract we consider a degenerate parabolic reaction diffusion equation with a monostable nonlinearity arising in population dynamics in some suitable scaling limit we prove the generation and propagation of an interface with constant normal velocity in the case that the initial condition has a convex compact support

weighted approximation for weak convex external fields abstract let w x 0 be a non negative continuous weight function which decays faster than 1 x near infinity and let q x l o g w x totik proved that if q x is convex then a continuous function f x can be approximated by weighted polynomials w x n p n x n 0 1 2 if and only if f x vanishes outside the support of the equilibrium measure associated with q x we prove a similar result in the case when q x is only weak convex

wavefront propagation for a class of degenerate reaction diffusion equation abstract we identify characterizations of wavefront locations of a class of degenerate reaction diffusion equations we give conditions under which the wavefront locations can be approximated by those of the nondegenerate ones convergence of wavefronts are proved and examples satisfying the conditions are calculated

diffusing passive tracers in random incompressible flows statistical topography aspects the paper studies statistical characteristics of the passive tracer concentrations and of its spatial gradient in random incompressible velocity fields from the viewpoint of statistical topography the statistics of interest include mean values probability distributions as well as various functionals characterizing topographic features of tracers the functional approach is used we consider the influence of the mean flow the linear shear flow and the molecular diffusion coefficient on the statistics of the tracer most of our analysis is carried out in the framework of the delta correlated in time approximation and conditions for its applicability are established but we also consider the diffusion approximation scheme for finite correlation radius the latter is applied to a diffusing passive tracer that undergoes sedimentation in a random velocity field

homogenization of a generalized stefan problem in the context of ergodic algebras abstract we address the deterministic homogenization in the general context of ergodic algebras of a doubly nonlinear problem which generalizes the well known stefan model and includes the classical porous medium equation it may be represented by the differential inclusion for a real valued function u x t t u x e x u x x e x t u u f x e x t u on a bounded domain r n t 0 t together with initial boundary conditions where z x is strictly convex and z x t u is a c 1 convex function both with quadratic growth satisfying some additional technical hypotheses as functions of the oscillatory variable x u x t u and f x t u belong to the generalized besicovitch space b 2 associated with an arbitrary ergodic algebra a the periodic case was addressed by visintin 2007 based on the two scale convergence technique visintin s analysis for the periodic case relies heavily on the possibility of reducing two scale convergence to the usual l 2 convergence in the cartesian product r n where is the periodic cell this reduction is no longer possible in the case of a general ergodic algebra to overcome this difficulty we make essential use of the concept of two scale young measures for algebras with mean value associated with bounded sequences in l 2

mathematical models with exact renormalization for turbulent transport the advection diffusion of a passive scalar by incompressible velocity fields which admit a statistical description and involve a continuous range of excited spatial and or temporal scales is very important in applications ranging from fully developed turbulence to the diffusion of tracers in heterogeneous porous media a variety of renormalization theories which typically utilize partial resummation of divergent perturbation series according to various recipes have been applied to this problem in various contexts in this paper a simple model problem for the advection diffusion of a passive scalar is introduced and the complete renormalization theory is developed with full mathematical rigor explicit formulas for the anomalous time scaling in various regimes as well as the green s function for the large scale long time ensemble average are developed here formulas for the renormalized higher order statistics are also developed the simple form of the model problem is deceptive the renormalization theory for this problem exhibits a remarkable range of different renormalization phenomena as parameters in the velocity statistics are varied these phenomena include the existence of several distinct anomalous scaling regimes as the spectral parameter tilde varepsilon is varied as well as explicit regimes in tilde varepsilon where the effective equation for the ensemble average is not a simple diffusion equation but instead involves an explicit random nonlocal eddy diffusivity we use fourier analysis and the feynman kac formula as main tools in the explicit exact renormalization theory developed here

