l 1 stability estimates for n x n conservation laws let u t f u x 0 be a strictly hyperbolic n times n system of conservation laws each characteristic field being linearly degenerate or genuinely nonlinear in this paper we explicitly define a functional phi phi u v equivalent to the l 1 distance which is almost decreasing i e phi big u t v t big phi big u s v s big leq o ve cdot t s quad hbox for all t s geq 0 for every pair of e approximate solutions u v with small total variation generated by a wave front tracking algorithm the small parameter e here controls the errors in the wave speeds the maximum size of rarefaction fronts and the total strength of all non physical waves in u and in v from the above estimate it follows that front tracking approximations converge to a unique limit solution depending lipschitz continuously on the initial data in the vec l 1 norm this provides a new proof of the existence of the standard riemann semigroup generated by a n n system of conservation laws

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

error bounds for the methods of glimm godunov and leveque the expected error in l 1 r at time t for glimm s scheme when applied to a scalar conservation law is bounded by left h frac 2 sqrt 3 left frac h delta t right 1 2 right left u 0 right bv bf r where h is the mesh size and delta t is the time step we show that the error in godunov s scheme is bounded by the same expression and use this result to analyze a variant of a large time step numerical method of leveque that may be viewed as a combination of godunov s scheme and a method of dafermos

sharp convergence rate of the glimm scheme for general nonlinear hyperbolic systems consider a general strictly hyperbolic quasilinear system in one space dimesion u t a u u x 0 quad quad quad quad quad quad 1 where u mapsto a u u in omega subset mathbb r n is a smooth matrix valued map given an initial datum u 0 with small total variation let u t be the corresponding unique vanishing viscosity solution of 1 obtained as a limit of solutions to the viscous parabolic approximation u t a u u x u xx as 0 for every t 0 we prove the a priori bound left u varepsilon t cdot u t cdot right mathbb l 1 o 1 cdot sqrt varepsilon log varepsilon quad quad quad quad quad quad 1 for an approximate solution u varepsilon of 1 constructed by the glimm scheme with mesh size delta x delta t varepsilon and with a suitable choice of the sampling sequence this result provides for general hyperbolic systems the same type of error estimates valid for glimm approximate solutions of hyperbolic systems of conservation laws u t f u x 0 satisfying the classical lax or liu assumptions on the eigenvalues k u and on the eigenvectors r k u of the jacobian matrix a u df u

decay estimate for a new measure of rarefaction waves in general systems of conservation laws abstract the sharp decay estimate of rarefaction waves in terms of a partial ordering among positive measures is not only interesting in itself but also crucial in the study of convergence rate of vanishing viscosity approximations cf bressan and yang 2004 10 such an estimate is well established for genuinely nonlinear system of conservation laws cf bressan and yang 2004 9 but similar result is not available for non genuinely nonlinear system in this paper we give a new measure about the rarefaction waves in addition a sharp decay estimate of the new measure is given for the cubic nonlinear system of conservation laws

error bounds for a deterministic version of the glimm scheme consider the hyperbolic system of conservation laws u t f u x 0 let u be the unique viscosity solution with initial condition u 0 x bar u x and let ue be an approximate solution constructed by the glimm scheme corresponding to the mesh sizes delta x delta t o delta x with a suitable choice of the sampling sequence we prove the estimate big u ve t cdot u t cdot big bf l 1 strut l 1 o 1 cdot sqrt delta x big ln delta x big

the boundary riemann solver coming from the real vanishing viscosity approximation we study the limit of the hyperbolic parabolic approximation left begin array l quad l quad l v varepsilon t tilde a left v varepsilon varepsilon v varepsilon x right v varepsilon x varepsilon tilde b v varepsilon v varepsilon xx quad v varepsilon in mathbb r n cr tilde rm ss v varepsilon t 0 equiv bar g cr v varepsilon 0 x equiv bar v 0 end array right the function tilde ss is defined in such a way as to guarantee that the initial boundary value problem is well posed even if tilde b is not invertible the data bar g and bar v 0 are constant when tilde b is invertible the previous problem takes the simpler form left begin array l quad l quad l v varepsilon t tilde a left v varepsilon varepsilon v varepsilon x right v varepsilon x varepsilon tilde b v varepsilon v varepsilon xx quad v varepsilon in mathbb r n cr v varepsilon t 0 equiv bar v b cr v varepsilon 0 x equiv bar v 0 end array right again the data bar v b and bar v 0 are constant the conservative case is included in the previous formulations convergence of the v varepsilon smallness of the total variation and other technical hypotheses are assumed and a complete characterization of the limit is provided the most interesting points are the following first the boundary characteristic case is considered that is one eigenvalue of tilde a can be 0 second as pointed out before we take into account the possibility that tilde b is not invertible to deal with this case we take as hypotheses conditions that were introduced by kawashima and shizuta relying on physically meaningful examples we also introduce a new condition of block linear degeneracy we prove that if this condition is not satisfied then pathological behaviors may occur

