approximate travelling waves for generalized kpp equations and classical mechanics we consider the existence of approximate travelling waves of generalized kpp equations in which the initial distribution can depend on a small parameter which in the limit 0 is the sum of some functions or a step function using the method of elworthy truman 1982 we construct a classical path which is the backward flow of a classical newtonian mechanics with given initial position and velocity before the time at which the caustic appears by the feynman kac formula and the maruyama girsanov cameron martin transformation we obtain an identity from which with a late caustic assumption we see the propagation of the global wave front and the shape of the trough our theory shows clearly how the initial distribution contributes to the propagation of the travelling wave finally we prove a huygens principle for kpp equations on complete riemannian manifolds without cut locus with some bounds on their volume element in particular cartan hadamard manifolds

a stable truncated series approximation of the reduction to the pole operator we combine a stabilized reduction to the pole and an upward continuation filter to produce meaningful reduced to the pole fields at low magnetic latitudes the stabilizing procedure is based on the development in taylor s series of the theoretical expression for the reduction to the pole filter in the wavenumber domain the filter instability is caused by the huge filter amplitudes along the magnetization azimuth which are expressed by the infinite sum of terms close to unity the stabilizing procedure reduces to simply truncating the infinite series the upward continuation filter attenuates the high wavenumber component of the noise and allows us to design a stabilized filter closer to the theoretical one besides quantitative interpretations of source depth based on the filtered field are more reliable when using upward continuation as compared with arbitrary low pass filters the proposed filter was applied to synthetic data of a single prism uniformly magnetized along a supposedly known direction

linear inviscid damping and enhanced dissipation for the kolmogorov flow abstract in this paper we prove the linear inviscid damping and vorticity depletion phenomena for the linearized euler equations around the kolmogorov flow these results confirm bouchet and morita s predictions based on numerical analysis by using the wave operator method introduced by li wei and zhang we solve beck and wayne s conjecture on the enhanced dissipation rate for the 2 d linearized navier stokes equations around the bar state called kolmogorov flow the same dissipation rate is proved for the navier stokes equations if the initial velocity is included in a basin of attraction of the kolmogorov flow with the size of 2 3 here is the viscosity coefficient

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

anomalous spreading in a system of coupled fisher kpp equations abstract in this article we report on the curious phenomena of anomalous spreading in a system of coupled fisher kpp equations when a single parameter is set to zero the system consists of two uncoupled fisher kpp equations which give rise to traveling fronts propagating with the unique minimal kpp speed when the coupling parameter is nonzero various behaviors can be observed anomalous spreading occurs when one component of the system spreads at a speed significantly faster in the coupled system than it does in isolation while the speed of the second component remains unchanged we study these anomalous spreading speeds and show that they arise due to poles of the pointwise green s function corresponding to the linearization about the unstable homogeneous state these poles lead to anomalous spreading in the linearized system and come in two varieties one that persists and leads to anomalous spreading for the nonlinear system and one that does not we describe mechanisms leading to these two behaviors and prove that one class of poles are irrelevant as far as nonlinear wavespeed selection is concerned finally we show that the same mechanism can give rise to anomalous spreading even when the slower component does not spread

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

front speed enhancement by incompressible flows in three or higher dimensions we study in dimensions n 3 the family of first integrals of an incompressible flow these are h 1 rm loc functions whose level surfaces are tangential to the streamlines of the advective incompressible field one main motivation for this study comes from earlier results proving that the existence of nontrivial first integrals of an incompressible flow q is the main key that leads to a linear speed up by a large advection of pulsating traveling fronts solving a reaction advection diffusion equation in a periodic heterogeneous framework the family of first integrals is not well understood in dimensions n 3 due to the randomness of the trajectories of q and this is in contrast with the case n 2 by looking at the domain of propagation as a union of different components produced by the advective field we provide more information about first integrals and we give a class of incompressible flows which exhibit ergodic components of positive lebesgue measure and hence are not shear flows and which under certain sharp geometric conditions speed up the kpp fronts linearly with respect to the large amplitude in the proofs we establish a link between incompressibility ergodicity first integrals and the dimension to give a sharp condition about the asymptotic behavior of the minimal kpp speed in terms of the configuration of ergodic components

