global existence of harmonic maps in minkowski space on considere la construction d une classe d applications harmoniques classiques globales de l espace de minkowski m 4 dans une variete de riemann complete a n dimensions

remarks on the global sobolev inequalities in the minkowski space rn 1 on demontre des inegalites de sobolev globales sur l espace de minkowski r n 1 le theoreme principal est valable sur des tranches a temps fixe

the generalized riemann problem for reactive flows abstract a generalized riemann problem is introduced for the equations of reactive non viscous compressible flow in one space dimension initial data are assumed to be linearly distributed on both sides of a jump discontinuity the resolution of the singularity is studied and the first order variation in time of flow variables is given in exact form

blow up for quasi linear wave equations in three space dimensions we consider equations of the form u phi x t u u u 1a for a function u u x 1 x n t u x t here denotes the d alembertian square frac partial 2 partial t 2 vartriangle 1b and u represents the vector of first derivatives u that of second derivatives of u with respect to the x k and t on linearizing 1a shall go over into the classical wave equation u 0 this means that and its first derivatives with respect to u u u shall vanish for u u u 0

multiresolution schemes for the reactive euler equations we present multiresolution mr schemes for the efficient numerical solution of the one dimensional system of the reactive euler equations which has possibly stiff source terms the original version of the method was developed by a harten 1995 comm pure appl math 48 12 1305 for homogeneous hyperbolic conservation laws by computing the cell average mr representation of the solution we obtain much information about the solution s regularity this description of smoothness is then used to reduce the number of direct flux computations as well as the expensive high order eno essentially nonoscillatory reconstruction both of which are now performed only near discontinuities thereby the numerical solution procedure becomes considerably more efficient in the present case of the reactive euler equations the average efficiency factor measured by counting the number of actual flux computations ranges from about 5 to 12 this is on the same order of and in some cases comes reasonably close to actual speed up factors obtained by code timings which were between 3 to 5 the mr overhead rate was about 10 for the eno and 36 for tvd schemes respectively the quality of the solution is shown to be the same as that of the finest grid detailed numerical and performance results are shown for up to fourth order accuracy for source terms ranging from moderate to extremely stiff

a bridging domain method for coupling continua with molecular dynamics abstract a bridging domain method for coupling continuum models with molecular models is described in this method the continuum and molecular domains are overlapped in a bridging subdomain where the hamiltonian is taken to be a linear combination of the continuum and molecular hamiltonians we enforce the compatibility in the bridging domain by lagrange multipliers or by the augmented lagrangian method an explicit algorithm for dynamic solutions is developed results show that this multiscale method can avoid spurious wave reflections at the molecular continuum interface without any additional filtering procedures even for problems with significant nonlinearities the method is also shown to naturally handle the coupling of the continuum energy equation with the molecular subdomain a multiple time step algorithm is also developed within this framework

resonance type behaviour in a system of nonlinear wave equations consider the cauchy problem for a system of two wave equations in r3 c12 t2u1 u1 u 1 u 2c22 t2 u2 u 1 u 2 where prime stands for any partial derivative in space or time variables if initial data is small in a suitable norm the following phenomenon occurs if c1 c2 the system possesses smooth solutions for all t 0 whereas if c1 c2 the solutions of the system might blowup the result is proved for a more general form of the above system in the case of a three space dimension also a similar result is proved for two space dimensions

discrete lagrange equations for reacting thermofluid dynamics in arbitrary lagrangian eulerian frames abstract interest in the development of multiscale methods has focused attention on the marked differences in the numerical modeling techniques typically applied at different scales most continuum dynamics models construct approximate solutions to partial differential equations while most nanoscale models employ a discrete hamiltonian approach previous research has demonstrated that the introduction of entropies or internal energies as generalized coordinates along with separation of the discretization and model formulation steps allows general thermomechanical models to be developed at the continuum scale using a nonholonomic hamiltonian or lagrange equation formulation with the introduction of additional state variables and nonholonomic constraints the latter work may be further extended in order to model reacting systems employing a finite element interpolation and a lagrangian an eulerian or an ale mesh the new formulation has been validated by solving several one dimensional reacting shock physics problems the result is a continuum dynamics modeling approach highly compatible with the discrete energy methods normally used at the nanoscale

