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

diffusion approximation for self similarity of stochastic advection in burgers equation self similarity of burgers equation with stochastic advection is studied in self similar variables a stationary solution is constructed which establishes the existence of a stochastically self similar solution for the stochastic burgers equation the analysis assumes that the stochastic coefficient of advection is transformed to a white noise in the self similar variables furthermore by a diffusion approximation the long time convergence to the self similar solution is proved in the sense of distribution

symmetry reduction for nonlinear relativistically invariant equations symmetry reduction is studied for the relativistically invariant scalar partial differential equation h u u 2 u 0 in n 1 dimensional minkowski space m n 1 the introduction of k symmetry variables 1 k as invariants of a subgroup g of the poincare group p n 1 having generic orbits of codimension k n in m n 1 reduces the equation to a pde in k variables all codimension 1 symmetry variables in m n 1 n arbitrary reducing the equation studied to an ode are found as well as all codimension 2 and 3 variables for the low dimensional cases n 2 3 the type of equation studied includes many cases of physical interest in particular nonlinear klein gordon equations such as the sine gordon equation and hamilton jacobi equations

reduction of the n dimensional static vacuum einstein equation and generalized schwarzschild solutions abstract we consider the static vacuum einstein spacetime when the spatial factor is conformal to a n dimensional pseudo euclidean space the most general ansatz that reduces the resulting system of partial differential equations to a system of ordinary differential equations is completely described we obtain the entire set of solutions of the reduced system where the classical schwarzschild solution arises as a particular solution in addition we show that the riemannian spatial factors associated to these solutions are foliated by parallel hypersurfaces of constant mean curvature

lp strong convergence of the averaging principle for slow fast spdes with jumps abstract the averaging principle is an important method to extract effective macroscopic dynamic from complex systems with slow component and fast component this paper concerns the l p strong convergence of the averaging principle for two time scales stochastic partial differential equations spdes driven by wiener processes and poisson jumps to achieve this a key step is to show the existence for an invariant measure with exponentially ergodic property for the fast equation where the dissipative conditions are needed furthermore it is shown that under suitable assumptions the slow component l p strongly converges to the solution of the averaged equation the rate of the convergence is also obtained as a byproduct

birational extensions in dimension two and integrally closed ideals the aim of this paper is to examine primarily from an algebraic point of view the structure of a 2 dimensional normal local domain s n which birationally dominates a 2 dimensinal regular local ring r m from the geometric point of view the sine qua non is lipman s paper lr on rational singularities for it follows from one of the early results in that paper that s has a rational singularity using elementary algebraic techniques we are able to recover much information concerning the structure of s we have particularly focused on the fact from lipman ll and artin ar that s must have minimal multiplicity we hope this approach will aid in understanding the algebra of rational singularities and in the exploration of many open questions in dimension two and in higher dimensions the framework of the paper is set up as follows in the first section we use basic algebraic techniques such as analytic independence and zariski s main theorem to prove for example that s has a regular height 1 prime if r m is infinite if s is not regular we may assume that r is maximally regular in s and when this is the case we show for instance that the

stratified structure of the universe and burgers equation a probabilistic approach the model of the potential turbulence described by the 3 dimensional burgers equation with random initial data was developped by zeldovich and shandarin in order to explain the existing large scale structure of the universe most of the recent probabilistic investigations of large time asymptotics of the solution deal with the central limit type results the gaussian scenario under suitable moment assumptions on the initial velocity field these results and some open questions are discussed in sect 2 where we concentrate on the gaussian model and the shot noise model in sect 3 we construct a probabilistic model of strong initial fluctuations a zero range shot noise field with high amplitudes which reveals an intermittent large time behaviour with the velocity vec v t x determined by the position of the largest initial fluctuation discounted by the heat kernelg t x in a neighborhood ofx the asymptoties of such local maximum ast can be analyzed with the help of the theory of records sect 4 finally in sect 5 we introduce a global definition of a point process oft local maxima and show the weak convergence of the suitably rescaled process to a non trivial limit ast

one dimensional stochastic burgers equation driven by levy processes abstract in this paper we proved the global existence and uniqueness of the strong weak and mild solutions for one dimensional burgers equation perturbed by a poisson form process a poisson form and q wiener process with the dirichlet bounded condition we also proved the existence of the invariant measure of these models

