initial boundary value problem for the system of compressible adiabatic flow through porous media abstract we study the initial value problem for the system of compressible adiabatic flow through porous media in the one space dimension with fixed boundary condition under the restriction on the oscillations in the initial data we establish the global existence and large time behavior for the classical solutions via the combination of characteristic analysis and energy estimate methods the time asymptotic states for the solution are found and the exponential convergence to the asymptotic states is proved it is also shown that this problem can be approximated very well time asymptotically by an initial boundary problem for the related diffusion equations obtained from the original hyperbolic system by darcy s law provided that the initial total mass is the same the difference between these two solutions tends to zero exponentially fast as time goes to infinity in the sense of h 1 the diffusive phenomena caused by damping mechanism with boundary effects are thus observed

boundary effect on asymptotic behaviour of solutions to the p system with linear damping we consider the asymptotic behaviour of solutions to the p system with linear damping on the half line r 0 vt ux 0 ut p v x u with the dirichlet boundary condition u x 0 0 or the neumann boundary condition ux x 0 0 the initial date v0 u0 x has the constant state v u at x l hsiao and t p liu commun math phys 143 1992 599 605 have shown that the solution to the corresponding cauchy problem behaves like diffusion wave and k nishihara j differential equations131 1996 171 188 137 1997 384 395 has proved its optimal convergence rate our main concern in this paper is the boundary effect in the case of null dirichlet boundary condition on u the solution v u is proved to tend to v 0 as t tends to infinity its optimal convergence rate is also obtained by using the green function of the diffusion equation with constant coefficients in the case of null neumann boundary condition on u v 0 t is conservative and v 0 t v0 0 by virtue of the first equation so that v x t is expected to tend to the diffusion wave v x t connecting v0 0 and v in fact the solution v u x t is proved to tend to v x t 0 in the special case v0 0 v the optimal convergence rate is also obtained however this is not known in the case v0 0 v

global bv solutions of compressible euler equations with spherical symmetry and damping abstract in this paper we study the global existence of bv solutions for compressible euler equations with spherical symmetry and damping using glimm s scheme the key point to consider is the combination of the geometric effects due to the spherical symmetry and the effects of the frictional damping on the total variation of the solutions by measuring the strength of the waves by r s where r s are the riemann invariants we construct a function which leads to the boundedness of the bv norm of the corresponding approximate solutions

convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping we consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time asymptotically the hyperbolic model may be viewed as isentropic euler equations with friction term added to the momentum equation to model gas flow through a porous media in this case our result justifies darcy s law time asymptotically our model may also be viewed as an elastic model with damping

existence and asymptotic behavior of global smooth solution for p system with damping and boundary effect abstract in this paper we study the existence and asymptotic behavior of global smooth solution for p system with damping on the strip 0 1 0 v t u x 0 u t p v x u with the null dirichlet boundary condition u 0 t u 1 t 0 first we prove the existence of the global smooth solution under the assumption that only the c 0 norm of the first derivatives of the initial data is sufficiently small while the c 0 norm of the initial data is not necessarily small second when the initial data is in c 2 and the c 0 norm of the first derivatives of the initial data is large enough then the first derivatives of the corresponding c 2 smooth solution must blow up in finite time finally we consider the asymptotic behavior of the global smooth solution precisely we show that the solution of this problem tends to the corresponding nonlinear diffusive wave governed by the classical darcy s law exponentially fast as time goes to infinity in the sense of h 1 norm provided the initial total mass is the same and thus justifies the validity of darcy s law in large time

best asymptotic profile for linear damped p system with boundary effect abstract this paper is devoted to the study of the 2 2 linear damped p system with boundary effect by a heuristic analysis we realize that the best asymptotic profile for the original solution is the parabolic solution of the ibvp for the corresponding porous media equation with a specified initial data in particular we further show the convergence rates of the original solution to its best asymptotic profile which are much better than the existing rates obtained in the previous works the approach adopted in the paper is the elementary weighted energy method with green function method together

