geometrical treatment of systems driven by coloured noise the authors use geometrical methods and functional derivative techniques to analyse the stochastic behaviour of multidegree of freedom nonlinear systems driven by gaussian noises with arbitrary correlation functions the method has the advantage of being also applicable to systems with an infinite number of degrees of freedom partial differential equations with some kind of hamiltonian structure and gives some insight into the nature of the approximations that lead to some of the main effective fokker planck equations used in the literature

one dimensional transport equations with discontinuous coefficients we consider one dimensional linear transport equations with bounded but possibly discontinuous coefficient a the cauchy problem is studied from two different points of view in the first case we assume that a is piecewise continuous we give an existence result and a precise description of the solutions on the lines of discontinuity in the second case we assume that a satisfies a one sided lipschitz condition we give existence uniqueness and general stability results for backward lipschitz solutions and forward measure solutions by using a duality method we prove that the flux associated to these measure solutions is a product by some canonical representative of a key words linear transport equations discontinuous coefficients weak stability duality product of a measure by a discontinuous function nonnegative solutions 1991 mathematics subject classification primary 35f10 35b35 34a12 to appear in nonlinear analysis tma departement de mathematiques et applications umr cnrs 8553 ecole normale superieure et cnrs 45 rue d ulm 75230 paris cedex 05 france francois bouchut ens fr mathematiques applications et physique mathematique d orleans umr cnrs 6628 universite d orleans 45067 orleans cedex 2 france james math cnrs fr

the riemann problem for the two dimensional zero pressure euler equations abstract the riemann problem for the two dimensional zero pressure euler equations is considered the initial data are constant values in each quadrant which satisfy an assumption that each initial discontinuity projects only one two dimensional wave the phenomenon of two dimensional delta shock wave with a dirac delta function in both density and internal energy is identified both generalized rankine hugoniot relation and entropy condition for this type of two dimensional delta shock wave are proposed the qualitative behavior of entropy solutions to this relation with certain special initial data is established based on these preparations we obtain twenty three explicit solutions and their corresponding criteria in particular the mach reflection like patterns arise in the exact solutions

a class of nonlinear nonhyperbolic systems of conservation laws with well posed initial value problems abstract with mild restrictions on the initial data we show well posedness of the initial value problem for systems of conservation laws in one space variable with real and equal characteristic speeds and a deficiency of one corresponding eigenvector the obtained weak solutions assume values as borel measures at each time after a smooth solution ceases to exist the concept of entropy density flux functions is extended to such systems finally we show that systems with well posed initial value problems admitting weak solutions of this type necessarily satisfy these structural assumptions

a fractional step solution method for the unsteady incompressible navier stokes equations in generalized coordinate systems abstract a fractional step method is developed for solving the time dependent three dimensional incompressible navier stokes equations in generalized coordinate systems the primitive variable formulation uses the pressure defined at the center of the computational cell and the volume fluxes across the faces of the cells as the dependent variables instead of the cartesian components of the velocity this choice is equivalent to using the contravariant velocity components in a staggered grid multiplied by the volume of the computational cell the governing equations are discretized by finite volumes using a staggered mesh system the solution of the continuity equation is decoupled from the momentum equations by a fractional step method which enforces mass conservation by solving a poisson equation this procedure combined with a consistent approximation of the geometric quantities is done to satisfy the discretized mass conservation equation to machine accuracy as well as to gain the favorable convergence properties of the poisson solver the momentum equations are solved by an approximate factorization method and a novel zebra scheme with four color ordering is devised for the efficient solution of the poisson equation several two and three dimensional laminar test cases are computed and compared with other numerical and experimental results to validate the solution method good agreement is obtained in all cases

the riemann problem for a resonant nonlinear system of conservation laws with dirac measure solutions the riemann problem for a resonant nonlinear system of conservation laws is considered here the riemann solution was constructed by employing the viscosity approximation approach one kind of new discontinuity which is called the dirac contact wave appeared in the riemann solution because the strict hyperbolicity as well as the genuine nonlinearity of the system considered failed the solution we obtained in this paper is not unique for some initial data an additional condition was explored to guarantee the uniqueness of the riemann problem