continuous dependence for 2 2 conservation laws with boundary on the domain 0 t x r t 0 and x 9 t for some boundary profile 9 r r as usual 1 1 is assumed to be strictly hyperbolic and with each characteristic field either linearly degenerate or genuinely non linear an initial data u 0 x u x having sufficiently small total variation is given we consider two different kinds of boundary conditions along x 9 t and in both cases we construct a lipschitzean flow whose trajectories are solutions of an initial boundary value problem for 1 1 thus we prove the continuous dependence of the solution upon the initial data upon the boundary condition and upon the boundary profile the existence theory for global bv solutions to the cauchy problem for 1 1 goes back to the fundamental paper 15 by glimm more recently in 6 a new approach has been introduced it relies on the construction of a lipschitzean semigroup the standard riemann semigroup srs whose trajectories extend the local standard lax 17 solutions of riemann problems at present such a srs has been constructed in the 2 2 case in 8 9 while for the general n n case see 7 10 and 3 in article no de973274

a contractive metric for systems of conservation laws with coinciding shock and rarefaction curves abstract we introduce two algorithms for the construction of weak entropy admissible solutions to a class of systems of conservation laws with coinciding shock and rarefaction curves global existence uniqueness and continuous dependence are proved for all solutions obtained by our constructive procedure the generated semigroup is contractive with respect to a riemann type metric defined in terms of glimm s wave interaction functional equivalent to the usual l 1 distance

decay estimation in nonlinear hyperbolic system of conservation laws abstract in this paper we study some decay estimates in nonlinear hyperbolic system of conservation laws this research is not only interesting in itself but also crucial in studying the large time behavior problem by introducing a proper glimm functional we obtain some useful decay estimates which are proved helpful in obtaining decay rates of the admissible solutions to nonlinear hyperbolic conservation laws as t

uniqueness of classical and nonclassical solutions for nonlinear hyperbolic systems abstract in this paper we establish a general uniqueness theorem for nonlinear hyperbolic systems of partial differential equations in one space dimension first of all we introduce a new notion of admissible solutions based on prescribed sets of admissible discontinuities and admissible speeds our definition unifies in a single framework the various notions of entropy solutions known for hyperbolic systems of conservation laws as well as for systems in nonconservative form for instance it covers the nonclassical undercompressive shock waves generated by a vanishing diffusion dispersion regularization and characterized via a kinetic relation it also covers dal maso lefloch and murat s definition of weak solutions of nonconservative systems under certain natural assumptions on the prescribed sets and and assuming the existence of a l 1 continuous semi group of admissible solutions we prove that for each cauchy datum at t 0 there exists at most one admissible solution to the cauchy problem depending l 1 continuously upon the initial data in particular our result shows the uniqueness of the l 1 continuous semi group of admissible solutions in short this paper proves that supplementing a hyperbolic system with the dynamics of elementary discontinuities characterizes at most one l 1 continuous and admissible solution

singular limits of stiff relaxation and dominant diffusion for nonlinear systems abstract we are concerned with singular limits of stiff relaxation and dominant diffusion for general 2 2 nonlinear systems of conservation laws that is the relaxation time tends to zero faster than the diffusion parameter e o e e 0 we establish the following general framework if there exists an a priori l bound that is uniformly with respect to e for the solutions of a system then the solution sequence converges to the corresponding equilibrium solution of this system our results indicate that the convergent behavior of such a limit is independent of either the stability criterion or the hyperbolicity of the corresponding inviscid quasilinear systems which is not the case for other type of limits this framework applies to some important nonlinear systems with relaxation terms such as the system of elasticity the system of isentropic fluid dynamics in eulerian coordinates and the extended models of traffic flows the singular limits are also considered for some physical models without l bounded estimates including the system of isentropic fluid dynamics in lagrangian coordinates and the models of traffic flows with stiff relaxation terms the convergence of solutions in l p to the equilibrium solutions of these systems is established provided that the relaxation time tends to zero faster than e