approximation schemes for viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of applications including the calculus of variations control theory and differential games recently m g crandall and p l lions established the correct notion of generalized solutions for these equations this article discusses the convergence of general approximation schemes to this solution and gives under certain hypotheses explicit error estimates these results are then applied to obtain various representations as limits of solutions of general explicit and implicit finite difference schemes with error estimates

convergence to strong nonlinear diffusion waves for solutions to p system with damping in this paper we consider the so called p system with linear damping and we will prove an optimal decay estimates without any smallness conditions on the initial error more precisely if we restrict the initial data v0 u0 in the space h3 r l1 r h2 r l1 r then we can derive faster decay estimates than those given in p marcati m mei b rubino optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping j math fluid mech 7 2 2005 224 240 h zhao convergence to strong nonlinear diffusion waves for solutions of p system with damping j differential equations 174 1 2001 200 236 and m jian c zhu convergence to strong nonlinear diffusion waves for solutions to p system with damping on quadrant j differential equations 246 1 2009 50 77

the spectral geometry of the hopf fibration we study the spectral geometry of the hopf fibration and determine the right invariant metrics on for which there exist eigenforms of the laplacian on which pull back to eigenforms of the laplacian on we show that the pull back of the volume form on can be an eigenform of the laplacian on with non zero eigenvalue we show that if is a principal bundle with a bundle metric and that if then eigenvalues cannot change thus eigenvalues do not change for the fibrations and if we also study the corresponding questions in the complex category for the fibration of the hopf manifold

multi phase computations of the semiclassical limit of the schrodinger equation and related problems whitham vs wigner abstract we present and compare two different techniques to obtain the multi phase solutions for the schrodinger equation in the semiclassical limit the first is whitham s averaging method which gives the modulation equations governing the evolution of multi phase solutions the second is the wigner transform a convenient tool to derive the semiclassical limit equation in the phase space the vlasov equation for the linear schrodinger equation motivated by the linear superposition principle we derive and prove the multi phase ansatz for the wigner function by the stationary phase method and then use the ansatz to close the moment equations of the vlasov equation and obtain the multi phase equations in the physical space we show that the multi phase equations so derived agree with those derived by whitham s averaging method which can be proved using different arguments generic way of obtaining and computing the multi phase equations by the wigner function is given and kinetic schemes are introduced to solve the multi phase equations the numerical schemes based on multiphase closure of the liouville equation are purely eulerian and only operate in the physical space several numerical examples are given to explore the validity of this approach similar studies are conducted for the linearized korteweg de vries equation and the linear wave equation

convergence to nonlinear diffusion waves for solutions of p system with time dependent damping abstract in this paper we study the cauchy problem for the hyperbolic p system with time gradually degenerate damping term 1 1 t u for 0 1 and show that the damped p system has a couple of global solutions uniquely and such solutions tend time asymptotically to the shifted nonlinear diffusion waves which are the solutions of the corresponding nonlinear parabolic equation governed by the darcy s law we further derive the convergence rates when the initial perturbations are in l 2 the approach adopted is the technical time weighted energy method

lie algebras under constraints and non bijective canonical transformations the concept of a lie algebra under constraints is developed in connection with the theory of non bijective canonical transformations a finite dimensional vector space m carrying a faithful linear representation of a lie algebra l is mapped into a lower dimensional space m in such a manner that a subalgebra l0 of l is mapped into d l0 0 the lie algebra l under the constraint d l0 0 is the largest subalgebra l1 of l that can be represented faithfully on m if l0 is semisimple then l1 is shown to be the centraliser cent ll0 if l is semisimple and l0 is a one dimensional diagonal subalgebra of a cartan subalgebra of l then l1 is shown to be the factor algebra cent ll0 l0 the later two results are applied to non bijective canonical transformations generalising the kustaanheimo stiefel transformation

an energy preserving nonlinear system modeling a string with input and output on the boundary abstract we study an energy conserving distributed parameter system described by a nonlinear string equation with the input and output at the boundary we prove the existence of global smooth solutions to this quasilinear hyperbolic system if the initial data and the boundary input are small if moreover the input function becomes zero after some finite time then the state trajectories decay exponentially