quenching of flames by fluid advection we consider a simple scalar reaction advection diffusion equation with ignitiontype nonlinearity and discuss the following question what kinds of velocity profiles are capable of quenching any given flame provided the velocity s amplitude is adequately large even for shear flows the answer turns out to be surprisingly subtle if the velocity profile changes in space so that it is nowhere identically constant or if it is identically constant only in a region of small measure then the flow can quench any initial data but if the velocity profile is identically constant in a sizable region then the ensuing flow is incapable of quenching large enough flames no matter how much larger the amplitude of this velocity is the constancy region must be wider across than a couple of laminar propagating front widths the proof uses a linear pde associated to the nonlinear problem and quenching follows when the pde is hypoelliptic the techniques used allow the derivation of new nearly optimal bounds on the speed of traveling wave solutions c 2001 john wiley sons inc

reduction to the pole at low latitudes by wiener filtering using simple estimates of the signal and noise power from gridded magnetic data we design regulated frequency domain operators for reduction to the pole at low magnetic latitudes these operators suppress the artifacts along the direction of the magnetic declination associated with the conventional reduction to the pole procedure with negligible increase in computational load the new procedure is applied to produce high quality reductions to the pole for noisy low latitude synthetic data and for magnetic data from the dixon seamount

a computational study of residual kpp front speeds in time periodic cellular flows in the small diffusion limit abstract the minimal speeds c of the kolmogorov petrovsky piskunov kpp fronts at small diffusion 1 in a class of time periodic cellular flows with chaotic streamlines is investigated in this paper the variational principle of c reduces the computation to that of a principle eigenvalue problem on a periodic domain of a linear advection diffusion operator with space time periodic coefficients and small diffusion to solve the advection dominated time dependent eigenvalue problem efficiently over large time a combination of spectral methods and finite element as well as the associated fast solvers are utilized to accelerate computation in contrast to the scaling c o 1 4 in steady cellular flows a new relation c o 1 as 1 is revealed in the time periodic cellular flows due to the presence of chaotic streamlines residual propagation speed emerges from the lagrangian chaos which is quantified as a sub diffusion process

speed up of combustion fronts in shear flows this paper is concerned with the analysis of speed up of reaction diffusion advection traveling fronts in infinite cylinders with periodic boundary conditions the advection is a shear flow with a large amplitude and the reaction is nonnegative with either positive or zero ignition temperature the unique or minimal speeds of the traveling fronts are proved to be asymptotically linear in the flow amplitude as the latter goes to infinity solving an open problem from berestycki nonlinear pdes in condensed matter and reactive flows kluwer doordrecht 2003 the asymptotic growth rate is characterized explicitly as the unique or minimal speed of traveling fronts for a limiting degenerate problem and the convergence of the regular traveling fronts to the degenerate ones is proved for positive ignition temperatures under an additional hormander type condition on the flow

pulsating type entire solutions of monostable reaction advection diffusion equations in periodic excitable media abstract we establish the existence of pulsating type entire solutions of reaction advection diffusion equations with monostable nonlinearities in a periodic framework here the nonlinearities include the classic kpp case the pulsating type entire solutions are defined in the whole space and for all time t r by studying a pulsating traveling front connecting a constant unstable stationary state to a stable stationary state which is allowed to be a positive function we proved that there exist pulsating type entire solutions behaving as two pulsating traveling fronts coming from both directions and approaching each other the key techniques are to characterize the asymptotic behavior of the solutions as t in terms of appropriate subsolutions and supersolutions

fast flow asymptotics for stochastic incompressible viscous fluids in mathbb r 2 and spdes on graphs fast advection asymptotics for a stochastic reaction diffusion advection equation are studied in this paper to describe the asymptotics we introduce a new class of spdes defined on a graph corresponding to the stream function of the underlying incompressible flow we prove that as the advection term becomes faster and faster the solution of the stochastic reaction diffusion advection equation on the plane converges in a suitable sense to the solution of such an spde defined on the graph this result is a consequence of averaging due to a probabilistic representation of solutions both for the spdes on the plane and for the spde on the graph