existence and uniqueness of entropy solution to initial boundary value problem for the inviscid burgers equation this paper is concerned with the existence and uniqueness of the entropy solution to the initial boundary value problem for the inviscid burgers equation to apply the method of vanishing viscosity to study the existence of the entropy solution we first introduce the initial boundary value problem for the viscous burgers equation and as in evans 1998 partial differential equations providence ri american mathematical society and hopf 1950 commun pure appl math 3 201 30 give the formula of the corresponding viscosity solutions by hopf cole transformation secondly we prove the convergence of the viscosity solution sequences and verify that the limiting function is an entropy solution finally we give an example to show how our main result can be applied to solve the initial boundary value problem for the burgers equation

on the front tracking algorithm in this paper we present an improved version of the front tracking algorithm for systems of conservation laws the formulation and the theoretical analysis are here somewhat simpler than in previous algorithms at the same time our version leads to a more efficient numerical scheme

uniqueness of weak solutions to systems of conservation laws consider a strictly hyperbolic system of conservation laws in one space dimension relying on the existence of the standard riemann semigroup generated by we establish the uniqueness of entropy admissible weak solutions to the cauchy problem under a mild assumption on the variation of along space like segments

a wavefront tracking algorithm for n n nongenuinely nonlinear conservation laws we introduce a wavefront tracking algorithm for n n hyperbolic systems of conservation lawsut f u x 0 that admits characteristic fields that are neither genuinely nonlinear nor linearly degenerate in the sense of lax instead we assume that for any nongenuinely nonlinear ith characteristic family the derivative of the ith eigenvalue i u of df u in the direction of the ith right eigenvector ri u vanishes on a single n 1 dimensional hypersurface in the u space transversal to the field ri u systems that fulfill this type of assumptions are of particular interest in studying elastodynamic or rigid heat conductors at low temperature the first proof of the existence of weak solutions for nongenuinely nonlinear systems was given by t p liu mem amer math soc 30 1981 no 240 based on a glimm scheme our construction here provides an alternative method for establishing the global existence of weak solutions for such systems moreover relying on the stability analysis developed in ancona and marson preprint s i s s a i s a s 27 99 11 1999 and preprint 2000 we show that these solutions are entropy admissible in the sense of lax

a new glimm functional and convergence rate of glimm scheme for general systems of hyperbolic conservation laws in this paper we introduce a new glimm functional for general systems of hyperbolic conservation laws this new functional is consistent with the classical glimm functional for the case when each characteristic field is either genuinely nonlinear or linearly degenerate so that it can be viewed as optimal in some sense with this new functional the consistency of the glimm scheme is proved clearly for general systems moreover the convergence rate of the glimm scheme is shown to be the same as the one obtained in bressan marson arch ration mech anal 142 2 155 176 1998 for systems with each characteristic field being genuinely nonlinear or linearly degenerate

well posedness for two dimensional steady supersonic euler flows past a lipschitz wedge abstract for a supersonic euler flow past a straight sided wedge whose vertex angle is less than the extreme angle there exists a shock front emanating from the wedge vertex and the shock front is usually strong especially when the vertex angle of the wedge is large in this paper we establish the l 1 well posedness for two dimensional steady supersonic euler flows past a lipschitz wedge whose boundary slope function has small total variation when the total variation of the incoming flow is small in this case the lipschitz wedge perturbs the flow and the waves reflect after interacting with the strong shock front and the wedge boundary we first obtain the existence of solutions in bv when the incoming flow has small total variation by the wave front tracking method and then establish the l 1 stability of the solutions with respect to the incoming flows in particular we incorporate the nonlinear waves generated from the wedge boundary to develop a lyapunov functional between two solutions containing strong shock fronts which is equivalent to the l 1 norm and prove that the functional decreases in the flow direction then the l 1 stability is established so is the uniqueness of the solutions by the wave front tracking method finally the uniqueness of solutions in a broader class the class of viscosity solutions is also obtained

