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

vanishing viscosity solutions of nonlinear hyperbolic systems we consider the cauchy problem for a strictly hyperbolic n n system in one space dimension ut a u ux 0 assuming that the initial data have small total variation we show that the solutions of the viscous approximations ut a u ux euxx are defined globally in time and satisfy uniform bv estimates independent of e moreover they depend continuously on the initial data in the l 1 distance with a lipschitz constant independent of t e letting e 0 these viscous solutions converge to a unique limit depending lipschitz continuously on the initial data in the conservative case where a df is the jacobian

the unique limit of the glimm scheme we introduce the definitions of a standard riemann semigroup and of a viscosity solution for a nonlinear hyperbolic system of conservation laws for a class including general 2 2 systems it is proved that the solutions obtained by a wavefront tracking algorithm or by the glimm scheme are precisely the semigroup trajectories in particular these solutions are unique and depend lipschitz continuously on the initial data in the l1 norm

probability density of the empirical wavelet coefficients of a noisy chaos we are interested in the random empirical wavelet coefficients of a noisy signal when this signal is a unidimensional or multidimensional chaos more precisely we provide an expression of the conditional probability density of such coefficients given a discrete observation grid the noise is assumed to be described by a symmetric alpha stable random variable if the noise is a dynamic noise then we present the exact expression of the probability density of each wavelet coefficient of the noisy signal if we face a measurement noise then the noise has a non linear influence and we propose two approximations the first one relies on a taylor expansion whereas the second one relying on an edgeworth expansion improves the first general taylor approximation if the cumulants of the noise are defined we give some illustrations of these theoretical results for the logistic map the tent map and a multidimensional chaos the henon map disrupted by a gaussian or a cauchy noise

multidimensional viscous shocks ii the small viscosity limit in this paper we prove the existence of curved multidimensional viscous shocks and also justify the small viscosity limit starting with a curved multidimensional inviscid shock solution to a system of hyperbolic conservation laws we show that the shock can be obtained as a small viscosity limit of solutions to an associated parabolic problem viscous shocks the two main hypotheses are a natural evans function assumption on the viscous profile together with a restriction on how much the shock can deviate from flatness the main tools are a conjugation lemma that removes xn dependence from the linearization of the parabolic problem about the viscous profile new degenerate kreiss type symmetrizers used to prove an l2 estimate for the linearized problem and a finite regularity calculus of semiclassical and mixed type classical semiclassical pseudodifferential operators 2003 wiley periodicals inc

characteristic boundary layers in real vanishing viscosity limits abstract in this paper we study the vanishing viscosity limit of initial boundary value problems for one dimensional mixed hyperbolic parabolic systems when the boundary is characteristic for both the viscous and the inviscid systems in particular we assume that an eigenvalue of the inviscid system vanishes uniformly we prove the stability of boundary layers expansions in small time i e before shocks for the inviscid system as long as the amplitude of the boundary layers remains sufficiently small in particular by using lagrangian coordinates we apply our result to physical systems like gasdynamics and magnetohydrodynamics with homogeneous dirichlet condition for the velocity at the boundary

the convergence rate of finite difference schemes in the presence of shocks finite difference approximations generically have cal o 1 pointwise errors close to a shock we show that this local error may effect the smooth part of the solution such that only first order is achieved even for formally higher order methods analytic and numerical examples of this form of accuracy are given we also show that a converging method will have the formal order of accuracy in domains where no characteristics have passed through a shock

l2 contraction of large planar shock waves for multi dimensional scalar viscous conservation laws abstract we consider a l 2 contraction a l 2 type stability of large viscous shock waves for the multi dimensional scalar viscous conservation laws up to a suitable shift by using the relative entropy methods quite different from the previous results we find a new way to determine the shift function which depends both on the time and space variables and solves a viscous hamilton jacobi type equation with source terms moreover we do not impose any conditions on the anti derivative variables of the perturbation around the shock profile more precisely it is proved that if the initial perturbation around the viscous shock wave is suitably small in l 2 norm then the l 2 contraction holds true for the viscous shock wave up to a suitable shift function note that bv norm or the l norm of the initial perturbation and the shock wave strength can be arbitrarily large furthermore as the time t tends to infinity the l 2 contraction holds true up to a spatially homogeneous time dependent shift function in particular if we choose some special initial perturbations then l 2 convergence of the solutions towards the associated shock profile can be proved up to a time dependent shift

