statistical theory for the stochastic burgers equation in the inviscid limit a statistical theory is developed for the stochastic burgers equation in the inviscid limit master equations for the probability density functions of velocity velocity difference and velocity gradient are derived no closure assumptions are made instead closure is achieved through a dimension reduction process namely the unclosed terms are expressed in terms of statistical quantities for the singular structures of the velocity field here the shocks master equations for the environment of the shocks are further expressed in terms of the statistics of singular structures on the shocks namely the points of shock generation and collisions the scaling laws of the structure functions are derived through the analysis of the master equations rigorous bounds on the decay of the tail probabilities for the velocity gradient are obtained using realizability constraints we also establish that the probability density function q of the velocity gradient decays as jj 7 2 as 1 c 2000 john wiley sons inc

symmetries of the turbulent state the emphasis of this review is on fundamental properties degree of universality and symmetries of the turbulent state the central questions are which symmetries remain broken even when the symmetry breaking factor reaches zero and which symmetries in contrast emerge in the state of developed turbulence we shall see that time reversibility is broken in all cases since turbulence is a far from equilibrium state accompanied by dissipation as far as scale invariance is concerned we argue that it is always broken in direct cascades toward small scales no matter how far one goes away from the pumping scale in contrast inverse cascades become scale invariant as they go toward large scales moreover some properties of the inverse cascades seem to be conformal invariant and related to schramm loewner evolution a class of random curves that can be mapped to a 1d brownian walk

statistical properties of shocks in burgers turbulence we consider the statistical properties of solutions of burgers equation in the limit of vanishing viscosity frac partial partial t u left x t right frac partial partial x left frac 1 2 u left x t right 2 right 0 with gaussian whitenoise initial data this system was originally proposed by burgers 1 as a crude model of hydrodynamic turbulence and more recently by zel dovichet al 12 to describe the evolution of gravitational matter at large spatio temporal scales with shocks playing the role of mass clusters we present here a rigorous proof of the scaling relationp s s 1 2 s 1 wherep s is the cumulative probability distribution of shock strengths we also show that the set of spatial locations of shocks is discrete i e has no accumulation points and establish an upper bound on the tails of the shock strength distribution namely 1 p s exp cs 3 fors 1 our method draws on a remarkable connection existing between the structure of burgers turbulence and classical probabilistic work on the convex envelope of brownian motion and related diffusion processes

statistics of shocks in solutions of inviscid burgers equation the purpose of this paper is to analyze statistical properties of discontinuities of solutions of the inviscid burgers equation having a typical realizationb y of the brownian motion as an initial datum this case was proposed and studied numerically in the companion paper by she aurell and frisch the description of the statistics is given in terms of the behavior of the convex hull of the random process w y int limits 0 y b eta eta d eta the hausdorff dimension of the closed set of thosey where the convex hull coincides withw is also studied

the one point statistics of viscous burgers turbulence initialized with gaussian data we study the statistics of the viscous burgers turbulence bt model initialized at time t 0 by a large class of gaussian data using a first principles analysis of the hopf cole formula for the burgers equation and the theory of large deviations for gaussian processes we characterize the tails of the probability distribution functions pdfs for the velocity u x t and the velocity derivatives the pdf tails have a non universal structure of the form where re is the reynolds number and p q and r depend on the order of differentiation and the infrared behavior of the initial energy spectrum

mean field approximation and a small parameter in turbulence theory numerical and physical experiments on two dimensional 2d turbulence show that the differences of transverse components of velocity field are well described by gaussian statistics and kolmogorov scaling exponents in this case the dissipation fluctuations are irrelevant in the limit of small viscosity in general one can assume the existence of a critical space dimensionality d d c at which the energy flux and all odd order moments of velocity difference change sign and the dissipation fluctuations become dynamically unimportant at dl d c the flow can be described by the mean field theory leading to the observed gaussian statistics and kolmogorov scaling of transverse velocity differences it is shown that in the vicinity of d d c the ratio of the relaxation and translation characteristic times decreases to zero thus giving rise to a small parameter of the theory the expressions for pressure and dissipation contributions to the exact equation for the generating function of transverse velocity differences are derived in the vicinity of d d c the resulting equation describes experimental data on two dimensional turbulence and demonstrates the onset of intermittency as d ensuremath d c g0 and r stackrel ensuremath rightarrow l 0 in three dimensional flows in close agreement with experimental data in addition some exact relations between correlation functions of velocity differences are derived it is also predicted that the single point probability density function of transverse velocity components in developing as well as in the large scale stabilized two dimensional turbulence is a gaussian

statistics of intense turbulent vorticity events we investigate statistical properties of vorticity fluctuations in fully developed turbulence which are known to exhibit a strong intermittent behavior taking as the starting point the navier stokes equations with a random force term correlated at large scales we obtain in the high reynolds number regime a closed analytical expression for the probability distribution function of an arbitrary component of the vorticity field the central idea underlying the analysis consists in the restriction of the velocity configurational phase space to a particular sector where the rate of strain and the rotation tensors can be locally regarded as slow and fast degrees of freedom respectively this prescription is implemented along the martin siggia rose functional framework whereby instantons and perturbations around them are taken into account within a steepest descent approach

constructive existence and uniqueness for two point boundary value problems with a linear gradient term constructive existence and uniqueness theorems are established for nonlinear two point boundary value problems governed by the equation y f x y p x y 0

minimum action method for the kardar parisi zhang equation we apply a numerical minimum action method derived from the wentzell freidlin theory of large deviations to the kardar parisi zhang equation for the height profile of a growing interface in one dimension we find that the transition pathway between different height configurations is determined by the nucleation and subsequent propagation of facets or steps corresponding to moving domain walls or growth modes in the underlying noise driven burgers equation this transition scenario is in accordance with recent analytical studies of the one dimensional kardar parisi zhang equation in the asymptotic weak noise limit we also briefly discuss transitions in two dimensions