homogenization of the nonlinear maxwell model of viscoelasticity and of the prandtl reuss model of elastoplasticity this paper deals with processes in nonlinear inelastic materials whose constitutive behaviour is represented by the inclusion here we denote by the stress tensor by e the linearized strain tensor by b x the compliance tensor and by x the subdifferential of a convex function x this relation accounts for elasto viscoplasticity including a nonlinear version of the classical maxwell model of viscoelasticity and the prandtl reuss model of elastoplasticity the constitutive law is coupled with the equation of continuum dynamics and well posedness is proved for an initial and boundary value problem the function and the tensor b are then assumed to oscillate periodically with respect to x and as this period vanishes a two scale model of the asymptotic behaviour is derived via nguetseng s notion of two scale convergence a fully homogenized single scale model is also retrieved and its equivalence with the two scale problem is proved this formulation is non local in time and is at variance with that based on so called analogical models that rest on a mean field type hypothesis

enhanced diffusivity and intercell transition layers in 2 d models of passive advection a class of two dimensional isotropic divergence free vector fields is introduced and the effective diffusivity of the corresponding advection diffusion equations is studied these examples are very idealized flows but they can be solved exactly in the limit pe 1 scaling laws d d0 pe are obtained where d0 molecular diffusion pe peclet number with exponents in the range 0 1 and examples of stream functions with logarithmic singularities for which d d0pe the exponent is related by a simple formula to the shape of the stream function along cell boundaries suggesting that similar scaling laws should hold for more general 2 d closed cell flows

diffusion coefficients as function of kubo number in random fields particle standard diffusion d k in 2d homogeneous isotropic stationary random velocity fields is studied numerically as a function of the dimensionless kubo number k defined as the ratio between the correlation time of the velocity field and its sweeping time the three cases corresponding to the energy spectra proportional in the inertial range respectively to k 3 k 5 3 and k 2 are considered and two different power law regimes are found for large but finite k

gradient systems with wiggly energies and related averaging problems gradient systems with wiggly energies of the form and a d were proposed by abeyaratne chu james 2 to study the kinetics of martensitic phase transitions their model may be recast in the framework of the theory of averaging as a dynamical system on d d with the slow variable x d and fast variable d however this problem lies completely outside the classical theory of averaging since the vertical flow on d is not ergodic for sets of positive measure and we must interpret averages to mean weak limits

a class of hamilton jacobi equations with constraint uniqueness and constructive approach we discuss a class of time dependent hamilton jacobi equations where an unknown function of time is intended to keep the maximum of the solution to the constant value 0 our main result is that the full problem has a unique viscosity solution which is in fact classical the motivation is a selection mutation model which in the limit of small diffusion exhibits concentration on the zero level set of the solution of the hamilton jacobi equation uniqueness is obtained by noticing that as a consequence of the dynamic programming principle the solution of the hamilton jacobi equation is classical it is then possible to write an ode for the maximum of the solution and treat the full problem as a nonstandard cauchy problem

homogenization and hydrodynamic limit for fermi dirac statistics coupled to a poisson equation this paper deals with the diffusion approximation of a boltzmann poisson system modeling fermi dirac statistics in the presence of an extra external oscillating electrostatic potential here we extend the analysis done in 19 to the case of a nonlinear collision operator in addition to the averaging lemma and control from entropy dissipation used in 19 here we use two scale young measures and renormalization techniques to prove the convergence this result rigorously justifies the formal analysis of 3 2015 wiley periodicals inc

a numerical study of turbulent flame speeds of curvature and strain g equations in cellular flows abstract we study front speeds of curvature and strain g equations arising in turbulent combustion these g equations are hamilton jacobi type level set partial differential equations pdes with non coercive hamiltonians and degenerate nonlinear second order diffusion the hamiltonian of a strain g equation is also non convex numerical computation is performed based on monotone discretization and weighted essentially nonoscillatory weno approximation of transformed g equations on a fixed periodic domain the advection field in the computation is a two dimensional hamiltonian flow consisting of a periodic array of counter rotating vortices or cellular flows depending on whether the evolution is predominantly in the hyperbolic or parabolic regimes suitable explicit and semi implicit time stepping methods are chosen the turbulent flame speeds are computed as the linear growth rates of large time solutions a new nonlinear parabolic pde is proposed for the reinitialization of level set functions to prevent piling up of multiple bundles of level sets on the periodic domain we found that the turbulent flame speed s t of the curvature g equation is enhanced as the intensity a of cellular flows increases at a rate between those of the inviscid and viscous g equations the s t of the strain g equation increases in small a decreases in larger a then drops down to zero at a large enough but finite value a the flame front ceases to propagate at this critical intensity a and is quenched by the cellular flow