hyperbolic limit of the jin xin relaxation model we consider the special jin xin relaxation model 0 1 u i a u u x u xx u tt we assume that the initial data u 0 u 0 t are sufficiently smooth and close to u 0 in l and have small total variation then we prove that there exists a solution u t u t t with uniformly small total variation for all t 0 and this solution depends lipschitz continuously in the l 1 norm with respect to time and the initial data letting 0 the solution u converges to a unique limit providing a relaxation limit solution to the quasi linear nonconservative system 0 2 u t a u u x 0 these limit solutions generate a lipschitz semigroup s on a domain d containing the functions with small total variation and close to u this is precisely the riemann semigroup determined by the unique riemann solver compatible with 0 1

plane shear waves under a periodic boundary disturbance in a saturated granular medium abstract we study an initial boundary value problem for a fully nonlinear system which describes the propagation of plane shear waves in a three dimensional soil assuming a hypoplastic flow rule this model was derived to study liquefaction of soils for a periodic square wave velocity disturbance on the boundary the solution is found to saturate away from the boundary we calculate the asymptotic state in terms of the initial and boundary data this formula exhibits fractal behavior as the shape of the square wave is varied our analysis is confirmed by numerical computations

some free boundary problems recast as nonlocal parabolic equations abstract in this work we demonstrate that a class of some one and two phase free boundary problems can be recast as nonlocal parabolic equations on a submanifold the canonical examples would be one phase hele shaw flow as well as its two phase analog we also treat nonlinear versions of both one and two phase problems in the special class of free boundaries that are graphs over r d we give a precise characterization that shows their motion is equivalent to that of a solution of a nonlocal fractional nonlinear parabolic equation for functions on r d our main observation is that the free boundary condition defines a nonlocal operator having what we call the global comparison property a consequence of the connection with nonlocal parabolic equations is that for free boundary problems arising from translation invariant elliptic operators in the positive and negative phases one obtains in a uniform treatment for all of the problems one and two phase a propagation of modulus of continuity for viscosity solutions of the free boundary flow

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

the semigroup generated by 2 2 conservation laws consider the cauchy problem for a strictly hyperbolic 2 2 system of conservation laws in one space dimension ie1 01 assuming that each characteristic field is either linearly degenerate or genuinely nonlinear this paper develops a new algorithm based on wave front tracking which yields a cauchy sequence of approximate solutions converging to a unique limit depending continuously on the initial data the solutions that we obtain constitute a semigroup s defined on a set ie1 02 of integrable functions with small total variation for some lipschitz constant l we have the estimate ie1 03

bv solutions of the semidiscrete upwind scheme consider the strictly hyperbolic system of conservation laws u t f u x 0 1 where u n f n n smooth the system is said to be strictly hyperbolic provided that the jacobian matrix a u df u has n distinct eigenvalues 1 u n u

smooth cuspidal automorphic forms and integrable discrete series in this paper we construct smooth cuspidal automorphic forms related to integrable discrete series of a connected semisimple lie group with finite center for classical and adelic situation as an application of the theory of schwartz spaces for automorphic forms developed by casselman in the classical situation smooth cuspidal automorphic forms are constructed via an explicit continuous map from the frechet space of smooth vectors of a banach realization inside l 1 g of an integrable discrete series into the space of smooth vectors of a strong topological dual of an appropriate schwartz space

a locally quadratic glimm functional and sharp convergence rate of the glimm scheme for nonlinear hyperbolic systems consider the cauchy problem for a strictly hyperbolic n n quasilinear system in one space dimension u t a u u x 0 qquad u 0 x bar u x quad quad quad quad 1 where u mapsto a u is a smooth matrix valued map and the initial data overline u is assumed to have small total variation we investigate the rate of convergence of approximate solutions of 1 constructed by the glimm scheme under the assumption that letting k u rk u denote the k th eigenvalue and a corresponding eigenvector of a u respectively for each k th characteristic family the linearly degenerate manifold mathcal m k doteq left u in omega nabla lambda k u cdot r k u 0 right is either the whole space or it is empty or it consists of a finite number of smooth n 1 dimensional connected manifolds that are transversal to the characteristic vector field rk we introduce a glimm type functional which is the sum of the cubic interaction potential defined in bianchini discrete contin dyn syst 9 133 166 2003 and of a quadratic term that takes into account interactions of waves of the same family with strength smaller than some fixed threshold parameter relying on an adapted wave tracing method and on the decrease amount of such a functional we obtain the same type of error estimates valid for glimm approximate solutions of hyperbolic systems satisfying the classical lax assumptions of genuine nonlinearity or linear degeneracy of the characteristic families