l 1 approximation of stationary hamilton jacobi equations we describe a nonlinear finite element technique to approximate the solutions of stationary hamilton jacobi equations in two space dimensions using continuous finite elements of arbitrary degree the method consists of minimizing a functional containing the l 1 norm of the hamiltonian plus a discrete entropy it is shown that the approximate sequence converges to the unique viscosity solution under appropriate hypotheses on the hamiltonian and the mesh family

convergence of a non monotone scheme for hamilton jacobi bellman equations with discontinous initial data we prove the convergence of a non monotonous scheme for a one dimensional first order hamilton jacobi bellman equation of the form v t max f x vx 0 v 0 x v0 x the scheme is related to the hjb ultrabee scheme suggested in bokanowski and zidani j sci comput 30 1 1 33 2007 we show for general discontinuous initial data a first order convergence of the scheme in l1 norm towards the viscosity solution we also illustrate the non diffusive behavior of the scheme on several numerical examples

asymptotic behavior of solutions of a nonlinear diffusion equation a nonlinear diffusion equation which arises in the theory of plasma physics is discussed it is shown that the solutions of a class of initial boundary value problems converge to a similarity solution as t to infty the optimal rate of convergence is established

generalized canonical transformations for time dependent systems we introduce the concept of generalized canonical transformations as symplectomorphisms of the extended phase space we prove that any such transformation factorizes in a standard canonical transformation times another one that changes only the time variable the theory of generating functions as well as that of hamilton jacobi is developed some further applications are developed

implementation of the entropy viscosity method with the discontinuous galerkin method the notion of entropy viscosity method introduced in guermond and pasquetti 21 is extended to the discontinuous galerkin framework for scalar conservation laws and the compressible euler equations

an algebraic and geometric approach to non bijective quadratic transformations this work is concerned with an extension of the so called levi civita and kustaanheimo stiefel transformations the extension is achieved along two lines firstly the authors examine the latter transformations in other dimensions than the ones originally considered by on the one hand levi civita and on the other hand by kustaanheimo and stiefel secondly they pass from the compact to the non compact case this leads to quadratic non bijective transformations that they refer to as hurwitz or kustaanheimo stiefel like and quasihurwitz or levi civita like transformations the hurwitz and quasihurwitz transformations are introduced and studied in an algebraic framework which relies on the use of eight dimensional cayley dickson algebras an explicit formulation of the hurwitz and quasihurwitz transformations is also given in terms of clifford algebras the hurwitz transformations are investigated from a geometrical viewpoint indeed they are connected to hopf and pseudohopf fibrations finally some differential aspects of the hurwitz and to a lesser extent quasihurwitz transformations are developed in view of future physical applications

convergence of an antidiffusion lagrange euler scheme for quasilinear equations this paper presents an explicit scheme which may be broken into several steps the first one is a transport phase for a time step with a projection on a lagrangian mesh then a new projection is performed on a fixed eulerian mesh lastly a corrected antidiffusion step of the shasta type occurs a stability condition appears to make the scheme easily computable this condition allows larger timesteps than the well known courant friedrichs lewy condition a more complex version of the scheme is proposed which works without any stability condition convergence towards the entropy solution is proved a few numerical experiments are reported showing the quality of the approximation of shocks or giving some information about the precision

on the convergence rate of operator splitting for hamilton jacobi equations with source terms we establish a rate of convergence for a semidiscrete operator splitting method applied to hamilton jacobi equations with source terms the method is based on sequentially solving a hamilton jacobi equation and an ordinary differential equation the hamilton jacobi equation is solved exactly while the ordinary differential equation is solved exactly or by an explicit euler method we prove that the l infty error associated with the operator splitting method is bounded by mathcal o delta t where delta t is the splitting or time step this error bound is an improvement over the existing mathcal o sqrt delta t bound due to souganidis nonlinear anal 9 1985 pp 217 257 in the one dimensional case we present a fully discrete splitting method based on an unconditionally stable front tracking method for homogeneous hamilton jacobi equations it is proved that this fully discrete splitting method possesses a linear convergence rate moreover numerical results are presented to illustrate the theoretical convergence results