exact relation between eulerian and lagrangian velocity increment statistics we present a formal connection between lagrangian and eulerian velocity increment distributions which is applicable to a wide range of turbulent systems ranging from turbulence in incompressible fluids to magnetohydrodynamic turbulence for the case of the inverse cascade regime of two dimensional turbulence we numerically estimate the transition probabilities involved in this connection in this context we are able to directly identify the processes leading to strongly non gaussian statistics for the lagrangian velocity increments

singularities and the distribution of density in the burgers adhesion model abstract we are interested in the tail behavior of the pdf of mass density within the one and d dimensional burgers adhesion model used e g to model the formation of large scale structures in the universe after baryon photon decoupling we show that large densities are localized near kurtoparabolic singularities residing on space time manifolds of codimension two d 2 or higher d 3 for smooth initial conditions such singularities are obtained from the convex hull of the lagrangian potential the initial velocity potential minus a parabolic term the singularities contribute universal power law tails to the density pdf when the initial conditions are random in one dimension the singularities are preshocks nascent shocks whereas in two and three dimensions they persist in time and correspond to boundaries of shocks in all cases the corresponding density pdf has the exponent 7 2 originally proposed by e et al phys rev lett 78 1997 1904 for the pdf of velocity gradients in one dimensional forced burgers turbulence we also briefly consider models permitting particle crossings and thus multi stream solutions such as the zel dovich approximation and the jeans vlasov poisson equation with single stream initial data they have singularities of codimension one yielding power law tails with exponent 3

towards a dynamical theory of multifractals in turbulence abstract making use of the exact equations for structure functions supplemented by the equations for dissipative anomaly as well as an estimate for the lagrangian acceleration of fluid particles we obtain a main result of the multifractal theory of turbulence the central element of the theory is a dissipation cut off that depends on the order of the structure function an expression obtained for the exponents s n in the scaling relations u x n u x 2 n 2 re s n between the velocity gradients u x and the reynolds number re agrees well with experimental data

a preliminary study of the burgers equation with symbolic computation a novel approach based on recursive symbolic computation is introduced for the approximate analytic solution of the burgers equation once obtained appropriate numerical values can be inserted into the symbolic solution to explore parametric variations the solution is valid for both inviscid and viscous cases covering the range of reynolds number from 500 to infinity whereas current direct numerical simulation dns methods are limited to reynolds numbers no greater than 4000 what further distinguishes the symbolic approach from numerical and traditional analytic techniques is the ability to reveal and examine direct nonlinear interactions between waves including the interplay between inertia and viscosity thus preliminary efforts suggest that symbolic computation may be quite effective in unveiling the anatomy of the myriad interactions that underlie turbulent behavior however due to the tendency of nonlinear symbolic operations to produce combinatorial explosion future efforts will require the development of improved filtering processes to select and eliminate computations leading to negligible high order terms indeed the initial symbolic computations present the character of turbulence as a problem in combinatorics at present results are limited in time evolution but reveal the beginnings of the well known saw tooth waveform that occurs in the inviscid case i e re future efforts will explore more fully developed 1 d flows and investigate the potential to extend symbolic computations to 2 d and 3 d potential applications include the development of improved subgrid scale sgs parameterizations for large eddy simulation les models and studies that complement dns in exploring fundamental aspects of turbulent flow behavior

instanton theory of burgers shocks and intermittency a lagrangian approach to burgers turbulence is carried out along the lines of the field theoretical martin siggia rose formalism of stochastic hydrodynamics we derive from an analysis based on the hypothesis of unbroken galilean invariance the asymptotic form of the probability distribution function of negative velocity differences the origin of burgers intermittency is found to rely on the dynamical coupling between shocks identified to instantons and noncoherent background fluctuations which char22 then char22 cannot be discarded in a consistent statistical description of the flow

phase diagram of optimal paths we show that by choosing appropriate distributions of the randomness the search for optimal paths links diverse problems of disordered media such as directed percolation invasion percolation and directed and nondirected spanning polymers we also introduce a simple and efficient algorithm which solves the d dimensional model numerically in o n 1 df d steps where df is the fractal dimension of the path using extensive simulations in two dimensions we identify the phase boundaries of the directed polymer universality class a new strong disorder phase occurs where the optimum paths are self affine with parameter dependent scaling exponents furthermore the phase diagram contains directed and nondirected percolation as well as the directed random walk models at specific points and lines

instantons in the burgers equation the instanton solution for the forced burgers equation is found this solution describes the exponential tail of the probability distribution function of velocity differences in the region where shock waves are absent that is for large positive velocity differences the results agree with the one found recently by polyakov who used the operator product conjecture if this conjecture is true then our wkb asymptotics of the wyld functional integral should be exact to all orders of perturbation expansion around the instanton solution we also generalized our solution for the arbitrary dimension of the burgers kpz equation as a result we found the asymptotics of the angular dependence of the velocity difference probability distribution function textcopyright 1996 the american physical society

dynamical pruning of rooted trees with applications to 1 d ballistic annihilation we introduce generalized dynamical pruning on rooted binary trees with edge lengths that encompasses a number of discrete and continuous pruning operations including the tree erasure and horton pruning the pruning removes parts of a tree t starting from the leaves according to a pruning function defined on descendant subtrees within t we prove the invariance of critical binary galton watson tree with exponential edge lengths with respect to the generalized dynamical pruning for an arbitrary admissible pruning function these results facilitate analysis of the continuum 1 d ballistic annihilation model a a rightarrow varnothing for a constant particle density and initial velocity that alternates between the values of pm 1 we show that the model s shock wave is isometric to the level set tree of the potential function and the model evolution is equivalent to the generalized dynamical pruning of the shock wave tree