stochastic dissipative pde s and gibbs measures we study a class of dissipative nonlinear pde s forced by a random force omega t x with the space variable x varying in a bounded domain the class contains the 2d navier stokes equations under periodic or dirichlet boundary conditions and the forces we consider are those common in statistical hydrodynamics they are random fields smooth in t and stationary short correlated in time t in this paper we confine ourselves to kick forces of the form where the k s are smooth bounded identically distributed random fields the equation in question defines a markov chain in an appropriately chosen phase space a subset of a function space that contains the zero function and is invariant for the random flow of the equation concerning this markov chain we prove the following main result see theorem 2 2 the markov chain has a unique invariant measure to prove this theorem we present a construction assigning to any invariant measure a gibbs measure for a 1d system with compact phase space and apply a version of ruelle perron frobenius uniqueness theorem to the corresponding gibbs system we also discuss ergodic properties of the invariant measure and corresponding properties of the original randomly forced pde

on the reduction of the nonlinear multi dimensional wave equations and compatibility of the d alembert hamilton system abstract the necessary conditions of the compatibility of the d alembert hamilton system in minkowsky space r 1 n are established the problem of reduction of p 1 n invariant wave equations to ordinary differential equations is discussed

spde in hilbert space with locally monotone coefficients the aim of this paper is to extend the usual framework of spde with monotone coecients to include a large class of cases with merely locally monotone coecients this new framework is conceptually not more involved than the classical one but includes many more fundamental examples not included previously thus our main result can be applied to various types of spdes such as stochastic reaction diusion equations stochastic burgers type equation stochastic 2 d navier stokes equation stochastic p laplace equation and stochastic porous media equation with non monotone perturbations

the exponential stability for stochastic delay partial differential equations abstract in this paper we consider by the energy equality the exponential stability of energy solutions to a non linear stochastic functional partial differential equation with finite delay r 0 in separable hilbert spaces our method is very simple and is powerful for concrete stochastic delay partial differential equations

stochastic functional partial differential equations existence uniqueness and asymptotic decay property existence and uniqueness of strong solutions for a class of stochastic functional differential equations in hilbert spaces are established sufficient conditions which guarantee the transference of mean square and pathwise exponential stability from stochastic partial differential equations to stochastic functional partial differential equations are studied the stability results derived are also applied to stochastic ordinary differential equations with hereditary characteristics in particular as a direct consequence our main results improve some of those by mao shah in which it was proved that under certain conditions pathwise exponential stability is transferred from non delay equations to delay equations if the constant time lag appearing in the problem is sufficiently small while in our treatment the transference actually holds for arbitrary bounded delay variables not only in finite but in infinite dimensions

the stochastic burgers equation we study burgers equation perturbed by a white noise in space and time we prove the existence of solutions by showing that the cole hopf transformation is meaningful also in the stochastic case the problem is thus reduced to the anaylsis of a linear equation with multiplicative half white noise an explicit solution of the latter is constructed through a generalized feynman kac formula typical properties of the trajectories are then discussed a technical result concerning the regularizing effect of the convolution with the heat kernel is proved for stochastic integrals

a finiteness condition on regular local overrings of a local domain the local factorization theorem of zariski and abhyankar implies that between a given pair of 2 dimensional regular local rings s d r having the same quotient field every chain of regular local rings must be finite it is shown in this paper that this property extends to every such pair of regular local rings regardless of dimension an example is given to show that this does not hold if regular is replaced by cohen macaulay by normal or by rational singularity more generally it is shown that the set r r of ndimensional regular local rings birationally dominating a given n dimensional local domain r and ordered by containment satisfies the descending chain condition an example is given to show that if r is regular the two examples of minimal elements of r r given by j sally do not exhaust the set of minimal elements of r r

stationary solutions of spdes and infinite horizon bdsdes with non lipschitz coefficients abstract we prove a general theorem that the l 2 r d r 1 l 2 r d r d valued solution of an infinite horizon backward doubly stochastic differential equation if exists gives the stationary solution of the corresponding stochastic partial differential equation we prove the existence and uniqueness of the l 2 r d r 1 l 2 r d r d valued solutions for backward doubly stochastic differential equations on finite and infinite horizon with linear growth without assuming lipschitz conditions but under the monotonicity condition therefore the solution of finite horizon problem gives the solution of the initial value problem of the corresponding stochastic partial differential equations and the solution of the infinite horizon problem gives the stationary solution of the spdes according to our general result