hydrodynamic limit of the boltzmann equation with contact discontinuities the hydrodynamic limit for the boltzmann equation is studied in the case when the limit system that is the system of euler equations contains contact discontinuities when suitable initial data is chosen to avoid the initial layer we prove that there exist a family of solutions to the boltzmann equation globally in time for small knudsen number and this family of solutions converge to the local maxwellian defined by the contact discontinuity of the euler equations uniformly away from the discontinuity as the knudsen number e tends to zero the proof is obtained by an appropriately chosen scaling and the energy method through the micro macro decomposition

wave front tracing and asymptotic stability of planar travelling waves for a two dimensional shallow river model abstract the propagation of surface water waves in a frictional channel with a uniformly inclined bed is governed by a two dimensional shallow river model in this paper we consider the time asymptotic stability of weak planar travelling waves for a two dimensional shallow river model with darcy s law we derive an effective parabolic equation to analyze the wave front motion by employing weighted energy estimates we show that weak planar travelling waves are time asymptotically stable under sufficiently small perturbations

relaxation limit for piecewise smooth solutions to systems of conservation laws abstract in this paper we study the asymptotic equivalence of a general system of 1 d conservation laws and the corresponding relaxation model proposed by s jin and z xin 1995 comm pure appl math 48 235 277 in the limit of small relaxation rate it is shown that if the relaxation system satisfies the subcharacteristic condition and the solution of the hyperbolic conservation laws is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition then there exist solutions of the relaxation systems that converge to the solution of the original conservation laws equilibrium system at a rate of order e as the rate of relaxation e goes to zero the proof uses a matched asymptotic analysis and an energy estimate related to the nonlinear stability theory for viscous shock profiles

multi symplectic runge kutta collocation methods for hamiltonian wave equations a number of conservative pdes like various wave equations allow for a multi symplectic formulation which can be viewed as a generalization of the symplectic structure of hamiltonian odes we show that gauss legendre collocation in space and time leads to multi symplectic integrators i e to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for hamiltonian odes we also discuss the issue of conservation of energy and momentum since time discretization by a gauss legendre method is computational rather expensive we suggest several semi explicit multi symplectic methods based on gauss legendre collocation in space and explicit or linearly implicit symplectic discretizations in time

convergence of the viscosity method for isentropic gas dynamics a convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

a remark on numerical errors downstream of slightly viscous shocks lower order errors downstream of a shock layer have been detected in computations with nonconstant solutions when using higher order shock capturing schemes in one and two dimensions b engquist and b sjogreen siam j numer anal 35 1998 pp 2464 2485 by analyzing the steady state solution of slightly viscous hyperbolic systems of conservation laws we find that the solution can have an cal o h dependence downstream of a shock layer although the viscous term in that region is of cal o h 2 numerical examples illustrate the analysis

the inviscid limit for an inflow problem of compressible viscous gas in presence of both shocks and boundary layers in this paper we study the inviscid limit problem for the navier stokes equations of one dimensional compressible viscous gas on half plane we prove that if the solution of the inviscid euler system on half plane is piecewise smooth with a single shock satisfying the entropy condition then there exist solutions to navier stokes equations which converge to the inviscid solution away from the shock discontinuity and the boundary at an optimal rate of e1 as the viscosity e tends to zero

zero dissipation limit to strong contact discontinuity for the 1 d compressible navier stokes equations abstract in this paper we study the zero dissipation limit problem for the one dimensional compressible navier stokes equations we prove that if the solution of the inviscid euler equations is piecewise constants with a contact discontinuity then there exist smooth solutions to the navier stokes equations which converge to the inviscid solution away from the contact discontinuity at a rate of 1 4 as the heat conductivity coefficient tends to zero provided that the viscosity is of higher order than the heat conductivity without loss of generality we set 0 here we have no need to restrict the strength of the contact discontinuity to be small