universality classes for self similarity of noiseless multidimensional burgers turbulence and interface growth the present work is devoted to the evolution of random solutions of the unforced burgers and kpz equations in d dimensions in the limit of vanishing viscosity we consider a cellular model and as initial condition assign a value for the velocity potential chosen independently within each cell we show that the asymptotic behavior of the turbulence at large times is determined by the tail of the initial potential probability distribution function three classes of initial distribution leading to self similar evolution are identified a distributions with a power law tail b compactly supported potential c stretched exponential tails in class c we find that the mean potential mean height of the surface increases logarithmically with time and the turbulence energy e t mean square gradient of the surface decays as 1 t times a logarithmic correction in classes a and b we find that the changes in the mean potential and energy have a power law time dependence in class c the roughness of the surface measured by its mean square gradient may either decrease or increase with time we discuss also the influence of finite viscosity and long range correlation on the late stage evolution of the burgers turbulence

a perturbation theory for ergodic markov chains and application to numerical approximations perturbations to markov chains and markov processes are considered the unperturbed problem is assumed to be geometrically ergodic in the sense usually established through the use of foster lyapunov drift conditions the perturbations are assumed to be uniform in a weak sense on bounded time intervals the long time behavior of the perturbed chain is studied applications are given to numerical approximations of a randomly impulsed ode an ito stochastic differential equation sde and a parabolic stochastic partial differential equation spde subject to space time brownian noise existing perturbation theories for geometrically ergodic markov chains are not readily applicable to these situations since they require very stringent hypotheses on the perturbations

kolmogorov turbulence in a random force driven burgers equation anomalous scaling and probability density functions high resolution numerical experiments described in this work show that velocity fluctuations governed by the one dimensional burgers equation driven by a white in time random noise with the spectrum ensuremath vert f k mathrm ensuremath vert 2 ifmmode bar else textasciimacron fi ensuremath propto mathit k mathrm ensuremath 1 exhibit a biscaling behavior all moments of velocity differences mathit s mathit n mathrm ensuremath le 3 r ensuremath vert u x r u x mathrm ensuremath vert mathit n ifmmode bar else textasciimacron fi ensuremath equiv ensuremath vert ensuremath delta u mathrm ensuremath vert mathit n ifmmode bar else textasciimacron fi ensuremath propto mathit r mathit n 3 while mathit s mathit n g3 r ensuremath propto mathit r mathit n ensuremath xi with ensuremath xi mathit n ensuremath approxeq 1 for real ng0 chekhlov and yakhot phys rev e 51 r2739 1995 the probability density function which is dominated by coherent shocks in the interval ensuremath delta u0 is scrp ensuremath delta u r ensuremath propto ensuremath delta u mathrm ensuremath mathit q with q ensuremath approxeq 4 a phenomenological theory describing the experimental findings is presented

a novel method for nonlinear boundary value problems in this paper a novel numerical method is proposed for solving nonlinear boundary value problems this method is based on a combination of the reproducing kernel and least squares methods so the accuracy of this method is improved we have proved the uniform convergence of the approximate solution u n by convergence of u n in w 3 a b numerical results obtained by using the scheme show that the numerical scheme is feasible and effective

nagumo type existence results for second order nonlinear dynamic bvps abstract in this paper we present a nagumo condition for time scales equations that includes as a particular case a type of the classical one for differential equations involving a certain restriction the established nagumo condition further allows us to prove the existence of at least one solution lying between a pair of lower and upper solutions of a nonlinear second order dynamic equation with nonlinear functional boundary value conditions

systems of difference equations associated with boundary value problems for second order systems of ordinary differential equations we establish existence results for solutions to boundary value problems for systems of second order difference equations associated with systems of second order ordinary differential equations subject to nonlinear boundary conditions

two results concerning asymptotic behavior of solutions of the burgers equation with force we consider the burgers equation with an external force for the case of the force periodic in space and time we prove the existence of a solution periodic in space and time which is the limit of a wide class of solutions ast if the force is the product of a periodic function ofx and white noise in time we prove the existence of an invariant distribution concentrated on the space of space periodic functions which is the limit of a wide class of distributions ast

velocity difference probability density functions for burgers turbulence in this paper the polyakov equation phys rev e bold 52 6183 1995 for the velocity difference probability density functions with the random gaussian external force with the correlation function kappa y approximately 1 minus y sup alpha is analyzed solutions for the cases alpha l brace 2 1 2 1 r brace are found which agree very well with available numerical results it is also argued that the stationary regime of burgers turbulence can depend not only on the distribution of the external force but also on the dissipative regularization copyright ital 1997 ital the american physical society

existence of solutions for a two point boundary value problem on time scales let t be a time scale let f 0 s 1 xr 2 r be continuous we study the existence of solutions for the nonlinear two point boundary value problems on time scalex t f t x t x t email protected 0 1 x 0 0 x s 1 0 we do not need any growth restrictions on f

sailing the deep blue sea of decaying burgers turbulence we study lagrangian trajectories and scalar transport statistics in decaying burgers turbulence we choose velocity fields solutions of the inviscid burgers equation whose probability distributions are specified by kida s statistics they are time correlated and neither time reversal invariant nor gaussian we discuss in some detail the effect of shocks on trajectories and transport equations we derive the inviscid limit of these equations using a formalism of operators localized on shocks we compute the probability distribution functions of the trajectories although they do not define markov processes as physically expected these trajectories are statistically well defined but collapse with probability one at infinite time we point out that the advected scalars enjoy inverse energy cascades we also make a few comments on the connection between our computations and persistence problems