existence and uniqueness of discontinuous solutions for a hyperbolic system in this paper we prove the global existence and uniqueness of solutions to the cauchy problem of a hyperbolic system which probably contains so called waves

a thermochemical instability ii inhomogeneous fluctuations the behavior of coupled stochastic fields near a cusp bifurcation is studied the results are applied to a thermochemical model and the possibility of observing nonclassical critidal behavior is discussed

distributions as initial values in a triangular hyperbolic system of conservation laws the present paper concerns the system ut u x 0 vt u v x 0 having distributions as initial conditions under certain conditions and supposing functions we explicitly solve this cauchy problem within a convenient space of distributions u v for this purpose a consistent extension of the classical solution concept defined in the setting of a distributional product not constructed by approximation processes is used shock waves shock waves and also waves defined by distributions that are not measures are presented explicitly as examples this study is carried out without assuming classical results about conservation laws for reader s convenience a brief survey of the distributional product is also included

numerical method for the bone regeneration model defined within the evolving 2d axisymmetric physical domain abstract a numerical algorithm used to obtain a solution for a peri implant osseointegration model is constructed the model is formulated in terms of a system of three nonlinear coupled time dependent advection diffusion reaction equations which are defined within the irregular two dimensional physical domain which evolves in time the embedded boundary method and the level set function which is approximated on the fixed regular rectangular grid are used to track the changes of the irregular geometry of the physical domain the method of lines is applied to separate the discretizations in time and in space the advection diffusion and reaction terms are discretized separately by means of the cell centered finite volume method the exact solution of the riemann problem for the nonstrictly hyperbolic system without genuine nonlinearity is obtained an approach for the determination of the gradients of the unknown variables on the edges of the irregular control volumes is proposed the explicit second order trapezoidal rule is used for the time integration since it allows to maintain positivity of the solution which is critical for the considered problem some results of the numerical simulations are presented contact and distance osteogenesis are predicted for micro rough and smooth implants

computing a non maxwellian velocity distribution from first principles we investigate a family of single particle anomalous velocity distribution by solving a particular class of stochastic liouville equations the stationary state is obtained analytically and the maxwell boltzmann distribution is reobtained in a particular limit we discuss the comparison with other different methods to obtain the stationary state extensions when the models cannot be solved in an exact way are also pointed out in connection with the one ficton approximation

a new method of analysis of the effect of weak colored noise in nonlinear dynamical systems a systematic method for obtaining the asymptotic behavior of a dynamical system forced by colored noise in the limit of small intensity is developed it is based on the search of wkb solutions to the fokker planck equation for the joint probability density of the system and noise in which the perturbation expansion is continued to the first correction beyond the hamilton jacobi limit the method can be applied to noise with correlation time of order unity it is illustrated on the normal form of a pitchfork bifurcation where it is pointed out that additive noise can induce a shift of the most probable value this prediction is confirmed by numerical simulation of the stochastic differential equations

effective potential and weak noise transitions we review the notion of effective potential for stochastic processes and discuss its possible applications we calculate this function up to first order in a parameter measuring the intensity of the noise for a general nonlinear system the result is applied exhibiting a transition induced by weak noise

delta shock wave formation in the case of triangular hyperbolic system of conservation laws abstract we describe shock wave generation from continuous initial data in the case of triangular conservation law system arising from generalized pressureless gas dynamics model we use smooth approximations in the weak sense that are more general than small viscosity approximations

ordered cumulant technique and differential equations for probability density abstract this paper is concerned with stochastic differential equations of langevin type in which the stochastic forces are gaussian and not of white noise type the problem of finding explicitly the differential equation for the probability density of the process is solved by means of an expansion in the small correlation time of the noise for this purpose we will use the cumulant technique different situations are studied such as markovian or fokker planck equation first correction to the markovian approximation but with the fokker planck form and the corrections of non fokker planck form