initial boundary value problem for p system with damping in half space abstract in this paper the initial boundary value problem for p system with damping in half space is studied with a posed mix boundary condition the fundamental solution green s function for cauchy problem is constructed with a sharp decaying structure and a precise capture of the singularity by a singularity removal process later the symbols of both fundamental solution and green s function for initial boundary value problem are obtained with transformed variable s and spatial variable x while in this level the green s function could be decomposed into fundamental solution and boundary operator thus with the study of boundary operator and previous precise information about the fundamental solution the pointwise structure as well as the singular structure of the green s function is established finally the nonlinear stability is obtained by the green s function and a priori estimate from energy method

global solution and large time behavior of the 3d compressible euler equations with damping national natural science foundation of china nsaf 10976026 national natural science foundation of china 11271305

asymptotic behaviors of solutions of quasilinear hyperbolic equations with linear damping ii this paper as a follow up to the recent work of k nishihara 1997 j differential equations133 384 395 is concerned with the asymptotic behaviors of the solution of quasilinear hyperbolic equation with linear damping vtt a vx x vt 0 a v 0forallv r which satisfies the following prescribed initial condition v t x vt t x t 0 v0 x v1 x x r compared with the results obtained by k nishihara the main novelties of our present paper lie in the following first we recover all the results obtained by k nishihara under some other smallness assumptions on the initial data v0 x v1 x which are much weaker than those needed in k nishihara s arguments second under some additional assumptions on the nonlinear function a v and on the initial data v0 x v1 x when 0 0 1 0 we succeed in proving that t x is still an asymptotic profile of v t x and this answers partially the open problem left by k nishihara

large time behavior of solutions for compressible euler equations with damping in r 3 we consider the long time behavior and optimal decay rates of global strong solutions for the isentropic compressible euler equations with damping in r3 in the present paper when the regular initial data belong to some sobolev space hl r3 b 1 s r3 with l 4 and s 0 1 we show that the density of the system converges to its equilibrium state at the rates 1 t 34 s2 in the l2 norm or 1 t 32 s2 in the l norm respectively the momentum of the system decays at the rates 1 t 54 s2 in the l2 norm or 1 t 2 s2 in the l norm respectively which are shown to be optimal for the compressible euler equations with damping

high order schemes and entropy condition for nonlinear hyperbolic systems of conservation laws on presente une procedure systematique de construction des approximations explicites des systemes strictement hyperboliques des lois de conservation on etablit la convergence

l 1 minimization methods for hamilton jacobi equations the one dimensional case a new approximation technique based on l 1 minimization is introduced it is proven that the approximate solution converges to the viscosity solution in the case of one dimensional stationary hamilton jacobi equation with convex hamiltonian

asymptotics in nonlinear evolution system with dissipation and ellipticity on quadrant abstract in this paper we consider an initial boundary value problem for some nonlinear evolution system with dissipation and ellipticity we establish the global existence and furthermore obtain the l p p 2 decay rates of solutions corresponding to diffusion waves the analysis is based on the energy method and pointwise estimates

global smooth solutions for the quasilinear wave equation with boundary dissipation abstract we consider the existence of global solutions of the quasilinear wave equation with a boundary dissipation structure of an input output in high dimensions when initial data and boundary inputs are near a given equilibrium of the system our main tool is the geometrical analysis the main interest is to study the effect of the boundary dissipation structure on solutions of the quasilinear system we show that the existence of global solutions depends not only on this dissipation structure but also on a riemannian metric given by the coefficients and the equilibrium of the system some geometrical conditions on this riemannian metric are presented to guarantee the existence of global solutions in particular we prove that the norm of the state of the system decays exponentially if the input stops after a finite time which implies the exponential stabilization of the system by boundary feedback