dynamical equations for high order structure functions and a comparison of a mean field theory with experiments in three dimensional turbulence two recent papers v yakhot phys rev e 63 026307 2001 and r j hill j fluid mech 434 379 2001 derive through two different approaches that have the navier stokes equations as the common starting point a set of dynamic equations for structure functions of arbitrary order in turbulence these equations are not closed yakhot proposed a mean field theory to close the equations for locally isotropic turbulence and obtained scaling exponents of structure functions and expressions for the peak in the probability density function of transverse velocity increments and for its behavior for intermediate amplitudes at high reynolds numbers some relevant experimental data on pressure gradient and dissipation terms are presented that are needed to provide closure as well as on other aspects predicted by the theory comparison between the theory and the data shows varying levels of agreement and reveals gaps inherent to the implementation of the theory

markovian solutions of inviscid burgers equation for solutions of inviscid forceless one dimensional burgers equation with random initial condition it is heuristically shown that a stationary feller markov property with respect to the space variable at some time is conserved at later times and an evolution equation is derived for the infinitesimal generator previously known explicit solutions such as frachebourg martin s white noise initial velocity and carraro duchon s levy process intrinsic statistical solutions including brownian initial velocity are recovered as special cases

velocity fluctuations in forced burgers turbulence burgers turbulence the tails of the velocity gradient distribution in the regions where there are no shocks we evaluate the right tail through a rather simple computation and compare it to the more sophisticated approaches developed recently 7 3 6 we then give a conjecture on the left tail which is based on a plausible argument requiring the system to reach a stationary state we shall first discuss the one dimensional case then turn to higher dimensions and compare our results with the previously available ones

beyond scaling and locality in turbulence an analytic perturbation theory is suggested in order to find finite size corrections to the scaling power laws in the frame of this theory it is shown that the first order finite size correction to the scaling power laws has following form s r cong cr alpha 0 ln r eta alpha 1 where is a finite size scale in particular for turbulence it can be the kolmogorov dissipation scale using data of laboratory experiments and numerical simulations it is shown shown that a degenerate case with 0 0 can describe turbulence statistics in the near dissipation range r where the ordinary power law scaling does not apply for moderate reynolds numbers the degenerate scaling range covers almost the entire range of scales of velocity structure functions the log corrections apply to finite reynolds number interplay between local and non local regimes has been considered as a possible hydrodynamic mechanism providing the basis for the degenerate scaling of structure functions and extended self similarity these results have been also expanded on passive scalar mixing in turbulence overlapping phenomenon between local and non local regimes and a relation between position of maximum of the generalized energy input rate and the actual crossover scale between these regimes are briefly discussed

topological transversality and boundary value problems on time scales this work presents some results for the existence of solutions to boundary value problems on time scales the ideas rely on the topological transversality of a granas

viscous limits for piecewise smooth solutions to systems of conservation laws in this paper we study the zero dissipation problem for a general system of conservation laws with positive viscosity it is shown that if the solution of the problem with zero viscosity is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition then there exist solutions to the corresponding system with viscosity that converge to the solutions of the system without viscosity away from shock discontinuities at a rate of order e as the viscosity coefficient e goes to zero the proof uses a matched asymptotic analysis and an energy estimate related to the stability theory for viscous shock profiles

decaying turbulence in the generalised burgers equation we consider the generalised burgers equation frac partial u partial t f u frac partial u partial x nu frac partial 2 u partial x 2 0 t geqq 0 x in s 1 where f is strongly convex and is small and positive we obtain sharp estimates for sobolev norms of u upper and lower bounds differ only by a multiplicative constant then we obtain sharp estimates for the dissipation length scale and the small scale quantities which characterise the decaying burgers turbulence i e the structure functions and the energy spectrum the proof uses a quantitative version of an argument by aurell et al j fluid mech 238 467 486 1992 note that we are dealing with decaying as opposed to stationary turbulence thus our estimates are not uniform in time however they hold on a time interval t1 t2 where t1 and t2 depend only on f and the initial condition and do not depend on the viscosity these results allow us to obtain a rigorous theory of the one dimensional burgers turbulence in the spirit of kolmogorov s 1941 theory in particular we obtain two results which hold in the inertial range on one hand we explain the bifractal behaviour of the moments of increments or structure functions on the other hand we obtain an energy spectrum of the form k 2 these results remain valid in the inviscid limit

statistical analysis and simulation of random shocks in stochastic burgers equation we study the statistical properties of random shock waves in stochastic burgers equation subject to random space time perturbations and random initial conditions by using the response excitation probability density function pdf method and the mori zwanzig mz formulation of irreversible statistical mechanics we derive exact reduced order equations for the one point and two point pdfs of the solution field in particular we compute the statistical properties of random shock waves in the inviscid limit by using an adaptive shock capturing discontinuous galerkin method in both physical and probability spaces we consider stochastic flows generated by high dimensional random initial conditions and random additive forcing terms yielding multiple interacting shock waves collapsing into clusters and settling down to a similarity state we also address the question of how random shock waves in space and time manifest themselves in probability space the proposed new mathematical framework can be applied to different conservation laws potentially leading to new insights into high dimensional stochastic dynamical systems and more efficient computational algorithms

inverse cascade and intermittency of passive scalar in one dimensional smooth flow random advection of a lagrangian tracer scalar field ensuremath theta t x by a one dimensional spatially smooth and short correlated in time velocity field is considered scalar fluctuations are maintained by a source concentrated at the integral scale l the statistical properties of both scalar differences and the dissipation field are analytically determined exploiting the dynamical formulation of the model the gaussian statistics known to be present at small scales for incompressible velocity fields emerges here at large scales x ensuremath gg l these scales are shown to be excited by an inverse cascade of ensuremath theta 2 and the probability distribution function pdf of the corresponding scalar differences to approach the gaussian form as larger and larger scales are considered small scale x ensuremath ll l statistics is shown to be strongly non gaussian a collapse of scaling exponents for scalar structure functions takes place moments of order pg 1 all scale linearly independently of the order p smooth scaling x p is found for ensuremath 1lpl1 tails of the scalar difference pdf are exponential while at the center a cusped shape tends to develop when smaller and smaller ratios x l are considered the same tendency is present for the scalar gradient pdf with respect to the inverse of the p eclet number the pumping to diffusion scale ratio the tails of the latter pdf are however much more extended decaying as a stretched exponential of exponent 2 3 smaller than unity this slower decay is physically associated with the strong fluctuations of the dynamical dissipative scale