a boundary layer theory for diffusively perturbed transport around a heteroclinic cycle we consider a two dimensional transport equation subject to small diffusive perturbations the transport equation is given by a hamiltonian flow near a compact and connected heteroclinic cycle we investigate approximately harmonic functions corresponding to the generator of the perturbed transport equation in particular we investigate such functions in the boundary layer near the heteroclinic cycle the space of these functions gives information about the likelihood of a particle moving a mesoscopic distance into one of the regions where the transport equation corresponds to periodic oscillations i e a well of the hamiltonian we find that we can construct such approximately harmonic functions which can be used as corrector functions in certain averaging questions when certain macroscopic gluing conditions are satisfied this provides a different perspective on some previous work of freidlin and wentzell on stochastic averaging of hamiltonian systems 2004 wiley periodicals inc

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

computing reactive front speeds in random flows by variational principle we study reactive front speeds in randomly perturbed cellular flows using a variational representation for the front speed we develop this representation into a computational tool for computing the front speeds without resorting to closure approximations we demonstrate that the front speeds depend on flow statistics and topologies in a complex and dramatic manner

on dynamical systems with an integral invariant on the torus abstract differential systems on the torus with an integral invariant and containing singular points are considered the main topological types of global structure of trajectories for this class of differential equations are studied and a simple criterion to determine to which type any concrete differential system belongs is also given

random perturbations of 2 dimensional hamiltonian flows we consider the motion of a particle in a periodic two dimensional flow perturbed by small molecular diffusion the flow is generated by a divergence free zero mean vector field the long time behavior corresponds to the behavior of the homogenized process that is diffusion process with the constant diffusion matrix effective diffusivity we obtain the asymptotics of the effective diffusivity when the molecular diffusion tends to zero

noise induced dissipation in lebesgue measure preserving maps on d dimensional torus we consider dissipative systems resulting from the gaussian and alpha stable noise perturbations of measure preserving maps on the d dimensional torus we study the dissipation time scale and its physical implications as the noise level e vanishes we show that nonergodic maps give rise to an o 1 e dissipation time whereas ergodic toral automorphisms including cat maps and their d dimensional generalizations have an o ln 1 e dissipation time with a constant related to the minimal dimensionally averaged entropy among the automorphism s irreducible blocks our approach reduces the calculation of the dissipation time to a nonlinear arithmetic optimization problem which is solved asymptotically by means of some fundamental theorems in theories of convexity diophantine approximation and arithmetic progression we show that the same asymptotic can be reproduced by degenerate noises as well as mere coarse graining we also discuss the implication of the dissipation time in kinematic dynamo

variational principles for complex conductivity viscoelasticity and similar problems in media with complex moduli linear processes in media with dissipation arising in conductivity optics viscoelasticity etc are considered time periodic fields in such media are described by linear differential equations for complex valued potentials the properties of the media are characterized by complex valued tensors for example by complex conductivity or complex elasticity tensors variational formulations are suggested for such problems the functionals whose euler equations coincide with the original ones are constructed four equivalent variational principles are obtained two minimax and two minimal ones the functionals of the obtained minimal variational principles are proportional to the energy dissipation averaged over the period of oscillation the last principles can be used in the homogenization theory to obtain the bounds on the effective properties of composite materials with complex valued properties tensors