theory of nonbijective canonical transformations in mechanics application to the coulomb problem it is shown that the study of nonbijective transformation requires a fiber bundle formulation of mechanics the conditions upon which nonbijective canonical point transformations can be defined are given then as an example we apply that theory to the study of the coulomb problem in two and three dimensions the hopf fibration leads to an inverse harmonic oscillator problem since the completion of this work a paper by g h ringwood and j t devreese has been published in j math phys 21 1390 1980 dealing with the same problem their work is based on the construction of propagators in quotient spaces the identity between the propagator prescriptions and nonlinear canonical transformations is not automatically fulfilled therefore it seems that the reliability of their results is not due to their method which in general is not correct but to an underlying property of the transformation used namely the kustaanheimo stiefel map see our results

the lp lq estimates of solutions to one dimensional damped wave equations and their application to the compressible flow through porous media we first obtain the lp lq estimates of solutions to the cauchy problem for one dimensional damped wave equation vtt vxx vt 0 v vt t 0 v0 v1 x x t r r corresponding to that for the parabolic equation t xx 0 t 0 v0 v1 x the estimates are shown by v t e t 2v0 t v0 t 2lp ct 121q 1p 1 v0 v1 lq t 1 etc for 1 q p to show the explicit formula of the damped wave equation will be used to apply the estimates to nonlinear problems is the second aim we will treat the system of a compressible flow through porous media the solution is expected to behave as the diffusion wave which is the solution to the porous media equation due to the darcy law when the initial data has the same constant state at a sharp lp convergence rate for p 2 has been recently obtained by nishihara proc roy soc edinburgh sect a 133a 2003 1 20 by choosing a suitably located diffusion wave we will show the l1 convergence applying

half space problem for euler equations with damping in 3 d abstract in this paper we consider the half space problem for euler equations with damping in 3 d we restudy the fundamental solution for the cauchy problem to obtain an exponentially sharp pointwise structure and a clear decomposition of the singular regular components later both green s function for initial boundary value problem and fundamental solutions for cauchy problems are investigated in the transformed domain after laplace transform the symbols are obtained and a connection between green s function and fundamental solutions are established for the pointwise space time structure of green s function finally the sharp estimates for green s function together with a priori estimates from the energy method for high order derivatives result in the nonlinear stability of the solution and also the decaying rates

one sided stability and convergence of the nessyahu tadmor scheme non oscillatory schemes are widely used in numerical approximations of nonlinear conservation laws the nessyahu tadmor nt scheme is an example of a second order scheme that is both robust and simple in this paper we prove a new stability property of the nt scheme based on the standard minmod reconstruction in the case of a scalar strictly convex conservation law this property is similar to the one sided lipschitz condition for first order schemes using this new stability we derive the convergence of the nt scheme to the exact entropy solution without imposing any nonhomogeneous limitations on the method we also derive an error estimate for monotone initial data

high resolution nonoscillatory central schemes for hamilton jacobi equations in this paper we construct second order central schemes for multidimensional hamilton jacobi equations and we show that they are nonoscillatory in the sense of satisfying the maximum principle thus these schemes provide the first examples of nonoscillatory second order godunov type schemes based on global projection operators numerical experiments are performed l 1 l infty errors and convergence rates are calculated for convex hamiltonians numerical evidence confirms that our central schemes converge with second order rates when measured in the l1 norm advocated in our recent paper numer math to appear the standard l infty norm however fails to detect this second order rate

large time behavior for the system of compressible adiabatic flow through porous media in r3 abstract the cauchy problem of the system of compressible adiabatic flow through porous media in r 3 is considered we obtain the global existence and large time behavior of the solution when the initial data is close to its equilibrium in h 3 norm this is a continuing work of 31 where due to non dissipative property of the entropy s an additional assumption that the initial data is bounded in l 1 norm plays a key role in closing the a priori energy estimates and hence establishing the global existence of the solution in this paper to remove this assumption we first decompose the solution u into high frequency part u h and low frequency part u l then by constructing delicate energy functionals for u h and optimal decay rates for u l we can get the desired energy estimates to close the a priori assumption furthermore we get the optimal l 2 l 2 decay rate of the solution and show that the velocity decays faster than the density