intermittency of height fluctuations in stationary state of the kardar parisi zhang equation with infinitesimal surface tension in 1 1 dimensions the kardar parisi zhang kpz equation with infinitesimal surface tension dynamically develops sharply connected valley structures within which the height derivative is not continuous we discuss the intermittency issue in the problem of stationary state forced kpz equation in 1 1 dimensions it is proved that the moments of height increments c a behave as x 1 x 2 xi a with xi a a for length scales x 1 x 2 sigma the length scale sigma is the characteristic length of the forcing term we have checked the analytical results by direct numerical simulation

turbulence without pressure we develop exact field theoretic methods to treat turbulence when the effect of pressure is negligible we find explicit forms of certain probability distributions demonstrate that the breakdown of galilean invariance is responsible for intermittency and establish the operator product expansion we also indicate how the effects of pressure can be turned on perturbatively

probability distribution functions of derivatives and increments for decaying burgers turbulence a lagrangian method is used to show that the power law with a ensuremath 7 2 exponent in the negative tail of the probability distribution function pdf of the velocity gradient and of velocity increments predicted by e et al phys rev lett 78 1904 1997 for forced burgers turbulence is also present in the unforced case the theory is extended to the second order space derivative whose pdf has power law tails with exponent ensuremath 2 at both large positive and negative values and to the time derivatives pdf s of space and time derivatives have the same asymptotic functional forms this is interpreted in terms of a random taylor hypothesis

kinetic theory and lax equations for shock clustering and burgers turbulence we study shock statistics in the scalar conservation law t u x f u 0 x t 0 with a convex flux f and spatially random initial data we show that the markov property in x is preserved for a large class of random initial data markov processes with downward jumps and derivatives of levy processes with downward jumps the kinetics of shock clustering is then described completely by an evolution equation for the generator of the markov process u x t x we present four distinct derivations for this evolution equation and show that it takes the form of a lax pair the lax equation admits a spectral parameter as in manakov funct anal appl 10 328 329 1976 and has remarkable exact solutions for burgers equation f u u 2 2 this suggests the kinetic equations of shock clustering are completely integrable

levy process intrinsic statistical solutions of a randomly forced burgers equation we consider burgers equation forced by a brownian in space and white noise in time process partial t u frac 1 2 partial x u 2 f x t with e f x t f y s frac 1 2 x y x y delta t s and we show that there exist intrinsic statistical solutions that are levy processes at any given positive time we give the evolution equation for the characteristic exponent of such solutions in particular we give the explicit solution in the case u0 x 0

the inviscid burgers equation with initial data of brownian type the solutions to burgers equation in the limit of vanishing viscosity are investigated when the initial velocity is a brownian motion or fractional brownian motion function i e a gaussian process with scaling exponent 0 h 1 typea or the derivative thereof with scaling exponent 1 h 0 typeb largesize numerical experiments are performed helped by the fact that the solution is essentially obtained by performing a legendre transform the main result is obtained for typea and concerns the lagrangian functionx a which gives the location at timet 1 of the fluid particle which started at the locationa it is found to be a complete devil s staircase the cumulative probability of lagrangian shock intervals a also the distribution of shock amplitudes follows a a h law for small a the remaining regular lagrangian locations form a cantor set of dimensionh in eulerian coordinates the shock locations are everywhere dense the scaling properties of various statistical quantities are also found heuristic interpretations are provided for some of these results rigorous results for the case of brownian motion are established in a companion paper by ya sinai for typeb initial velocities e g white noise there are very few small shocks and shock locations appear to be isolated finally it is shown that there are universality classes of random but smooth non scaling initial velocities such that the long time large scale behavior is after rescaling the same as for typea orb

stochastic heat and burgers equations and their singularities i geometrical properties arnol d and thom s beautiful classification of caustics shockwaves for the burgers equation suggests a similar one for the wavefronts of the corresponding heat equation we give here a general theorem for hamiltonian systems characterizing how the level surfaces of hamilton s principal function wavefronts meet the caustic surface in both the deterministic and stochastic cases we further show how these results can be applied to the stochastic burgers equation by using earlier results of truman and zhao the generic example of a caustic appearing in the two dimensional case is the semicubical parabolic cusp with the corresponding zero level surface being a combination of a generalized hypocycloid and a line pair we refer to these as the cusp and tricorn the analogous butterfly caustic in the three dimensional case has a cusped zero level surface the fish which meets the butterfly caustic in three cusped curves and touches it along a straight line our results explain in terms of classical mechanics the properties of the caustic and wavefront for these two archetypal examples and characterize the caustic wavefront intersection for the general stochastic case we discuss the application of these results to turbulence for the burgers velocity field

multidimensional potential burgers turbulence we consider the multidimensional generalised stochastic burgers equation in the space periodic setting frac partial mathbf u partial t nabla f mathbf u cdot nabla mathbf u nu delta mathbf u nabla eta quad t geq 0 mathbf x in mathbb t d mathbb r mathbb z d under the assumption that u is a gradient here f is strongly convex and satisfies a growth condition is small and positive while is a random forcing term smooth in space and white in time for solutions u of this equation we study sobolev norms of u averaged in time and in ensemble each of these norms behaves as a given negative power of these results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence namely the averages of the increments and of the energy spectrum these quantities behave as a power of the norm of the relevant parameter which is respectively the separation l in the physical space and the wavenumber k in the fourier space our bounds do not depend on the initial condition and hold uniformly in nu we generalise the results obtained for the one dimensional case in 10 confirming the physical predictions in 4 30 note that the form of the estimates does not depend on the dimension the powers of nu mathbf k ell are the same in the one and the multi dimensional setting