witt rings and orderings of skew fields while ordered skew fields have been known for a long time they have received very little attention one purpose of this paper is to show that studying orderings and associated structures is one way of obtaining information about skew fields which are infinite dimensional over their centers all of our work is restricted to skew fields of characteristic not equal to 2 of course ordered skew fields have characteristic zero since they induce orderings on their centers the first example of a noncommutative ordered field appears to be due to d hilbert in grundlagen der geometrie 115 331 the example follows a proof of the fact that if a skew lield d has an archimedean ordering every positive element of d is less than some rational integer then d is commutative 15 321 another early result of a general nature was proved by a a albert i 1 based on earlier work of l dickson albert showed that an ordered skew field which is algebraic over its center must be commutative we shall generalize this result in the next section in 1952 t szele 28 extended to skew fields the basic results of artin and schreier on sums of squares 2 3 sums of products of squares for skew fields and a criterion for extension of orderings due to serre 1271 the second and third sections of our paper are devoted to extending these results and others to orderings of higher level in skew fields orderings of higher level were recently defined by e becker in 4 for commutative fields and appear to be of great importance in extending results for quadratic forms to forms of higher degree in particular 2 th powers take the place of squares in the artin schreier theory in fact in 5 1 he has extended this to arbitrary even powers but the valuation theory needed exceeds that which we have developed in section 3 for skew fields section 2 covers the extension to skew

laplace asymptotics for reaction diffusion equations we obtain sharp i e non logarithmic asymptotics for the solution of non homogeneous kolmogorov petrovski piskunov equation depending on a small parameter e for points ahead of the freidlin kpp front

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

localized axial green s function method for the convection diffusion equations in arbitrary domains a localized axial green s function method lagm is proposed for the convection diffusion equation the axial green s function method agm enables us to calculate the numerical solution of a multi dimensional problem using only one dimensional green s functions for the axially split differential operators this agm has been developed not only for the elliptic boundary value problems but also for the steady stokes flows however this paper is concerned with the localization of the agm this localization of the method is needed for practical purpose when computing the axial green s function specifically for the convection diffusion equation on a line segment that we call the local axial line although our focus is mainly on the convection dominated cases in arbitrary domains this method can solve other cases in a unified way numerical results show that despite irregular types of discretization on an arbitrary domain we can calculate the numerical solutions using the lagm without loss of accuracy even in cases of large convection in particular it is also shown that randomly distributed axial lines are available in our lagm and complicated domains are not a burden

using an equivalent source with positivity for low latitude reduction to the pole without striation we present a reformulation of reduction to the pole rtp of magnetic data at low latitudes and the equator using equivalent sources the proposed method addresses both the theoretical difficulty of low latitude instability and the practical issue of computational cost we prove that a positive equivalent source exists when the magnetic data are produced by normal induced magnetization and we show that the positivity is sufficient to overcome the low latitude instability in the space domain we further apply a regularization term directly to the recovered rtp field to improve the solution the use of equivalent source also naturally enables the processing of data acquired on uneven surface the result is a practical algorithm that is effective at the equatorial region and can process large scale data sets with uneven observation heights

averaging of hamiltonian flows with an ergodic component we consider a process on t2 which consists of 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 random perturbations of dynamical systems 2nd ed springer new york 1998 and mem amer math soc 109 1994 that if the stream function of the flow is periodic then the corresponding process on the graph weakly converges to a markov process we consider the situation where the stream function is not periodic and the flow when considered on the torus has an ergodic component of positive measure we show that if the rotation number is diophantine then the process on the graph still converges to a markov process which spends a positive proportion of time in the vertex corresponding to the ergodic component of the flow

the travelling wave fronts of nonlinear reaction diffusion systems via friedlin s stochastic approaches in this paper we study the asymptotic behaviour of reaction diffusion systems with a small parameter by using the n dimensional feynman kac formula and large deviation theory the generalised solutions are introduced in section 2 we obtain the travelling wave joining an unstable steady state and an asymptotically stable steady state of a diffusionless dynamical system in a reaction diffusion system with nonlinear ergodic interactions and a special case with nonlinear reducible interactions