explicit time scale splitting algorithm for stiff problems auto ignition of gaseous mixtures behind a steady shock a new explicit algorithm based on the computational singular perturbation csp method is presented this algorithm is specifically designed to solve stiff problems and its performance increases with stiffness the key concept in its structure is the splitting of the fast from the slow time scales in the problem realized by embedding csp concepts into an explicit scheme in simple terms the algorithm marches in time with only the terms producing the slow time scales while the contribution of the terms producing the fast time scales is taken into account at the end of each integration step as a correction the new algorithm is designed for the integration of stiff systems of pdes by means of explicit schemes for simplicity in the presentation and discussion of the different features of the new algorithm a simple test case is considered involving the auto ignition of a methane air mixture behind a normal shock wave which is described by a system of odes the performance of the new algorithm accuracy and computational efficiency is then compared with the well known lsode package its merits when used for the solution of systems of pdes are discussed although when dealing with a stiff system of odes the new algorithm is shown to provide equal accuracy with that delivered by lsode at the cost of higher execution time the results indicate that its performance could be superior when facing a stiff system of pdes

solution of counter diffusion problem with position dependent diffusion coefficent by using variational methods unsteady state counter diffusion problem with position dependent diffusion coefficient can be modeled using fick s second law a mathematical model was constructed and solved to quantitatively describe the dynamic behavior of solute diffusion through non homogeneous materials where diffusion coefficient is a function of position the eigenfunction expansion approach was utilized to solve the model the eigenvalues and eigenfunction of the system were obtained using a variational method it has been shown that position dependency of the material can be neglected if the thickness of the material is relatively small mathematical models were solved for different thicknesses and different diffusion coefficient functions

discrete hamilton s equations for viscous compressible fluid dynamics abstract lagrange s and hamilton s equations are used extensively in numerical modeling of rigid body dynamics and continuum solid dynamics problems the use of energy methods in viscous compressible flow problems has been by contrast rather limited largely confined to the development of basic balance laws in partial differential equation form however finite element interpolation of the modeled flow field allows for the direct application of the discrete form of hamilton s equations to viscous compressible fluid dynamics in eulerian frames the resulting model is a true energy formulation developed without reference to the partial differential balance equations which underlie conventional finite difference weighted residual finite element and finite volume methods

almost global existence to nonlinear wave equations in three space dimensions on considere l existence a t grand de solutions de petite amplitude pour des equations d onde d ordre 2 non lineaires a 4 dimensions d espace temps

nonlinear waves in friedman robertson walker space times in this paper we consider the initial value problem for the nonlinear wave equation u f u u in friedman robertson walker space time being the d alambertian in local coordinates of space time we obtain decay estimates and show that the equation has global solutions for small initial data we do it by reducing the problem to an initial value problem for the wave equation over hyperbolic space as byproduct we derive decay and global existence for solutions of the wave equation over the hyperbolic space with small initial data the same technique with some auxiliary lemmas similar to the ones proved in 6 7 can be used to generalize the result to the case when f depends also on second derivatives of u in a certain way

surfaces of prescribed mean curvature vector in semi riemannian manifolds abstract solving a second order quasilinear initial value problem we prove existence and uniqueness of both space like and time like immersions of prescribed mean curvature vector in general semi riemannian manifolds such surfaces arise for example as critical points of generalized area functionals as well as self similar solutions of the mean curvature flow

global existence of small amplitude solutions to nonlinear klein gordon equations in four space time dimensions consider the following class of nonlinear wave equations square text u u text text f u u u n k g where t 2 1 2 2 2 3 2 is the d alembertian of the 4 dimensional minkowski space time and f a smooth function of u u t x and its first and second partial derivates vanishing together with its first derivatives at u u u 0 in other words f is at least quadratic in u u u in a neighborhood of the trivial solution u 0 we subject u to the pure initial value problem text u mkern 1mu varepsilon text f x u t varepsilon g x at t 0 i v p with f g c 0 3 and e 0 a small parameter

weak solutions and development of singularities of the su 2 σ model pour toutes donnees initiales d energie finie le modele sigma o k a des solutions faibles globales en chaque dimension d espace temps

comparison between fixed and gaussian steplength in monte carlo simulations for diffusion processes we analyze the different degrees of accuracy of two monte carlo methods for the simulation of one dimensional diffusion processes with homogeneous or spatial dependent diffusion coefficient that we assume correctly described by a differential equation the methods analyzed correspond to fixed and gaussian steplengths for a homogeneous diffusion coefficient it is known that the gaussian steplength generates exact results at fixed time steps dt for spatial dependent diffusion coefficients the symmetric character of the gaussian distribution introduces an error that increases with time as an example we consider a diffusion coefficient with constant gradient and show that the error is not present for fixed steplength with appropriate asymmetric jump probabilities