stochastic heat and burgers equations and their singularities ii analytical properties and limiting distributions we study the inviscid limit 0 of the stochastic viscous burgers equation for the velocity field v x t t 0 x rd v t v v c x t k x t wt 2 2 v for small with v x 0 s0 x for some given s0 wt representing white noise here we use the hopf cole transformation v 2 lnu where u satisfies the stochastic heat equation of stratonovich type and the feynmac kac truman zhao formula for u where dut x 2 2 ut x 2c x t ut x dt 2k x t ut x dwt with u0 x t0 x exp s0 x 2 s0 as before and t0 a smooth positive function in an earlier paper davies truman and zhao j math phys 43 3293 2002 an exact solution of the stochastic viscous burgers equation was used to show how the formal blow up of the burgers velocity field occurs on random shockwaves for the v 0 solution of burgers equation coinciding with the caustics of a corresponding hamiltonian system with classical flow map moreover the u 0 solution of the stochastic heat equation has its

spontaneous stochasticity and anomalous dissipation for burgers equation we develop a lagrangian approach to conservation law anomalies in weak solutions of inviscid burgers equation motivated by previous work on the kraichnan model of turbulent scalar advection we show that the entropy solutions of burgers possess markov stochastic processes of generalized lagrangian trajectories backward in time for which the burgers velocity is a backward martingale this property is shown to guarantee dissipativity of conservation law anomalies for general convex functions of the velocity the backward stochastic burgers flows with these properties are not unique however we construct infinitely many such stochastic flows both by a geometric construction and by the zero noise limit of the constantin iyer stochastic representation of viscous burgers solutions the latter proof yields the spontaneous stochasticity of lagrangian trajectories backward in time for burgers at unit prandtl number it is conjectured that existence of a backward stochastic flow with the velocity as martingale is an admissibility condition which selects the unique entropy solution for burgers we also study linear transport of passive densities and scalars by inviscid burgers flows we show that shock solutions of burgers exhibit spontaneous stochasticity backward in time for all finite prandtl numbers implying conservation law anomalies for linear transport we discuss the relation of our results for burgers with incompressible navier stokes turbulence especially lagrangian admissibility conditions for euler solutions and the relation between turbulent cascade directions and time asymmetry of lagrangian stochasticity

existence of solutions to boundary value problems for dynamic systems on time scales abstract here we investigate systems of boundary value problems for dynamic equations on time scales using a generalized relationship between the boundary conditions and a certain subset of the solution space the existence of solutions is established through topological arguments the main tools used are leray schauder and brouwer degree theory

hyperbolic asymptotics in burgers turbulence and extremal processes large time asymptotics of statistical solutionu t x 1 2 of the burgers equation 1 1 is considered where x l x is a stationary zero mean gaussian process depending on a large parameterl 0 so that xi l x sim sigma l eta x l l to infty where sigma l l 2 2 log l 1 2 and x is a given standardized stationary gaussian process we prove that asl the hyperbolicly scaled random fieldsu l2t l2x converge in distribution to a random field with saw tooth trajectories defined by means of a poisson process on the plane related to high fluctuations of x which corresponds to the zero viscosity solutions at the physical level of rigor such asymptotics was considered before by gurbatov malakhov and saichev 1991

sle on doubly connected domains and the winding of loop erased random walks two dimensional loop erased random walks lerws are random planar curves whose scaling limit is known to be a schramm loewner evolution sle with parameter 2 in this note some properties of an sle trace on doubly connected domains are studied and a connection to passive scalar diffusion in a burgers flow is emphasised in particular the endpoint probability distribution and winding probabilities for sle2 on a cylinder starting from one boundary component and stopped when hitting the other are found a relation of the result to conditioned one dimensional brownian motion is pointed out moreover this result permits to study the statistics of the winding number for sle2 with fixed endpoints a solution for the endpoint distribution of sle4 on the cylinder is obtained and a relation to reflected brownian motion pointed out

burgers turbulence and random lagrangian systems we consider a spatially periodic inviscid random forced burgers equation in arbitrary dimension and the random time dependent lagrangian system related to it we construct a unique stationary distribution for viscosity solutions of the burgers equation we also show that with probability 1 there exists a unique minimizing trajectory for the random lagrangian system which generates a non trivial ergodic invariant measure for the non random skew product extension of the lagrangian system

stationary measures for a randomly forced burgers equation we study a randomly forced burgers equation and its corresponding hamilton jacobi equation on the line the forcing is of the form of a randomly modulated and shifted potential we prove the existence of invariant probability measures and provide examples for which these measures are not unique these measures do exhibit certain ergodic behavior the methods we use are closely connected to the lax ole inik variational principle 2004 wiley periodicals inc

a numerical solution to nonlinear second order three point boundary value problems in the reproducing kernel space abstract in this paper a new numerical algorithm is provided to solve nonlinear three point boundary value problems in a very favorable reproducing kernel space which satisfies all boundary conditions its reproducing kernel function is discussed in detail we also prove that the approximate solution and its first and second order derivatives all converge uniformly the numerical experiments show that the algorithm is quite accurate and efficient for solving nonlinear second order three point boundary value problems

basic qualitative and quantitative results for solutions to nonlinear dynamic equations on time scales with an application to economic modelling abstract this article investigates both basic qualitative and basic quantitative properties of solutions to first and higher order dynamic equations on time scales and thus provides a foundation and framework for future advanced nonlinear studies in the field particular focus lies in the existence uniqueness dependency approximation and explicit representation of solutions to nonlinear initial value problems the main tools used are from modern areas of nonlinear analysis including the fixed point theorems of banach and schafer the method of successive approximations a novel definition of measuring distance in metric spaces and normed spaces and a separation of variables technique is introduced to the general time scale setting the new results compliment and extend those of stefan hilger s seminal paper of 1990 as an application of the new results we present and analyse a simple model from economics known as the keynesian cross model with lagged income in the general time scale environment ideas suggesting further applications and possible new directions for the novel results are also presented