almost sure enhanced dissipation and uniform in diffusivity exponential mixing for advection diffusion by stochastic navier stokes we study the mixing and dissipation properties of the advection diffusion equation with diffusivity 0 kappa ll 1 and advection by a class of random velocity fields on mathbb t d d 2 3 including solutions of the 2d navier stokes equations forced by sufficiently regular in space non degenerate white in time noise we prove that the solution almost surely mixes exponentially fast uniformly in the diffusivity kappa namely that there is a deterministic exponential rate independent of kappa such that all mean zero h 1 initial data decays exponentially fast in h 1 at this rate with probability one this implies almost sure enhanced dissipation in l 2 specifically that there is a deterministic uniform in kappa exponential decay in l 2 after time t rsim left log kappa right both the mathcal o left log kappa right time scale and the uniform in kappa exponential mixing are optimal for lipschitz velocity fields this work is also a major step in our program on scalar mixing and lagrangian chaos necessary for a rigorous proof of the batchelor power spectrum of passive scalar turbulence

asymmetric potentials and motor effect a large deviation approach we provide a mathematical analysis for the appearance of concentrations as dirac masses in the solutions to fokker planck systems with asymmetric potentials this problem has been proposed as a model to describe motor proteins moving along molecular filaments the components of the system describe the densities of the different conformations of the proteins our results are based on the study of a hamilton jacobi equation arising at the zero diffusion limit after an exponential transformation change of the phase function that yields a viscous hamilton jacobi equation we consider different classes of conformation transitions coefficients bounded unbounded and locally vanishing

front propagation and phase field theory the connection between the weak theories for a class of geometric equations and the asymptotics of appropriately rescaled reaction diffusion equations is rigorously established two different scalings are studied in the first the limiting geometric equation is a first order equation in the second it is a generalization of the mean curvature equation intrinsic definitions for the geometric equations are obtained and uniqueness under a geometric condition on the initial surface is proved in particular in the case of the mean curvature equation this condition is satisfied by surfaces that are strictly starshaped that have positive mean curvature or that satisfy a condition that interpolates between the positive mean curvature and the starshape conditions

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

on the euler equations of incompressible fluids euler equations of incompressible fluids use and enrich many branches of mathematics from integrable systems to geometric analysis they present important open physical and mathematical problems examples include the stable statistical behavior of ill posed free surface problems such as the rayleightaylor and kelvin helmholtz instabilities the paper describes some of the open problems related to the incompressible euler equations with emphasis on the blowup problem the inviscid limit and anomalous dissipation some of the recent results on the quasigeostrophic model are also mentioned

sharp interface limit of the fisher kpp equation we investigate the singular limit as ep to 0 of the fisher equation partial t u ep delta u ep 1 u 1 u in the whole space we consider initial data with compact support plus possibly perturbations very small as vert x vert to infty by proving both generation and motion of interface properties we show that the sharp interface limit moves by a constant speed which is the minimal speed of some related one dimensional travelling waves we obtain an estimate of the thickness of the transition layers we also exhibit initial data lq lq not so small at infinity which do not allow the interface phenomena

boundary layers for cellular flows at high peclet numbers we analyze the behavior of solutions of steady advection diffusion problems in bounded domains with prescribed dirichlet data when the peclet number pe 1 is large we show that the solution converges to a constant in each flow cell outside a boundary layer of width o 1 2 pe 1 around the flow separatrices we construct an dependent approximate water pipe problem purely inside the boundary layer that provides a good approximation of the solution of the full problem but has independent computational cost we also define an asymptotic problem on the graph of streamline separatrices and show that solution of the water pipe problem itself may be approximated by an asymptotic independent problem on this graph finally we show that the dirichlet to neumann map of the water pipe problem approximates the dirichlet to neumann map of the separatrix problem with an error independent of the flow outside the boundary layers 2004 wiley periodicals inc

existence and nonexistence of traveling waves and reaction diffusion front propagation in periodic media we review the existence results of traveling wave solutions to the reaction diffusion equations with periodic diffusion convection coefficients and combustion bistable nonlinearities we prove that whenever traveling waves exist the solutions of the initial value problem with either frontlike or pulselike data propagate with the constant effective speeds of traveling waves in all suitable directions in the case of bistable nonlinearity and one space dimension we give an example of nonexistence of traveling waves which causes quenching localization of wavefront propagation quenching localization only occurs when the variations of the media from their constant mean values are large enough our related numerical results also provide evidence for this phenomenon in the parameter regimes not covered by the analytical example finally we comment on the role of the effective wave speeds in determining the effective wavefront equation hamilton jacobi equation of the reactiondiffusion equations under the small diffusion fast reaction limit with a formal geometric optics expansion