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

amplitude phase representation for solutions of nonlinear d alembert equations we consider the nonlinear complex d alembert equation square operator psi f mod psi mod psi with psi represented in terms of amplitude and phase in 1 n dimensional minkowski space we exploit a compatible d alembert hamilton system to construct new types of exact solutions for some nonlinearities

time asymptotics for solutions of the burgers equation with a periodic force we consider the burgers equation with a periodic force frac partial partial t u u cdot nabla u frac12 delta u nabla v x which presents a simplified model for turbulence we are interested in the asymptotic behaviour of solutions for t to infty this problem has been studied by sinai who uses a probabilistic and very technical approach using methods from spectral theory we get similar results this functional analytic approach gives an easier proof for certain initial data periodic or some random perturbations of those we show time convergence towards a deterministic periodic limit solution related to the ground state of a certain schrodinger operator

the existence and uniqueness of energy solutions to local non lipschitz stochastic evolution equations abstract let h v and k be separable hilbert spaces in this paper we consider the existence and uniqueness of energy solutions to the following stochastic evolution equation d x t a t x t f t x t d t g t x t d w t t 0 t x 0 x 0 h where a t v v is a linear bounded operator with coercivity monotone condition and hemicontinuity f 0 h h and g 0 h l 2 0 k h are measurable functions and satisfy the local non lipschitz condition proposed by the author t taniguchi successive approximations to solutions of stochastic differential equations j differential equations 96 1992 152 169

stability of semilinear stochastic evolution equations stability of moments of the mild solution of a semilinear stochastic evolution equation is studied and sufficient conditions are given for the exponential stability of the pth moment in terms of liapunov function sufficient conditions for sample continuity of the solution are also obtained and the exponential stability of sample paths is proved three examples are given to illustrate the theory

a dynamical polynomial chaos approach for long time evolution of spdes abstract we propose a dynamical generalized polynomial chaos dgpc method to solve time dependent stochastic partial differential equations spdes with white noise forcing the long time simulation of spde solutions by polynomial chaos pc methods is notoriously difficult as the dimension of the stochastic variables increases linearly with time exploiting the markovian property of white noise dgpc 1 implements a restart procedure that allows us to expand solutions at future times in terms of orthogonal polynomials of the measure describing the solution at a given time and the future white noise the dimension of the representation is kept minimal by application of a karhunen loeve kl expansion using frequent restarts and low degree polynomials on sparse multi index sets the method allows us to perform long time simulations including the calculation of invariant measures for systems which possess one we apply the method to the numerical simulation of stochastic burgers and navier stokes equations with white noise forcing our method also allows us to incorporate time independent random coefficients such as a random viscosity we propose several numerical simulations and show that the algorithm compares favorably with standard monte carlo methods

stochastic boundary control design for extensible marine risers in three dimensional space this paper presents a new design of boundary controllers for global practical k exponential p stabilization of vibration motions of extensible marine risers in three dimensional 3d space under both stochastic and deterministic sea loads the control design and analysis of well posedness and stability of the closed loop system are carried out based on a new lyapunov type theorem which is developed for studying well posedness and p stability of a class of stochastic evolution systems sess in hilbert space since this theorem eases difficulties in verification of the coercivity condition but requires conditions of a form similar to lyapunov type theorems for stochastic lumped parameter systems it has a potential application to other stochastic distributed parameter systems

on a reduction and solutions of nonlinear wave equations with broken symmetry a generalised definition for invariance of partial differential equations is proposed exact solutions of the equations with broken symmetry are obtained