convergence to strong nonlinear diffusion waves for solutions of p system with damping this paper is concerned with the p system with frictional damping and our main purpose is two fold first we show that for a certain class of given large initial data nu 0 x u 0 x the cauchy problem 1 1 1 2 admits a unique global smooth solution nu t x u t x and such a solution tends time asymptotically at the optimal l p 2 less than or equal to p less than or equal to infinity decay rates to the corresponding nonlinear diffusion wave over bar t x u over bar t x governed by the classical darcy s law provided that the corresponding prescribed initial error function v 0 x u 0 x ties in h 3 x h 2 r boolean and l 1 x l 1 r compared with former results in this direction obtained by l hsiao and t p liu 1992 comm math phys 143 599 605 k nishihara 1996 j differential equations 131 171 188 and k nishihara w k wang and t yang 2000 j differential equations 161 191 218 our main novelty lies in the facts that the nonlinear diffusion wave over bar t x u over bar t x need not to be weak and v 0 x u 0 x can be chosen arbitrarily large secondly we show that the nonlinear diffusion waves over bar t x u over bar t x are nonlinear stable provided that the strength or the nonlinear diffusion waves is weak and that the initial disturbance v 0 x u 0 x satisfies the assumption that parallel tov 0xx x parallel to l infinity parallel tou 0x x parallel to l infinity is sufficiently small we also show that the smallness assumption imposed oil the strength of the diffusion waves is a necessary condition to guarantee the nonlinear stability result and compared with the corresponding results obtained by l hsiao and t p liu 1992 comm math phys 143 599 605 the smallness conditions we imposed on the initial disturbance are much more weaker c 2001 academic press

global existence and large time behavior for the system of compressible adiabatic flow through porous media in r 3 abstract the system of compressible adiabatic flow through porous media is considered in r 3 in the present paper the global existence and uniqueness of classical solutions are obtained when the initial data is near its equilibrium we also show that the pressure of the system converges to its equilibrium state at the same l 2 rate 1 t 3 4 as the navier stokes equations without heat conductivity but the velocity of the system decays at the l 2 rate 1 t 5 4 which is faster than the l 2 rate 1 t 3 4 for the navier stokes equations without heat conductivity duan and ma 2008 3

entropy viscosity method for nonlinear conservation laws a new class of high order numerical methods for approximating nonlinear conservation laws is described entropy viscosity method the novelty is that a nonlinear viscosity based on the local size of an entropy production is added to the numerical discretization at hand this new approach does not use any flux or slope limiters applies to equations or systems supplemented with one or more entropy inequalities and does not depend on the mesh type and polynomial approximation various benchmark problems are solved with finite elements spectral elements and fourier series to illustrate the capability of the proposed method

darcy s law as long time limit of adiabatic porous media flow abstract it is conjectured that darcy s law governs the motion of compressible porous media flow in large time this has been justified for one dimensional isentropic flows in this work we show the conjecture is true for one dimensional adiabatic flows with generic small smooth initial data

convergence of semi lagrangian approximations to convex hamilton jacobi equations under very large courant numbers we consider a class of semi lagrangian high order approximation schemes for convex hamilton jacobi equations in this framework we prove that under certain restrictions on the relationship between delta x and delta t the sequence of approximate solutions is uniformly lipschitz continuous and hence by consistency that it converges to the exact solution the argument is suitable for most reconstructions of interest including high order polynomials and eno reconstructions

l 1 stability and error estimates for approximate hamilton jacobi solutions we study the l 1 stability and error estimates of general approximate solutions for the cauchy problem associated with multidimensional hamilton jacobi h j equations for strictly convex hamiltonians we obtain a priori error estimates in terms of the truncation errors and the initial perturbation errors we then demonstrate this general theory for two types of approximations approximate solutions constructed by the vanishing viscosity method and by godunov type finite difference methods if we let epsilon denote the small scale of such approximations the viscosity amplitude epsilon the spatial grad size delta x etc then our l 1 error estimates are of cal o epsilon and are sharper than the classical l infty results of order one half cal o sqrt epsilon the main building blocks of our theory are the notions of the semi concave stability condition and l 1 measure of the truncation error the whole theory could be viewed as a multidimensional extension of the lip prime stability theory for one dimensional nonlinear conservation laws developed by tadmor et al 34 24 25 in addition we construct new godunov type schemes for h j equations which consist of an exact evolution operator and a global projection operator here we restrict our attention to linear projection operators first order schemes we note however that our convergence theory applies equally well to nonlinear projections used in the context of modern high resolution conservation laws we prove semi concave stability and obtain l 1 bounds on their associated truncation errors l 1 convergence of order one then follows second order central godunov type schemes are also constructed numerical experiments are performed errors and orders are calculated to confirm our l 1 theory