solving boundary value problems of ordinary differential equations with non separated boundary conditions abstract this paper focuses on solving the two point boundary value problem in which boundary conditions are systems of nonlinear equations the shooting method was used together with a combination of newton s method and broyden s method to update the initial values of the differential equations the experiments showed that the proposed method performed well in the sense that the overall amount of work was less than that of the newton shooting method

instantons and intermittency we describe the method for finding the non gaussian tails of the probability distribution function pdf for solutions of a stochastic differential equation such as the convection equation for a passive scalar the random driven navier stokes equation etc the existence of such tails is generally regarded as a manifestation of the intermittency phenomenon our formalism is based on the wkb approximation in the functional integral for the conditional probability of large fluctuation we argue that the main contribution to the functional integral is given by a coupled field force configuration char22 the instanton as an example we examine the correlation functions of the passive scalar u advected by a large scale velocity field ensuremath delta correlated in time we find the instanton determining the tails of the generating functional and show that it is different from the instanton that determines the probability distribution function of high powers of u we discuss the simplest instantons for the navier stokes equation textcopyright 1996 the american physical society

universal power law tails for singularity dominated strong fluctuations integrals of the type i t identical to integral integral dxdy delta t h x y are considered where h is a function from an ensemble labelled by parameters i can represent circulation times of fluid particles in the plane orbital periods or semiclassical densities of states for one dimensional hamiltonian systems spectral densities for two dimensional crystals or the strength of a wave pulse produced by propagation of a deformed step discontinuity as h varies over the ensemble i develops strong fluctuations associated with legendre singularities for unrestricted functions h the probability distribution p i describing the fluctuations of i decays according to a universal i 9 law obtained by a scaling argument involving arnold s classification of catastrophes if h is only quadratic in y p varies as i 10 if the integral defining i is one dimensional p varies as i 3 if it is three dimensional i decays no faster than i 48 the probability distribution of i identical to mod delta i delta t mod decays as i 2

complete integrability of shock clustering and burgers turbulence we consider scalar conservation laws with convex flux and random initial data the hopf lax formula induces a deterministic evolution of the law of the initial data in a recent article we derived a kinetic theory and lax equations to describe the evolution of the law under the assumption that the initial datum is a spectrally negative markov process here we show that i the lax equations are hamiltonian and describe a principle of least action on the markov group ii the lax equations are completely integrable and linearized via a loop group factorization of operators iii the associated zero curvature equations can be solved via inverse scattering our results are rigorous for n dimensional approximations of the lax equations and yield formulas for the limit n the main observation is that the lax equations and zero curvature equations are a markovian analog of known integrable systems geodesic flow on lie groups and the n wave model respectively this allows us to introduce a variety of methods from the theory of integrable systems

lower tail of the kpz equation we provide the first tight bounds on the lower tail probability of the one point distribution of the kardar parisi zhang kpz equation with narrow wedge initial data our bounds hold for all sufficiently large times t and demonstrates a crossover between superexponential decay with exponent 52 and leading prefactor 415 t1 3 for tail depth greater than t2 3 and exponent 3 with leading prefactor at least 112 for tail depth less than t2 3

probability distribution functions for the random forced burgers equation statistical properties of solutions of the random forced burgers equation have been a subject of intensive studies recently see refs 1 6 of particular interest are the asymptotic properties of probability distribution functions associated with velocity gradients and velocity increments aside from the fact that such issues are of direct interest to a large number of problems such as the growth of random surfaces 1 it is also hoped that the field theoretic techniques developed for the burgers equation will eventually be useful for understanding more complex phenomena such as turbulence in this paper we propose a new and direct approach for analyzing the scaling properties of the various distribution functions for the random forced burgers equation we will consider the problem

dynamic boundary value problems of the second order bernstein nagumo conditions and solvability abstract this article analyzes the solvability of second order nonlinear dynamic boundary value problems bvps on time scales new bernstein nagumo conditions are developed that guarantee an a priori bound on the delta derivative of potential solutions to the bvps under consideration topological methods are then employed to gain solvability

taylor s frozen flow hypothesis in burgers turbulence by detailed analytical treatment of the shock dynamics in the burgers turbulence with large scale forcing we calculate the velocity structure functions between pairs of points displaced both in time and space our analytical treatment verifies the so called taylor s frozen flow hypothesis without relying on any closure and under very general assumptions we discuss the limitation of the hypothesis and show that it is valid up to time scales smaller than the correlation time scale of temporal velocity correlation function we support the analytical calculation by performing numerical simulation of the periodically kicked burgers equation

constructive proof for existence of nonlinear two point boundary value problems owing to the importance of differential equations in physics the existence of solutions for differential equations has been paid much attention in this paper the existence of solution are obtained for the nonlinear second order two point boundary value problem in the reproducing kernel space under certain assumptions on right hand side we propose constructive proof for the existence result and a method is presented to obtain the exact solution expressed by the form of series this paper is a extension of previous paper wei jiang minggen cui the exact solution and stability analysis for integral equation of third or first kind with singular kernel appl math comput 202 2 2008 666 674 which extends a method of solving linear problems to present method for solving nonlinear problems

wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics in this paper we propose a numerical method based on wiener chaos expansion and apply it to solve the stochastic burgers and navier stokes equations driven by brownian motion the main advantage of the wiener chaos approach is that it allows for the separation of random and deterministic effects in a rigorous and effective manner the separation principle effectively reduces a stochastic equation to its associated propagator a system of deterministic equations for the coefficients of the wiener chaos expansion simple formulas for statistical moments of the stochastic solution are presented these formulas only involve the solutions of the propagator we demonstrate that for short time solutions the numerical methods based on the wiener chaos expansion are more efficient and accurate than those based on the monte carlo simulations