zero relaxation limit for piecewise smooth solutions to a rate type viscoelastic system in the presence of shocks abstract we study a rate type viscoelastic system proposed in i suliciu int j engng sci 28 1990 827 841 which is a 3 3 hyperbolic system with relaxation as the relaxation time tends to zero this system converges to the well known p system formally in the case that the solutions of the p system are piecewise smooth including finitely many noninteracting shock waves we show that there exist smooth solutions for suliciu s model which converge to those of the p system strongly as the relaxation time goes to zero the method used here is the so called matched asymptotic analysis suggested in j goodman and z p xin arch ration mech anal 121 1992 235 265 which includes two parts the matched asymptotic expansion and stability analysis

zero dissipation limit to rarefaction waves for the one dimensional navier stokes equations of compressible isentropic gases we study the zero dissipation limit problem for the one dimensional navier stokes equations of compressible isentropic gases in the case that the corresponding euler equations have rarefaction wave solutions we prove that the solutions of the navier stokes equations with centered rarefaction wave data exist for all time and converge to the centered rarefaction waves as the viscosity vanishes uniformly away from the initial discontinuities in the case that either the effects of initial layers are ignored or the rarefaction waves are smooth we then obtain a rate of convergence which is valid uniformly for all time our method of proof consists of a scaling argument and elementary energy analysis based on the underlying wave structure 1993 john wiley sons inc

existence and stability of multidimensional shock fronts in the vanishing viscosity limit in this paper we present a new approach to the study of linear and nonlinear stability of inviscid multidimensional shock waves under small viscosity perturbation yielding optimal estimates and eventually an extension to the viscous case of the celebrated theorem of majda on existence and stability of multidimensional shock waves more precisely given a curved lax shock solution u0 to a hyperbolic system of conservation laws we construct nearby viscous shock solutions ue to a parabolic viscous perturbation of the hyperbolic system which converge to u0 as viscosity e 0 and satisfy an appropriate conormal version of majda s stability estimate

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

convergence to the superposition of rarefaction waves and contact discontinuity for the 1 d compressible navier stokes korteweg system abstract this paper is concerned with the vanishing capillarity viscosity limit for the one dimensional compressible navier stokes korteweg system to the riemann solution of the euler system that consists of the supposition of two rarefaction waves and a contact discontinuity it is shown that there exists a family of smooth solutions to the compressible navier stokes korteweg system which converge to the riemann solution away from the initial time t 0 and the contact discontinuity located at x 0 as the coefficients of capillarity viscosity and heat conductivity tend to zero moreover a uniform convergence rate in terms of the above physical parameters is also obtained here the strengths of both the rarefaction waves and the contact discontinuity are not required to be small

viscous standing asymptotic states of isentropic compressible flows through a nozzle in this work we consider a viscous regularization of a well known one dimensional model for isentropic viscous compressible flows through a nozzle for the existence and multiplicity of standing asymptotic states for a certain type of ducts a complete analysis in a framework of dynamical systems is provided as an application of the geometric singular perturbation theory we show that all standing asymptotic states admit viscous profiles

nonlinear stability of discrete shocks for systems of conservation laws in this paper we study the asymptotic nonlinear stability of discrete shocks for the lax friedrichs scheme for approximating general m systems of nonlinear hyperbolic conservation laws it is shown that weak single discrete shocks for such a scheme are nonlinearly stable in the lp norm for all p 1 provided that the sums of the initial perturbations equal zero these results should shed light on the convergence of the numerical solution constructed by the lax friedrichs scheme for the single shock solution of system of hyperbolic conservation laws if the riemann solution corresponding to the given far field states is a superposition of m single shocks from each characteristic family we show that the corresponding multiple discrete shocks are nonlinearly stable in l p p 2 these results are proved by using both a weighted estimate and a characteristic energy method based on the internal structures of the discrete shocks and the essential monotonicity of the lax friedrichs scheme

zero viscosity limit for analytic solutions of the navier stokes equation on a half space ii construction of the navier stokes solution this is the second of two papers on the zero viscosity limit for the incompressible navier stokes equations in a half space in either 2d or 3d under the assumption of analytic initial data we construct solutions of navier stokes for a short time which is independent of the viscosity the navier stokes solution is constructed through a composite asymptotic expansion involving the solutions of the euler and prandtl equations which were constructed in the first paper plus an error term this shows that the navier stokes solution goes to an euler solution outside a boundary layer and to a solution of the prandtl equations within the boundary layer the error term is written as a sum of first order euler and prandtl corrections plus a further error term the equation for the error term is weakly nonlinear its linear part is the time dependent stokes equation this error equation is solved by inversion of the stokes equation through expressing the solution as a regular euler like part plus a boundary layer prandtl like part the main technical tool in this analysis is the abstract cauchy kowalewski theorem