stability in distribution of mild solutions to stochastic partial differential delay equations with jumps the existence uniqueness and some sufficient conditions for stability in distribution of mild solutions to stochastic partial differential delay equations with jumps are presented the principle technique of our investigation is to construct a proper approximating strong solution system and carry out a limiting type of argument to pass on stability of strong solutions to mild ones as a consequence stability results of basak et al basak et al 1999 j math anal appl 202 604 622 and yuan et al yuan et al 2003 syst control lett 50 195 207 are generalized to cover a class of much more general stochastic partial differential delay equations with jumps in infinite dimensions in contrast to the almost sure exponential stability in ichikawa ichikawa 1982 j math anal appl 90 12 44 and luo liu luo liu 2008 stoch proc appl 118 864 895 and the moment exponential stability in luo liu we present a new result on the stability in distribution of mild solutions finally an example is given to demonstrate the applicability of our work

on the general solution of the d alembert equation with a nonlinear eikonal constraint and its applications we construct the general solutions of the system of nonlinear differential equations nu 0 u u 0 in the four and five dimensional complex pseudo euclidean spaces the results obtained are used to reduce the multidimensional nonlinear d alembert equation 4u f u to ordinary differential equations and to construct its new exact solutions

a certain class of solutions of the nonlinear wave equation in this paper are investigated some differential geometry methods in the theory of the nonlinear wave equation 2u u u u a special class of solutions is discussed for which u u is constant on each level of the function u it is proved that levels of such solutions form in the space of independent variable s hypersurfaces with all principal curvatures constant the general form of such hypersurfaces is given then it is proved that via the method of characteristics it is possible to construct in principle all the solutions of the discussed class they may be obtained by integration of an ode of second order using a special class of the polynomial functions some new solutions are given for equations v 4av3 3bv2 2cv d v exp v v sin v v cosh v and v sinh v

almost sure asymptotic stability of stochastic partial differential equations with jumps in this paper we investigate stochastic partial differential equations with jumps in infinite dimensions the key motivation of this paper is under a local lipschitz condition but without a linear growth condition to give an existence and uniqueness theorem khasminskii type theorem where the classical existence and uniqueness result can be regarded as a special case and then to discuss the almost sure asymptotic stability of the solutions moreover as a by product we also derive that the solutions are weakly attracted finally several examples are constructed to demonstrate our results

orthogonal and non orthogonal separation of variables in the wave equation utt uxx v x u 0utt uxx v x u 0 we develop a direct approach to the separation of variables in partial differential equations within the framework of this approach the problem of the separation of variables in the wave equation with time independent potential reduces to solving an over determined system of nonlinear differential equations we have succeeded in constructing its general solution and as a result all potentials v x permitting variable separation have been found for each of them we have constructed all inequivalent coordinate systems providing separability of the equation under study it should be noted that the above approach yields both orthogonal and non orthogonal systems of coordinates

a precise definition of reduction of partial differential equations we give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry classical and non classical reductions of partial differential equations using the introduced definition of reduction of differential equations we establish equivalence of the non classical conditional symmetry and direct ansatz approaches to reduction of partial differential equations as an illustration we give an example of non classical reduction of the nonlinear wave equation in 1 3 dimensions the conditional symmetry approach when applied to the equation in question yields a number of non lie reductions which are far reaching generalizations of the well known symmetry reductions of the nonlinear wave equations

stability of nonlinear stochastic evolution equations abstract criteria for boundedness asymptotic stability of sample paths given by solutions to nonlinear stochastic evolution equations are presented the analysis is based on a functional ito formula liapunov and related functionals and generalization of methods developed in finite dimensions applications to parabolic ito equations are given

pathwise random periodic solutions of stochastic differential equations abstract in this paper we study the existence of random periodic solutions for semilinear stochastic differential equations we identify these as the solutions of coupled forward backward infinite horizon stochastic integral equations in general cases we then use the argument of the relative compactness of wiener sobolev spaces in c 0 0 t l 2 and generalized schauder s fixed point theorem to prove the existence of a solution of the coupled stochastic forward backward infinite horizon integral equations the condition on f is then further weakened by applying the coupling method of forward and backward gronwall inequalities the results are also valid for stationary solutions as a special case when the period can be an arbitrary number

stochastic burgers equations and their semi classical expansions in this paper we use the hopf cole logarithmic transformation and the stochastic hamilton jacobi theory to study stochastic heat equations and burgers equations before the caustics the stochastic inviscid burgers equation gives the first term of our semi classical expansion i e the inviscid limit of the viscous stochastic burgers equation in order to push our results beyond the inviscid limit we construct solutions for iterated stochastic hamilton jacobi continuity equations then we give the semi classical asymptotic expansions for stochastic heat equations and burgers equations by using nelson s stochastic mechanical processes with drifts given by the solution of the iterated stochastic hamilton jacobi continuity equations the explicit formula for the remainder term is given by a path integral