eulerian droplet model delta shock waves and solution of the riemann problem abstract we study an eulerian droplet model which can be seen as the pressureless gas system with a source term a subsystem of this model and the inviscid burgers equation with source term the condition for loss of regularity of a solution to burgers equation with source term is established the same condition applies to the eulerian droplet model and its subsystem the riemann problem for the eulerian droplet model is constructively solved by going through the solution of the riemann problems for the inviscid burgers equation with a source term and the subsystem respectively under suitable generalized rankine hugoniot relations and entropy condition the existence of delta shock solution is established the existence of a solution to the generalized rankine hugoniot conditions is proven some numerical illustrations are presented

two dimensional riemann problems for zero pressure gas dynamics with three constant states abstract the riemann problems for two dimensional zero pressure gas dynamics are solved completely when the initial data take three constant states having discontinuities on x y positive and x negative axes with the help of characteristic analysis by studying interactions among delta shocks vacuums and contact discontinuities the riemann solutions constructed exhibit nine different explicit configurations the mach reflection like configurations appear in some solutions

control volume representation of molecular dynamics a molecular dynamics md parallel to the control volume cv formulation of fluid mechanics is developed by integrating the formulas of irving and kirkwood j chem phys 18 817 1950 over a finite cubic volume of molecular dimensions the lagrangian molecular system is expressed in terms of an eulerian cv which yields an equivalent to reynolds transport theorem for the discrete system this approach casts the dynamics of the molecular system into a form that can be readily compared to the continuum equations the md equations of motion are reinterpreted in terms of a lagrangian to control volume mathcal lcv conversion function ensuremath vartheta i for each molecule i the mathcal lcv function and its spatial derivatives are used to express fluxes and relevant forces across the control surfaces the relationship between the local pressures computed using the volume average lutsko j appl phys 64 1152 1988 techniques and the method of planes todd et al phys rev e 52 1627 1995 emerges naturally from the treatment numerical experiments using the md cv method are reported for equilibrium and nonequilibrium start up couette flow model liquids which demonstrate the advantages of the formulation the cv formulation of the md is shown to be exactly conservative and is therefore ideally suited to obtain macroscopic properties from a discrete system

ω expansion of van kampen through functional integrals we give a functional integral version of the van kampen expansion method for the master equation explicit expressions are given for the generating functional of correlation functions

a new approximate stationary probability distribution for processes driven by ornstein uhlenbeck noise a novel approximate probability distribution for a class of nonlinear processes driven by multiplicative linear ornstein uhlenbeck noise is presented a comparison is made with other approximate distributions reported in the literature

finite correlation time effects in nonequilibrium phase transitions i dynamic equation and steady state properties abstract we determine an approximate renormalized equation of evolution for an arbitrary nonlinear single degree of freedom system externally driven by gaussian parametric fluctuations of finite correlation time the renormalization scheme used here gives a second order equation with a time and state dependent diffusion coefficient we are able to calculate the diffusion coefficient in closed form the steady state distribution can easily be obtained from the evolution equation we are thus able to determine the parameter dependence of the steady state distribution and in particular the influence of a correlation time of the fluctuations which does not vanish on the steady state distribution

the riemann problem with delta initial data for a class of coupled hyperbolic systems of conservation laws abstract in this paper we solve the riemann problem with the initial data containing dirac delta functions for a class of coupled hyperbolic systems of conservation laws under suitably generalized rankine hugoniot relation and entropy condition the existence and uniqueness of solutions involving delta shock waves are proved further four kinds of different structure for solutions are established uniquely

langevin like equation with colored noise consider a stochastic differential equation of the form of a langevin equation but in which the noise source is not white if it is nearly white i e its autocorrelation time is short a systematic approximation method is known it leads to a fokker planck equation with successive higher order corrections to obtain the coefficients more explicitly a secondary expansion may be employed the validity of the resulting double series approximation is discussed and confronted with the various results given in the literature in addition an alternative approximation method is obtained using the technique for eliminating fast variables it produces the same terms in a different sequence

stochastic analysis of symmetry breaking bifurcations master equation approach the multivariate master equation for a general reaction diffusion system is solved perturbatively in the vicinity of a bifurcation point leading to symmetry breaking transitions the possibility to express the result through a brazovskii type of potential is examined and a comparison with the langevin analysis of walgraefet al adv chem phys 49 311 1982 is performed