diffusion advection in cellular flows with large peclet numbers for periodic two dimensional cellular flows in incompressible fluids we provide explicit upper and lower estimates of the effective diffusivity which have the correct scaling behavior for large peclet numbers we demonstrate that all allowed scaling laws can occur the bounds prove that there is no residual diffusion in the infinite peclet number limit for holder continuous flows answering a problem posed by kozlov

on the loss of continuity for super critical drift diffusion equations we show that there exist solutions of drift diffusion equations in two dimensions with divergence free super critical drifts that become discontinuous in finite time we consider classical as well as fractional diffusion however in the case of classical diffusion and time independent drifts we prove that solutions satisfy a modulus of continuity depending only on the local l 1 norm of the drift which is a super critical quantity

sum rules for jacobi matrices and their applications to spectral theory we discuss the proof of and systematic application of case s sum rules for jacobi matrices of special interest is a linear combination of two of his sum rules which has strictly positive terms among our results are a complete classification of the spectral measures of all jacobi matrices j for which j j 0 is hilbert schmidt and a proof of nevai s conjecture that the szego condition holds if j j 0 is trace class

convection enhanced diffusion for random flows we analyze the effective diffusivity of a passive scalar in a two dimensional steady incompressible random flow that has mean zero and a stationary stream function we show that in the limit of small diffusivity or large peclet number with convection dominating there is substantial enhancement of the effective diffusivity our analysis is based on some new variational principles for convection diffusion problems and on some facts from continuum percolation theory some of which are widely believed to be correct but have not been proved yet we show in detail how the variational principles convert information about the geometry of the level lines of the random stream function into properties of the effective diffusivity and substantiate the result of isichenko and kalda that the effective diffusivity behaves like 3 13 when the molecular diffusivity is small assuming some percolation theoretic facts we also analyze the effective diffusivity for a special class of convective flows random cellular flows where the facts from percolation theory are well established and their use in the variational principles is more direct than for general random flows

a new approach to variational problems with multiple scales we introduce a new concept the young measure on micropatterns to study singularly perturbed variational problems that lead to multiple small scales depending on a small parameter e this allows one to extract in the limit e 0 the relevant information at the macroscopic scale as well as the coarsest microscopic scale say e and to eliminate all finer scales to achieve this we consider rescaled functions rx t x s e t viewed as maps of the macroscopic variable s with values in a suitable function space the limiting problem can then be formulated as a variational problem on the young measures generated by rex as an illustration we study a one dimensional model that describes the competition between formation of microstructure and highest gradient regularization we show that the unique minimizer of the limit problem is a young measure supported on sawtooth functions with a given period 2001 john wiley sons inc

a first order asymptotic preserving scheme for front propagation in a one dimensional kinetic reaction transport equation abstract in this work we propose an asymptotic preserving scheme for a nonlinear kinetic reaction transport equation in the regime of sharp interface with a nonlinear reaction term of kpp type a phenomenon of front propagation was proven in 9 this behaviour can be highlighted by considering a suitable hyperbolic limit of the kinetic equation using a hopf cole transform it was proven in 6 8 11 that the logarithm of the distribution function then converges to the viscosity solution of a constrained hamilton jacobi equation the hyperbolic scaling and the hopf cole transform make the kinetic equation stiff thus the numerical resolution of the problem is challenging since the standard numerical methods usually lead to high computational costs in these regimes asymptotic preserving ap schemes have typically been introduced to deal with this difficulty since they are designed to be stable along the transition to the macroscopic regime the scheme we propose is adapted to the non linearity of the problem enjoys a discrete maximum principle and solves the limit equation in the sense of viscosity it is based on a dedicated micro macro decomposition attached to the hopf cole transform as it is well adapted to the singular limit our scheme is able to cope with singular behaviours in space sharp interface and possibly in velocity concentration in the velocity distribution various numerical tests are proposed to illustrate the properties and the efficiency of our scheme