second order numerical methods for first order hamilton jacobi equations we present practical numerical methods which produce provably second order approximations for a class of stationary first order hamilton jacobi partial differential equations using probabilistic methods we derive high order asymptotic expansions for a first order method and then use those results to design second order methods we prove second order convergence for the solution and for its gradient on a subset of the domain where the solution is smooth although we limit our attention to second order schemes in principle the techniques in this paper can be extended to arbitrarily high order methods examples illustrate the rate of convergence as well as global sharp resolution of discontinuities the hamilton jacobi equations we consider correspond to deterministic optimal control problems and our rate of convergence results are valid for the value functions and for the optimal feedback controls

the mixed initial boundary value problem for quasilinear hyperbolic systems with linearly degenerate characteristics abstract in this paper we investigate the mixed initial boundary value problem with small bv data for linearly degenerate quasilinear hyperbolic systems with general nonlinear boundary conditions in the half space t x t 0 x 0 as the result in a bressan contractive metrics for nonlinear hyperbolic systems indiana univ math j 37 1988 409 421 suggests that one may achieve global smoothness even if the c 1 norm of the initial data is large we prove that if the c 1 norm of the initial and boundary data is bounded but possibly large and the bv norm of the initial and boundary data is sufficiently small then the solution remains c 1 globally in time applications to quasilinear hyperbolic systems arising in physics and mechanics particularly to the system describing the motion of the relativistic string in the minkowski space time r 1 n are also given

existence and asymptotic behavior of c1 solutions to the multi dimensional compressible euler equations with damping abstract in this paper the existence and asymptotic behavior of c 1 solutions to the multi dimensional compressible euler equations with damping on the framework of besov space are considered comparing with the well posedness results of sideris thomases wang t sideris b thomases d h wang long time behavior of solutions to the three dimensional compressible euler with damping comm partial differential equations 28 2003 953 978 we weaken the regularity assumptions on the initial data the global existence lies on a crucial a priori estimate which is obtained by the spectral localization method the main analytic tools are the littlewood paley decomposition and bony s paraproduct formula

half space problem of the system of compressible adiabatic flow through porous media in multi d abstract in this paper we consider the half space problem for the system of compressible adiabatic flow through porous media in 2 d and 3 d combining energy method green s function method and bootstrap method one establishes the decaying energy estimates which lead to global existence and optimal l 2 decaying rate of the solution finally for more large time information especially in spatial variables one applies green s function solution representation obtained by duhamel s principle and a priori estimate from energy method to obtain the space time pointwise structure of the solution

nonlinear diffusion waves for hyperbolic p system with nonlinear damping abstract this paper is concerned with the p system of hyperbolic conservation laws with nonlinear damping when the constant states are small the solutions of the cauchy problem for the damped p system globally exist and converge to their corresponding nonlinear diffusion waves which are the solutions of the corresponding nonlinear parabolic equation given by the darcy s law the optimal convergence rates are also obtained in order to overcome the difficulty caused by the nonlinear damping a couple of correction functions have been technically constructed the approach adopted is the elementary energy method together with the technique of approximating green function on the other hand when the constant states are large the solutions of the cauchy problem for the p system will blow up at a finite time