pressure tensor for inhomogeneous fluids we develop a simple efficient and general statistical mechanical technique for calculating the pressure tensor of an atomic fluid the method is applied to the case of planar poiseuille flow through a narrow slit pore and the results indicate that our technique is accurate and relatively efficient a second method to calculate shear stress is derived from the momentum continuity equation this mesoscopic method again is seen to be accurate with low statistical uncertainty using both approaches the viscosity is calculated as a function of position across the pore and is seen to oscillate because of a wall induced local structure in the fluid we discuss these methods in relation to the well known ambiguity of the pressure tensor

sensitivity of a hopf bifurcation to multiplicative colored noise the location of a hopf bifurcation in parameter space may be postponed or advanced by multiplicative colored noise depending on the interplay between the time scale of the noise and the rotation period of the phase variable

comment on noise and bifurcations we calculate in exact form the first correction in a parameter measuring the strength of the noise to the effective potential for one variable diffusion processes the use of this potential to study transitions is discussed

vanishing viscosity limit for a system of balance laws with general type initial data arising from 1d saint venant model we study the vanishing viscosity limit of a non strictly hyperbolic system of balance laws known as the 1d saint venant model whose solution admits waves the balance term depends on both the variables x and t and is unbounded in the space variable x unlike the work of olenik usp math nauk 12 3 3 73 1957 in russian we also studied the large time behavior when the viscosity parameter is positive when the balance term depends only on the time variable we obtained an explicit formula for the solution

on the stationary distribution of self sustained oscillators around bifurcation points a double expansion in powers of the damping coefficient and noise intensity is shown to be a powerful method for obtaining the stationary distribution of systems that after rescaling become weakly damped conservative ones systems undergoing hopf bifurcations belong to this class as an illustrative example the generalized van der pol oscillator is considered around its bifurcation point a calculation is carried out up to third order in both the noise intensity and the bifurcation parameter damping coefficient

delta shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws abstract for simple models of hyperbolic systems of conservation laws we study a new type of nonlinear hyperbolic wave a delta shock wave which is a dirac delta function supported on a shock we prove that delta shock waves are w limits in l1 of solutions to some reasonable viscous perturbations as the viscosity vanishes further we solve the multiplication problem of a delta function with a discontinuous function to show that delta shock waves satisfy the equations in the sense of distributions under suitable generalized rankine hugoniot and entropy conditions we establish the existence and uniqueness of solutions involving delta shock waves for the riemann problems the existence of solutions to the cauchy problem is also investigated

hybrid atomistic continuum method for the simulation of dense fluid flows we present a hybrid atomistic continuum method for multiscale simulations of dense fluids in this method the atomistic part is described using a molecular dynamics description while the continuum flow is described by a finite volume discretization of the incompressible navier stokes equations the two descriptions are combined in a domain decomposition formulation using the schwarz alternating method a novel method has been proposed in order to impose non periodic velocity boundary conditions from the continuum to the atomistic domain based on an effective boundary potential consistent body forces a particle insertion algorithm and specular walls the extraction of velocity boundary conditions for the continuum from the atomistic domain is formulated by taking into account the associated statistical errors the advantages and drawbacks of the proposed schwarz decomposition method as compared to related flux based schemes are discussed the efficiency and applicability of the method is demonstrated by considering hybrid and full molecular dynamics simulations of the flow of a lennard jones fluid past a carbon nanotube

weak noise expansions through functional integrals for colored noise we use path integral methods to obtain expansions for the correlation functions of the non markovian stochastic processes generated by stochastic differential equations with colored noise

comparison of several staggered atomistic to continuum concurrent coupling strategies abstract in this contribution several staggered schemes used to couple continuum mechanics cm and molecular mechanics mm are proposed the described approaches are based on the atomistic to continuum correspondence obtained by spatial averaging in the spirit of irving and kirkwood and noll similarities between this and other concurrent coupling schemes are indicated thus providing a broad overview of different approaches in the field the schemes considered here are decomposed into the surface type displacement or traction boundary conditions and the volume type the latter restricts the continuum displacement field and possibly its gradient in some sense to the atomistic discrete displacements using lagrange multipliers a large strain cm formulation incorporating lagrange multipliers and a strategy to solve the resulting coupled linear system using an iterative solver is presented finally the described coupling methods are numerically examined using two examples uniaxial deformation and a plate with a hole relaxed under surface tension accuracy and convergence rates of each method are reported it was found that the displacement surface coupling scheme and the lagrangian volume scheme based on either discrete displacements or the h 1 norm derived from continuous displacement fields provide the best performance