theoretical and numerical structure for unstable one dimensional detonations the spatio temporal structure of unstable detonations in a single space dimension is studied through a combination of numerical and asymptotic methods a new high resolution numerical method for computing unstable detonations is developed this method combines the piecewise parabolic method ppm with conservative shock tracking and adaptive mesh refinement a new nonlinear asymptotic theory for the spatio temporal growth of instabilities is also developed this asymptotic theory involves a nonclassical hopf bifurcation because resonant acoustic scattering states with exponential growth in space cross the imaginary axis and become nonlinear eigenmodes in a complex free boundary problem for a nonlinear hyperbolic equation an interplay between the asymptotic theory and numerical simulation is used to elucidate the spatio temporal mechanisms of nonlinear stability near the transition boundary in particular a quantitative qualitative explanation is developed for the experimentally observed instabilities

high accuracy numerical methods for thermally perfect gas flows with chemistry the compressible navier stokes equations can be extended to model multi species chemically reacting gas flows the result is a large system of convection diffusion equations with stiff source terms in this paper we develop the framework needed to apply modern high accuracy numerical methods from computational gas dynamics to this extended system we also present representative computational results using one such method the framework developed here is useful for many modern numerical schemes we first present an enthalpy based form of the equations that is well suited both for physical modeling and for numerical implementation we show how to treat the stiff reactions via time splitting and in particular how to increase accuracy by avoidng the common practice of approximating the temperature we derive simple exact formulas for the characteristics of the convective part of the equations which are essential for application of all characteristic based schemes we also show that the common practice of using approximate analytical expressions for the characteristics can potentially produce spurious oscillations in computations we implement these developments with a particular high accuracy characteristic based method the finite difference eno space discretization with the 3rd order tvd runge kutta time discretization combined with the second order accurate strang time splitting of the reaction terms we illustrate the capabilities of this approach with calculations of a 1 d reacting shock tube and a 2 d combustor

harmonic maps in three dimensional minkowski space with image near geodesics shatah i sh has constructed global weak solutions of the nonlinear sigma model in h1 2 for arbitrary dimension moreover for the case of maps from four dimensional minkowski space r 3 1 to sphere s 3 he has found an example of smooth symmetric initial data for which the corresponding solution develops singularities on the second order derivatives in finite time showing that the large initial data may contribute the formation of singularities to avoid this situation sideris i si proved the existence of global smooth solution from r 3 1 into an n dimensional complete riemannian manifold n as a perturbation of a special class of harmonic maps geodesic geometrically the solution lies in a small tubular region of a given geodesic on n instead of a small neighborhood of a point on n therefore the initial data is partly small the purpose of this paper is to extend sideris s result to the case where domain is the minkowski space r 2 1 in this case the existence of smooth solution of small amplitude was proved by kovalyov ko but ours is more general since the parameter along the geodesic may vary freely formally a harmonic map from the minkowski space r 2 1 into an n dimensional complete riemannian manifold n h is described as a critical point r 2 1 n of the energy integral

uniform decay estimates and the lorentz invariance of the classical wave equation on etudie le comportement asymptotique des solutions de u 0 ou t 2 1 2 n 2 pour des conditions initiales u 0 u t g x en t 0 avec g reguliere a support compact dans r n

global solutions of nonlinear hyperbolic equations for small initial data on considere des systemes quasilineaires d equations hyperboliques d ordre 2 qui sont des deformations non lineaires de l equation d onde on etudie l existence globale et le comportement asymptotique des solutions du probleme de cauchy quand la donnee initiale est suffisamment petite

group theoretic approach for solving the problem of diffusion of a drug through a thin membrane the transformation group theoretic approach is applied to study the diffusion process of a drug through a skinlike membrane which tends to partially absorb the drug two cases are considered for the diffusion coefficient the application of one parameter group reduces the number of independent variables by one and consequently the partial differential equation governing the diffusion process with the boundary and initial conditions is transformed into an ordinary differential equation with the corresponding conditions the obtained differential equation is solved numerically using the shooting method and the results are illustrated graphically and in tables

discrete hamilton s equations for arbitrary lagrangian eulerian dynamics of viscous compressible flow abstract a number of different arbitrary lagrangian eulerian ale formulations of continuum fluid and solid dynamics problems have been developed to address applications where more conventional lagrangian or eulerian modeling techniques are difficult to apply in general these ale formulations are based on finite difference or weighted residual finite element solutions of the partial differential equations for the system an alternative energy based ale model for fluid dynamics simulations may be obtained by direct application of hamilton s canonical equations to a finite element discretization of an open deforming control volume formulated in terms of convected coordinates and incorporating an adaptive mesh scheme this modeling approach yields a simple but general description of viscous compressible flows numerical application of the method demonstrates accurate results in the solution of several shock problems whether the calculations are performed using a lagrangian an eulerian or an ale mesh