boundary layers and kpp fronts in a cellular flow we study an eigenvalue problem associated with a reaction diffusion advection equation of the kpp type in a cellular flow we obtain upper and lower bounds on the eigenvalues in the regime of a large flow amplitude a 1 it follows that the minimal pulsating traveling front speed c a satisfies the upper and lower bounds c1a1 4 c a c2a1 4 physically the speed enhancement is related to the boundary layer structure of the associated eigenfunction accordingly we establish an averaging along the streamlines principle for the unique positive eigenfunction

anomalous diffusion in fast cellular flows at intermediate time scales it is well known that on long time scales the behaviour of tracer particles diffusing in a cellular flow is effectively that of a brownian motion this paper studies the behaviour on intermediate time scales before diffusion sets in various heuristics suggest that an anomalous diffusive behaviour should be observed we prove that the variance on intermediate time scales grows like o sqrt t hence on these time scales the effective behaviour can not be purely diffusive and is consistent with an anomalous diffusive behaviour

invading wave fronts and their oscillatory wakes are linked by a modulated travelling phase resetting wave abstract periodic wave trains are the generic solution from for oscillatory reaction diffusion equations in one space dimension it has been shown previously that invasive wavefronts generate behind them a wave train with a different speed from that of the invasion sherratt physica d 70 1994 370 382 in this paper the mechanism of wave train generation is studied in detail for systems of two reaction diffusion equations close to a supercritical hopf bifurcation in the kinetics with equal diffusion coefficients a combination of analytical and numerical evidence is presented suggesting that the invasive front and wave train are separated by a modulated travelling wave of phase gradient in which phase singularities occur periodically this calculation leads to a prediction of the amplitude and speed of the wave train generated by invasion

strictly physical global weak solutions of a navier stokes q tensor system with singular potential we study the existence regularity and so called strict physicality of global weak solutions of a beris edwards system which is proposed as a model for the incompressible flow of nematic liquid crystal materials an important contribution to the dynamics comes from a singular potential introduced by john ball and apala majumdar which replaces the commonly employed landau de gennes bulk potential this is built into our model to ensure that a natural physical constraint on the eigenvalues of the q tensor order parameter is respected by the dynamics of this system moreover by a maximum principle argument we are able to construct global strong solutions in dimension two

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

elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena this paper is concerned with the asymptotic behaviour of the principal eigenvalue of some linear elliptic equations in the limit of high first order coefficients roughly speaking one of the main results says that the principal eigenvalue with dirichlet boundary conditions is bounded as the amplitude of the coefficients of the first order derivatives goes to infinity if and only if the associated dynamical system has a first integral and the limiting eigenvalue is then determined through the minimization of the dirichlet functional over all first integrals a parabolic version of these results as well as other results for more general equations are given some of the main consequences concern the influence of high advection or drift on the speed of propagation of pulsating travelling fronts

solvability via viscosity solutions for a model of phase transitions driven by configurational forces abstract this article is concerned with an initial boundary value problem for an elliptic parabolic coupled system arising in martensitic phase transition theory of elastically deformable solid materials e g steel this model was proposed in alber and zhu 2007 4 and investigated in alber and zhu 2006 3 the existence of weak solutions which are defined in a standard way however the key technique used in alber and zhu 2006 3 is not applicable to multi dimensional problem one of the motivations of this study is to solve this multi dimensional problem and another is to investigate the sharp interface limits thus we define weak solutions in a way which is different from alber and zhu 2006 3 by using the notion of viscosity solution we do prove successfully the existence of weak solutions in this sense for one dimensional problem yet the multi dimensional problem is still open

exit time problems in optimal control and vanishing viscosity method the authors study the connections between deterministic exit time control problems and possibly discontinuous viscosity solutions of a first order hamilton jacobi hj equation up to the boundary this equation admits a maximum and a minimum solution that are the value functions associated to stopping time problems on the boundary when these solutions are equal they can be obtained through the vanishing viscosity method finally when the hj equation has a continuous solution it is proved to be the value function for the first exit time of the domain it is also the vanishing viscosity limit arising in particular in some large deviations problems