analytical and numerical study of coupled atomistic continuum methods for fluids the stability and convergence rate of coupled atomistic continuum methods are studied analytically and numerically these methods couple a continuum model with molecular dynamics through the exchange of boundary conditions in the continuum particle overlapping region different coupling schemes including velocity velocity flux velocity velocity flux and flux flux are studied it is found that the velocity velocity and flux velocity schemes are stable the flux flux scheme is weakly unstable the stability of the velocity flux scheme depends on the parameter t c which is the length of the time interval between successive exchange of boundary conditions it is stable when t c is small and unstable when t c is large for steady state problems the flux velocity scheme converges faster than the other coupling schemes

generalized plane delta shock waves for n dimensional zero pressure gas dynamics abstract with the help of a generalized plane wave solution we study a type of generalized plane delta shock wave for the n dimensional zero pressure gas dynamics and refine its generalized rankine hugoniot relation which is a system of ordinary equations this relation describes accurately the character of the generalized plane delta shock location propagation speed and weight under a suitable entropy condition four different explicit constructions of solutions for a kind of riemann problem with radon measure as initial data are established uniquely the overtaking of two plane delta shocks forming a new generalized plane delta shock is also investigated finally the 2 d riemann problem with four pieces of initial data is solved in a simplified situation

well posedness for pressureless flow we study the uniqueness problem for pressureless gases previous results on this topic are only known for the case when the initial data is assumed to be a bounded measurable function this assumption is unnatural because the solution is in general a radon measure in this paper the uniqueness of the weak solution is proved for the case when the initial data is a radon measure we show that besides the oleinik entropy condition it is also important to require the energy to be weakly continuous initially our uniqueness result is obtained in the same functional space as the existence theorem

an approximate master equation for systems driven by linear ornstein uhlenbeck noise abstract by use of an operator method we construct a novel approximate evolution equation for a one dimensional probability distribution of a single degree of freedom system driven by ornstein uhlenbeck noise this equation is an integro differential equation of the time convolutionless type and its steady state solution is presented a class of non linear equations with multiplicative noise is considered

concentration and cavitation in the vanishing pressure limit of solutions to the euler equations for nonisentropic fluids numerical simulations 2 d riemann problem in gas dynamics and formation of spiral in nonlinear problems in engineering and science numerical and analytical approach beijing 1991 science press beijing 1992 pp 167 179 discrete contin dyn syst 1 1995 555 584 6 2000 419 430 for the euler equations for gas dynamics in the regime of small pressure showed that for one case the particles seem to be more sticky and tend to concentrate near some shock locations and for the other case in the region of rarefaction waves the particles seem to be far apart and tend to form cavitation in the region in this paper we identify and analyze the phenomena of concentration and cavitation by studying the vanishing pressure limit of solutions of the full euler equations for nonisentropic compressible fluids with a scaled pressure it is rigorously shown that any riemann solution containing two shocks and possibly one contact discontinuity to the euler equations for nonisentropic fluids tends to a shock solution to the corresponding transport equations and the intermediate densities between the two shocks tend to a weighted measure that along with the two shocks and possibly contact discontinuity forms the shock as the pressure vanishes by contrast it is also shown that any riemann solution containing two rarefaction waves and possibly one contact discontinuity to the euler equations for nonisentropic fluids tends to a two contact discontinuity solution to the transport equations and the nonvacuum intermediate states between the two rarefaction waves tend to a vacuum state as the pressure vanishes some numerical results exhibiting the processes of concentration and cavitation are presented as the pressure decreases 2003 elsevier b v all rights reserved msc 35l65 35b30 76e19 35q35 35l67 35b25 65m06