statistical properties of shocks in burgers turbulence ii tail probabilities for velocities shock strengths and rarefaction intervals this paper studies the structure of the random sawtooth profile corresponding to the solution of the inviscid burgers equation with white noise initial data this function consists of a countable sequence of rarefaction waves separated by shocks we are concerned here with calculating the probabilities of rare events associated with the occurrence of very large values of the normalized velocity shock strength and rarefaction intervals we find that these quantities have tail probabilities of the form exp cx3 x 1 this cubic exponential decay of probabilities was conjectured in the companion paper 1 the calculations are done using a representation of the shock strength and length of rarefaction intervals in terms of the statistics of certain conditional diffusion processes

a dynamically adaptive wavelet approach to stochastic computations based on polynomial chaos capturing all scales of random modes on independent grids in stochastic computations or uncertainty quantification methods the spectral approach based on the polynomial chaos expansion in random space leads to a coupled system of deterministic equations for the coefficients of the expansion the size of this system increases drastically when the number of independent random variables and or order of polynomial chaos expansions increases this is invariably the case for large scale simulations and or problems involving steep gradients and other multiscale features such features are variously reflected on each solution component or random uncertainty mode requiring the development of adaptive methods for their accurate resolution in this paper we propose a new approach for treating such problems based on a dynamically adaptive wavelet methodology involving space refinement on physical space that allows all scales of each solution component to be refined independently of the rest we exemplify this using the convection diffusion model with random input data and present three numerical examples demonstrating the salient features of the proposed method thus we establish a new elegant and flexible approach for stochastic problems with steep gradients and multiscale features based on polynomial chaos expansions

what shape is your conjugate a survey of computational convex analysis and its applications computational convex analysis algorithms have been rediscovered several times in the past by researchers from different fields to further communications between practitioners we review the field of computational convex analysis which focuses on the numerical computation of fundamental transforms arising from convex analysis current models use symbolic numeric and hybrid symbolic numeric algorithms our objective is to disseminate widely the most efficient numerical algorithms useful for applications in image processing computing the distance transform the generalized distance transform and mathematical morphology operators partial differential equations solving hamilton jacobi equations and using differential equations numerical schemes to compute the convex envelope max plus algebra computing the equivalent of the fast fourier transform multifractal analysis etc the fields of applications include among others computer vision robot navigation thermodynamics electrical networks medical imaging and network communication

role of pressure in turbulence there is very limited knowledge of the kinematical relations for the velocity structure functions higher than three instead the dynamical equations for the structure functions of the velocity increment are obtained from the navier stokes equation under the assumption of the local homogeneity and isotropy these equations contain the correlation between the velocity and pressure gradient increments which is very difficult to know theoretically and experimentally we have examined these dynamical relations by using direct numerical simulation data at very high resolution at large reynolds numbers and found that the contribution of the pressure term is important to the dynamics of the longitudinal velocity with large amplitudes the pressure term is examined from the view point of the conditional average and the role of the pressure term in the turbulence dynamics is discussed

probability densities in strong turbulence abstract in this work we using mellin s transform combined with the gaussian large scale boundary condition calculate probability densities pdfs of velocity increments p r u r velocity derivatives p u r and the pdf of the fluctuating dissipation scales q re where re is the large scale reynolds number the resulting expressions strongly deviate from the log normal pdf p l r u r often quoted in the literature it is shown that the probability density of the small scale velocity fluctuations includes information about the large integral scale dynamics which is responsible for the deviation of p r u r from p l r u r an expression for the function d h of the multifractal theory free from spurious logarithms recently discussed in u frisch m martins afonso a mazzino v yakhot j fluid mech 542 2005 97 is also obtained

on a class of implicit solutions of the continuity and euler s equations for 1d systems with long range interactions results of computer simulations and of theoretical analysis done to investigate and interpret the space time evolution of the mass density and the velocity field of the inviscid self gravitating attractive and repulsive coulomb liquids in 1d with correlated initial conditions namely proportionality between the mass density and the divergence of the velocity field are reported here numerical data gathered for both models in a collisionless regime reveal an evolution with a time dependent proportionality factor feeding this result in the continuity and div euler equations leads to the introduction of another field which is shown to satisfy a burgers type of implicit equation a thorough description of regular implosion followed by singular collapses in the attractive case and of regular explosion in the repulsive case is obtained time inversion symmetry is investigated energy conservation and stability properties are shown to apply in the regular regions of smooth solutions the velocity potential satisfies a new local and inhomogeneous pde

three dimensional forced burgers turbulence supplemented with a continuity equation we investigate turbulent limit of the forced burgers equation supplemented with a continuity equation in three dimensions the scaling exponent of the conditional two point correlation function of density i e r x1 r x2 udu ux12x2u 2a 3 is calculated self consistently in the nonuniversal region from which we obtain a 353 also we derive an equation governing the evolution of the probability density function pdf of longitudinal velocity increments in length scale from which a possible mechanism for the dependence of the inertial pdf to one point urms is developed

normal and anomalous scaling of the fourth order correlation function of a randomly advected passive scalar for a delta correlated velocity field simultaneous correlation functions of a passive scalar satisfy closed equations we analyze the equation for the four point function to describe a solution completely one has to solve the matching problems at the scale of the source and at the diffusion scale we solve both the matching problems and thus find the dependence of the four point correlation function on the diffusion and pumping scale for large space dimensionality d it is shown that anomalous scaling appears in the first order of 1 d perturbation theory anomalous dimensions are found analytically both for the scalar field and for it s derivatives in particular for the dissipation field