vanishing curvature viscosity for front propagation in this paper we study the front propagation with constant speed and small curvature viscosity we first investigate two related problems of conservation laws one of which is on the nonlinear viscosity methods for the conservation laws and the other one is on the structure of solutions to conservation laws with l 1 initial data we show that the nonlinear viscosity methods approaching the piecewise smooth solutions with finitely many discontinuity for convex conservation laws have the first order rate of l 1 convergence the solutions of conservation laws with l 1 initial data are shown to be bounded after t 0 if all singular points of initial data are from shocks these results suggest that the front propagation with constant speed and a small curvature viscosity will approach the front movements with a constant speed as the small parameter goes to zero after the front breaks down the cusps will disappear promptly and corners will be formed 2000 academic press

multidimensional viscous shocks i degenerate symmetrizers and long time stability received by the editors september 18 2002 2000 mathematics subject classification primary 35l60 secondary 35b35 35b45 35k55 35l65 76l05

viscous limits for piecewise smooth solutions of the p system abstract we study the zero dissipation problem for a one dimensional model system for the isentropic flow of a compressible viscous gas the so called p system with viscosity when the solution of the inviscid problem is piecewise smooth and having finitely many noninteracting shocks satisfying the entropy condition there exists unique solution to the viscous problem which converges to the given inviscid solution away from shock discontinuities at a rate of order e as the viscosity coefficient e goes to zero the proof is given by a matched asymptotic analysis and an elementary energy method and we do not need the smallness condition on the shock strength

viscous limit to contact discontinuity for the 1 d compressible navier stokes equations abstract in this paper we study the zero dissipation limit problem for the one dimensional compressible navier stokes equations we prove that if the solution of the inviscid euler equations is piecewise constants with a contact discontinuity then there exist smooth solutions to the navier stokes equations which converge to the inviscid solution away from the contact discontinuity at a rate of 3 4 as the heat conductivity coefficient tends to zero provided that the viscosity is higher order than the heat conductivity or the same order as here we have no need to restrict the strength of the contact discontinuity to be small

vanishing viscosity limit of the compressible navier stokes equations for solutions to a riemann problem we study the vanishing viscosity limit of the compressible navier stokes equations to the riemann solution of the euler equations that consists of the superposition of a shock wave and a rarefaction wave in particular it is shown that there exists a family of smooth solutions to the compressible navier stokes equations that converges to the riemann solution away from the initial and shock layers at a rate in terms of the viscosity and the heat conductivity coefficients this gives the first mathematical justification of this limit for the navier stokes equations to the riemann solution that contains these two typical nonlinear hyperbolic waves

symmetric euler and navier stokes shocks in stationary barotropic flow on a bounded domain abstract we construct stationary solutions to the barotropic compressible euler and navier stokes equations in several space dimensions with spherical or cylindrical symmetry for given dirichlet data on a sphere or a cylinder we first construct smooth and radially symmetric solutions to the euler equations in the exterior domain on the other hand stationary smooth solutions in the interior domain necessarily become sonic and cannot be continued beyond a critical inner radius we then use these solutions to construct entropy satisfying shocks for the euler equations in the region between two concentric spheres or cylinders next we construct smooth navier stokes solutions converging to the previously constructed euler shocks in the small viscosity limit in the process we introduce a new technique for constructing smooth solutions which exhibit a fast transition in the interior to a class of two point boundary problems

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

pointwise error estimates for scalar conservation laws with piecewise smooth solutions we introduce a new approach to obtain sharp pointwise error estimates for viscosity approximation and in fact more general approximations to scalar conservation laws with piecewise smooth solutions to this end we derive a transport inequality for an appropriately weighted error function the key ingredient in our approach is a one sided interpolation inequality between classical l1 error estimates and lip stability bounds the one sided interpolation interesting for its own sake enables us to convert a global l1 result into a nonoptimal local estimate this in turn provides the necessary bounds on the coefficients of the above mentioned transport inequality estimates on the weighted error then follow from the maximum principle and a bootstrap argument yields optimal pointwise error bound for the viscosity approximation unlike previous works in this direction our method can deal with finitely many waves with possible collisions moreover in our approach one does not follow the characteristics but instead makes use of the energy method and hence this approach could be extended to other types of approximate solutions