sharp asymptotics for kpp pulsating front speed up and diffusion enhancement by flows we study kpp pulsating front speed up and effective diffusivity enhancement by general periodic incompressible flows we prove the existence of and determine the limits c e a a and de a a2 as the flow amplitude a with c e a the minimal front speed and de a the effective diffusivity in direction e

quenching of combustion by shear flows we consider a model describing premixed combustion in the presence of fluid flow reaction diffusion equation with passive advection and ignition type nonlinearity what kinds of velocity profiles are capable of quenching suppressing any given flame provided the velocity s amplitude is adequately large even for shear flows the solution turns out to be surprisingly subtle in this paper we provide a sharp characterization of quenching for shear flows the flow can quench any initial data if and only if the velocity profile does not have an interval larger than a certain critical size where it is identically constant the efficiency of quenching depends strongly on the geometry and scaling of the flow we discuss the cases of slowly and quickly varying flows proving rigorously scaling laws that have been observed earlier in numerical experiments the results require new estimates on the behavior of the solutions to advectionenhanced diffusion equation also known as passive scalar in physical literature a classical model describing a wealth of phenomena in nature the technique involves probabilistic and pde estimates in particular applications of malliavin calculus and central limit theorem for martingales

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

long time behaviour of heat flow global estimates and exact asymptotics we obtain global upper and lower bounds on the heat kernel of an elliptic second order differential operator which become sharp in certain long time and large space asymptotics we prove a generalization of aronson s gaussian bounds which identifies correctly an effective drift for heat flow in the case of periodic coefficients we give variational characterizations of the effective conductivity which is then made to appear in heat kernel bounds these results are for heat kernels with measurable coefficients for differentiable coefficients we prove tighter estimates in which the rate of homogenization is known to be optimal

modified prufer and efgp transforms and the spectral analysis of one dimensional schrodinger operators using control of the growth of the transfer matrices wediscuss the spectral analysis of continuum and discrete half line schrodinger operators with slowly decaying potentials among our results we show if where w has compact support and then h has purely a c resp purely s c spectrum on 0 if for n 1 2 a n potentials where a n are independent identically distributed random variables with e a n 0 e a 2 n 1 and 2 we find singular continuous spectrum with explicitly computable fractional hausdorff dimension

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

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

wave front propagation and large deviations for diffusion transmutation process we study systems of reaction diffusion equations of kpp type with the coefficients and nonlinear terms slowly varying in the space variables the long time behavior of the solution to such systems can be characterized by the motion of wave fronts we describe the wave front motion using the feynman kac formula and the large deviation principle for the corresponding diffusion transmutation process we give a geometrical description of the motion in the examples and show some effects which appear in case of systems but not in the single rde s

a pde approach to small stochastic perturbations of hamiltonian flows abstract in this note we present a unified approach based on pde methods for the study of averaging principles for small stochastic perturbations of hamiltonian flows in two space dimensions such problems were introduced by freidlin and wentzell and have been the subject of extensive study in the last few years using probabilistic arguments when the hamiltonian flow has critical points it exhibits complicated behavior near the critical points under a small stochastic perturbation asymptotically the slow averaged motion takes place on a graph the issues are to identify both the equations on the sides and the boundary conditions at the vertices of the graph our approach is very general and applies also to degenerate anisotropic elliptic operators which could not be considered using the previous methodology

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

extinction versus persistence in strong oscillating flows in this paper we give some conditions for finite time extinction or persistence of the solutions of diffusion advection equations in strong and oscillating flows under dirichlet boundary conditions the enhancement of the diffusion rate depends on the interplay between strong advection and time homogenization and in particular on the ratio between the strength of the flow and its frequency parameter quantitative estimates of this ratio which depend on the geometry of the domain are provided in the case of a uniform flow in the general time space dependent case the finite time behavior of the solutions is related to the existence of first integrals of the flow

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