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

the burgers equation with a random force and a general model for directed polymers in random environments the study of the burgers equation with a random force leads via a hopf cole type transformation to a stochastic heat equation having a white noise with spatial parameters type potential the latter can be studied by means of a general model of directed polymers in random environments with two point random potentials these models exhibit a gaussian behavior at large times and have certain stationary distributions which yield the corresponding results for the above stochastic heat and burgers equations

practical exponential stability of stochastic age dependent capital system with levy noise abstract it is well known that stability is one of most important topics in economy and control in this paper we focus on the practical stability problem for a class of stochastic age dependent vintage capital system with levy noise compared with the classical lyapunov stability theory practical stability can depict not only qualitative behavior but also quantitative properties such as specific trajectory bounds and specific transient behavior with the help of the stochastic analysis theory and ito s formula we establish some novel criteria for the mean square and almost sure practical exponential stability of the addressed system the obtained results improve some previous works given in the literature moreover an example is provided to illustrate the theoretical results

stationary solutions of spdes and infinite horizon bdsdes in this paper we study the existence of stationary solutions for stochastic partial differential equations we establish a new connection between l 2 rd r1 l 2 rd rd valued solutions of backward doubly stochastic differential equations bdsdes on infinite horizon and the stationary solutions of the spdes moreover we prove the existence and uniqueness of the solutions of bdsdes on both finite and infinite horizons so obtain the solutions of initial value problems and the stationary solutions independent of any initial value of spdes the connection of the weak solutions of spdes and bdsdes has independent interests in the areas of both spdes and bsdes

stabilization of stochastic nonlinear systems driven by noise of unknown covariance this paper poses and solves a new problem of stochastic nonlinear disturbance attenuation where the task is to make the system solution bounded by a monotone function of the supremum of the covariance of the noise this is a natural stochastic counterpart of the problem of input to state stabilization in the sense of sontag 1989 our development starts with a set of new global stochastic lyapunov theorems for an exemplary class of stochastic strict feedback systems with vanishing nonlinearities where the equilibrium is preserved in the presence of noise we develop an adaptive stabilization scheme based on tuning functions that requires no a priori knowledge of a bound on the covariance next we introduce a control lyapunov function formula for stochastic disturbance attenuation finally we address optimality and solve a differential game problem with the control and the noise covariance as opposing players for strict feedback systems the resulting isaacs equation has a closed form solution

solutions of a system of forced burgers equation in this article we obtain explicit solutions of a system of forced burgers equation subject to some classes of bounded and compactly supported initial data and also subject to certain unbounded initial data in a series of papers rao and yadav 2010 1 3 obtained explicit solutions of a nonhomogeneous burgers equation in one dimension subject to certain classes of bounded and unbounded initial data earlier kloosterziel 1990 4 represented the solution of an initial value problem for the heat equation with initial data in l 2 r n e x 2 2 as a series of self similar solutions of the heat equation in r n here we express the solutions of certain classes of cauchy problems for a system of forced burgers equation in terms of self similar solutions of some linear partial differential equations

the symmetry and some exact solutions of the nonlinear many dimensional liouville d alembert and eikonal equations multiparametrical exact solutions of the many dimensional nonlinear d alembert liouville sine gordon and eikonal equations are obtained the maximally extensive local invariance groups of the equations are determined and invariants of the extended poincare group are found

scaling laws for the multidimensional burgers equation with quadratic external potential the reordering of the multidimensional exponential quadratic operator in coordinate momentum space see x wang c h oh and l c kwek 1998 j phys a math gen 31 4329 4336 is applied to derive an explicit formulation of the solution to the multidimensional heat equation with quadratic external potential and random initial conditions the solution to the multidimensional burgers equation with quadratic external potential under gaussian strongly dependent scenarios is also obtained via the hopf cole transformation the limiting distributions of scaling solutions to the multidimensional heat and burgers equations with quadratic external potential are then obtained under such scenarios