adaptive finite element methods for conservation laws based on a posteriori error estimates we prove a posteriori error estimates for a finite element method for systems of strictly hyperbolic conservation laws in one space dimension and design corresponding adaptive methods the proof of the a posteriori error estimates is based on a strong stability estimate for an associated dual problem together with the galerkin orthogonality of the finite element method the strong stability estimate uses the entropy condition for the system in an essential way 1995 john wiley sons inc

on the convergence rate of vanishing viscosity approximations given a strictly hyperbolic genuinely nonlinear system of conservation laws we prove the a priori boundu t u e t l1 o 1 1 t e ln e on the distance between an exact bv solution u and a viscous approximation u e letting the viscosity coefficient e 0 in the proof starting from u we construct an approximation of the viscous solution u e by taking a mollification u e and inserting viscous shock profiles at the locations of finitely many large shocks for each fixed e error estimates are then obtained by introducing new lyapunov functionals that control interactions of shock waves in the same family and also interactions of waves in different families c 2004 wiley periodicals inc

on the l2 and l1 convergence rates of viscous solutions of the keyfitz kranzer system with piecewise smooth and large bv data abstract we study the zero dissipation problem of the keyfitz kranzer system in l 2 and l 1 spaces when the solution of the inviscid problem is piecewise smooth and has finitely many noninteracting shocks with finite strength there exists for each e the viscosity unique solution to the viscous problem with modified initial data and it converges to the given inviscid solution away from shock discontinuities as e tends to zero convergence rates are given in terms of e the proof is given by a matched asymptotic analysis and a weighted elementary energy method

galloping instability of viscous shock waves abstract motivated by physical and numerical observations of time oscillatory galloping spinning and cellular instabilities of detonation waves we study poincare hopf bifurcation of traveling wave solutions of viscous conservation laws the main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum preventing standard reduction to a finite dimensional center manifold we overcome this by direct lyapunov schmidt reduction using detailed pointwise bounds on the linearized solution operator to carry out a nonstandard implicit function construction in the absence of a spectral gap the key computation is a space time stability estimate on the transverse linearized solution operator reminiscent of duhamel estimates carried out on the full solution operator in the study of nonlinear stability of spectrally stable traveling waves

viscosity solutions with shocks a solution of single nonlinear first order equations may develop jump discontinuities even if initial data is smooth typical examples include a crude model equation describing some bunching phenomena observed in epitaxial growth of crystals as well as conservation laws where jump discontinuities are called shocks conventional theory of viscosity solutions does not apply we introduce a notion of proper viscosity solutions to track whole evolutions for such equations in multi dimensional spaces we establish several versions of comparison principles we also study the vanishing viscosity method to construct a unique global proper solution at least when the evolution is monotone in time or the initial data is monotone in some sense under additional technical assumptions in fact we prove that the graph of approximate solutions converges to that of a proper solution in the hausdorff distance topology such a convergence is also established for conservation laws with monotone data in particular local uniform convergence outside shocks is proved 2001 john wiley sons inc

existence and asymptotic stability of relaxation discrete shock profiles kin this paper we study the asymptotic nonlinear stability of discrete shocks of the relaxing scheme for approximating the general system of nonlinear hyperbolic conservation laws the existence of discrete shocks is established by suitable manifold construction and it is shown that weak single discrete shocks for such a scheme are nonlinearly stable in l 2 provided that the sums of the initial perturbations equal zero these results should shed light on the convergence of the numerical solution constructed by the relaxing scheme for the single shock solution of the system of hyperbolic conservation laws these results are proved by using both a weighted norm estimate and a characteristic energy method based on the internal structures of the discrete shocks

viscosity methods for piecewise smooth solutions to scalar conservation laws it is proved that for scalar conservation laws if the flux function is strictly convex and if the entropy solution is piecewise smooth with finitely many discontinuities which includes initial central rarefaction waves initial shocks possible spontaneous formation of shocks in a future time and interactions of all these patterns then the error of viscosity solution to the inviscid solution is bounded by o log in the l 1 norm which is an improvement of the o upper bound if neither central rarefaction waves nor spontaneous shocks occur the error bound is improved to o