oleinik type estimates and uniqueness for n n conservation laws abstract let ut f u x 0 be a strictly hyperbolic n n system of conservation laws in one space dimension relying on the existence of a semigroup of solutions we first establish the uniqueness of entropy admissible weak solutions to the cauchy problem under a mild assumption on the local oscillation of u in a forward neighborhood of each point in the t x plane in turn this yields the uniqueness of weak solutions which satisfy a decay estimate on positive waves of genuinely nonlinear families thus extending a classical result proved by oleinik in the scalar case

an improved convergence rate of glimm scheme for general systems of hyperbolic conservation laws we study the convergence rate of glimm scheme for general systems of hyperbolic conservation laws without the assumption that each characteristic field is either genuinely nonlinear or linearly degenerate we first give a sharper estimate of the error arising from the wave tracing argument by a careful analysis of the interaction between small waves with this key estimate the convergence rate is shown to be o 1 s13 lns 1 which is sharper compared to o 1 s14 lns given in t yang convergence rate of glimm scheme for general systems of hyperbolic conservation laws taiwanese j math 7 2 2003 195 205 however it is still slower than o 1 s12 lns given in a bressan a marson error bounds for a deterministic version of the glimm scheme arch ration mech anal 142 2 1998 155 176 for systems with each characteristic field being genuinely nonlinear or linearly degenerate here s is the mesh size and is any positive constant

stability of transonic jet with strong shock in two dimensional steady compressible euler flows abstract for steady supersonic flow past a solid convex corner surrounded by quiescent gas if the pressure of the upcoming supersonic flow is lower than the pressure of the quiescent gas there may appear a strong shock to increase the pressure and then a transonic characteristic discontinuity to separate the supersonic flow behind the shock front from the still gas in this paper we prove the global existence uniqueness and stability of such flow patterns under suitable conditions on the upstream supersonic flow and the pressure of the surrounding quiescent gas for the two dimensional steady complete compressible euler system mathematically a global weak solution to a characteristic free boundary problem of hyperbolic conservation laws is constructed and shown to be unique and stable under the framework of front tracking method

multivalued equations for granular avalanches we study existence and uniqueness for the scalar equation of first order hyperbolic type the rhs is in the form of a multifunction which is physically motivated by an avalanche flow model therefore some kind of monotonicity to the rhs has to be assumed

continuous dependence of entropy solutions to the euler equations on the adiabatic exponent and mach number we establish the l1 estimates for continuous dependence of entropy solutions to the full euler equations away from the vacuum on two physical parameters the adiabatic exponent 1 that passes from the non isentropic to isothermal euler equations and the mach number tt m to 0 that passes from the compressible to incompressible euler equations our analysis involves the effective approach developed in our earlier work and additional new techniques that generalize this approach to the setting of the full euler equations

regularity and lyapunov stabilization of weak entropy solutions to scalar conservation laws we consider the problem of lyapunov boundary stabilization of the weak entropy solution to a scalar conservation law with strictly convex flux in one dimension of space around a uniform equilibrium we show that for a specific class of boundary conditions the solution to the initial boundary value problem for an initial condition with bounded variations can be approximated arbitrarily closely in the l 1 norm by a piecewise smooth solution with finitely many discontinuities the constructive method we present designs explicit boundary conditions in this class which guarantee lyapunov stability of the weak entropy solution to the initial boundary value problem we show how the greedy control obtained by maximizing the decrease of the natural lyapunov function may fail to asymptotically stabilize and a brute force control generates unbounded variation of traces we then design a stabilizing control which avoid oscillations and propose a nonlocal technique depending on time and the whole initial datum which optimizes the convergence time controllers performance is illustrated on numerical benchmarks using the godunov scheme