unconditionally stable methods for hamilton jacobi equations we present new numerical methods for constructing approximate solutions to the cauchy problem for hamilton jacobi equations of the form ut h dxu 0 the methods are based on dimensional splitting and front tracking for solving the associated non strictly hyperbolic system of conservation laws pt dx h p 0 where p dx u in particular our methods depend heavily on a front tracking method for one dimensional scalar conservation laws with discontinuous coefficients the proposed methods are unconditionally stable in the sense that the time step is not limited by the space discretization and they can be viewed as large time step godunov type or front tracking methods we present several numerical examples illustrating the main features of the proposed methods we also compare our methods with several methods from the literature

existence of global smooth solution to the initial boundary value problem for p system with damping abstract this paper is concerned with the initial boundary value problem for the p system with damping we prove the existence of the global smooth solution under the assumption that only the c 0 norm of the derivative of the initial data is sufficiently small while the c 0 norm of the initial data is not necessarily small the proof is based on several key a priori estimates the maximum principle and the characteristic method

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

numerical schemes for the hamilton jacobi and level set equations on triangulated domains borrowing from techniques developed for conservation law equations numerical schemes which discretize the hamilton jacobi h j level set and eikonal equations on triangulated domains are presented the first scheme is a provably monotone discretization for the h j equations unfortunately the basic scheme lacks lipschitz continuity of the numerical hamiltonian by employing a virtual edge flipping technique local lipschitz continuity of the numerical flux is restored on acute triangulations next schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous hamiltonians these schemes possess a discrete maximum principle on arbitrary triangulations and exhibit local lipschitz continuity of the numerical hamiltonian finally a class of petrov galerkin approximations is considered these schemes are stabilized via a least squares bilinear form the petrov galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy discretization of the level set equation also requires the numerical approximation of a mean curvature term a simple mass lumped galerkin approximation is presented in section 6 and analyzed using maximum principle analysis the use of unstructured meshes permits several forms of mesh adaptation which have been incorporated into numerical examples these numerical examples include discretizations of convex and nonconvex forms of the h j equation the eikonal equation and the level set equation

generalized monotone schemes discrete paths of extrema and discrete entropy conditions solutions of conservation laws satisfy the monotonicity property the number of local extrema is a non increasing function of time and local maximum minimum values decrease increase monotonically in time this paper investigates this property from a numerical standpoint we introduce a class of fully discrete in space and time high order accurate difference schemes called generalized monotone schemes convergence toward the entropy solution is proven via a new technique of proof assuming that the initial data has a finite number of extremum values only and the flux function is strictly convex we define discrete paths of extrema by tracking local extremum values in the approximate solution in the course of the analysis we establish the pointwise convergence of the trace of the solution along a path of extremum as a corollary we obtain a proof of convergence for a muscl type scheme that is second order accurate away from sonic points and extrema

convergence of muscl and filtered schemes for scalar conservation laws and hamilton jacobi equations this paper considers the questions of convergence of i muscl type i e second order tvd finite difference approximations towards the entropic weak solution of scalar one dimensional conservation laws with strictly convex flux and ii higher order schemes filtered to preserve an upper bound on some weak second order finite differences towards the viscosity solution of scalar multi dimensional hamilton jacobi equations with convex hamiltonians

the 3d compressible euler equations with damping in a bounded domain abstract we proved global existence and uniqueness of classical solutions to the initial boundary value problem for the 3d damped compressible euler equations on bounded domain with slip boundary condition when the initial data is near its equilibrium time asymptotically the density is conjectured to satisfy the porous medium equation and the momentum obeys to the classical darcy s law based on energy estimate we showed that the classical solution converges to steady state exponentially fast in time we also proved that the same is true for the related initial boundary value problem of porous medium equation and thus justified the validity of darcy s law in large time

numerical schemes for conservation laws via hamilton jacobi equations we present some difference approximation schemes which converge to the entropy solution of a scalar conservation law having a convex flux the numerical methods described here take their origin from approximation schemes for hamilton jacobi bellman equations related to optimal control problems and exhibit several interesting features the convergence result still holds for quite arbitrary time steps the main assumption for convergence can be interpreted as a discrete analogue of oleinik s entropy condition numerical diffusion around the shocks is very limited some tests are included in order to compare the performances of these methods with other classical methods godunov tvd