exactly solvable potentials and the concept of shape invariance the author analyses the relationship between shape invariance and exactly solvable potentials using earlier results he shows that all actually known shape invariant potentials with a single parameter can be recovered quite simply extension to the case of shape invariant potentials with two parameters as well as to non shape invariant but exactly solvable potentials are also possible

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 of the fractional step lax friedrichs scheme and godunov scheme for the isentropic system of gas dynamics a convergence theorem of the fractional step lax friedrichs scheme and godunov scheme for an inhomogeneous system of isentropic gas dynamics 1 5 3 is established by using the framework of compensated compactness meanwhile a corresponding existence theorem of global solutions with large data containing the vacuum is obtained

global bv entropy solutions and uniqueness for hyperbolic systems of balance laws we consider the cauchy problem for n n strictly hyperbolic systems of nonresonant balance laws

a bound on the total variation of the conserved quantities for solutions of a general resonant nonlinear balance law we introduce a new potential interaction functional and use it to define a new glimm type functional that bounds the total variation of the conserved quantities at time t 0 by the total variation at time t 0 in glimm approximate solutions of a general resonant nonlinear balance law

shape invariance raising and lowering operators in hypergeometric type equations the schrodinger equations which are exactly solvable in terms of associated special functions are directly related to some self adjoint operators defined in the theory of hypergeometric type equations the fundamental formulae occurring in a supersymmetric approach to these hamiltonians are consequences of some formulae concerning the general theory of associated special functions we use this connection in order to obtain a general theory of schrodinger equations exactly solvable in terms of associated special functions and to extend certain results known in the case of some particular potentials

isentropic gas flow for the compressible euler equation in a nozzle we study the motion of isentropic gas in a nozzle nozzles are used to increase the thrust of engines or to accelerate a flow from subsonic to supersonic nozzles are essential parts for jet engines rocket engines and supersonicwind tunnels in the present paper we consider unsteady flow which is governed by the compressible euler equation and prove the existence of global solutions for the cauchy problem for this problem the existence theorem has already been obtained for initial data away from the sonic state liu in commun math phys 68 141 172 1979 here we are interested in the transonic flow which is essential for engineering and physics although the transonic flow has recently been studied tsuge in j math kyoto univ 46 457 524 2006 lu in nonlinear anal real world appl 12 2802 2810 2011 these papers assume monotonicity of the cross section area here we consider the transonic flow in a nozzle with a general cross section area when we prove global existence the most difficult point is obtaining a bounded estimate for approximate solutions to overcome this we employ a new invariant region that depends on the space variable moreover we introduce a modified godunov scheme the corresponding approximate solutions consist of piecewise steady state solutions of an auxiliary equation which yield a desired bounded estimate in order to prove their convergence we use the compensated compactness framework

quantum solvable models with gl 2 c lie algebra symmetry embedded into the extension of unitary parasupersymmetry introducing p 1 new parameters into the multilinear relations we extend the standard unitary parasupersymmetry algebra of order p so that by embedding the quantum solvable models possessing gl 2 c lie algebra symmetry into it the partitions of integer numbers p 1 and 1 p p 1 are established these two partitions are performed by the new parameters and the product of new parameters with their labels respectively the former partition is just necessary for the real form h4 however both of them are essential for the real forms u 2 and u 1 1 by occupying these parameters with arbitrary values the energy spectra are determined by the mean value of proposed parameters for the real form h4 with their label weight function as well as for the real forms u 2 and u 1 1 with the weight function of their squared label so for the given energies the multilinear behaviour of parasupercharges is not specified uniquely by varying the new parameters continuously

a functional integral approach to shock wave solutions of the euler equations with spherical symmetry ii abstract this is a continuation of 15 as is well known one dimensional conservation laws without source term have been extensively investigated after the foundamental paper of j glimm 3 and those with integrable source terms were solved by liu 6 7 etc for higher dimensional case with spherical symmetry makino 8 first proved a linear growth rate for solutions when p 2 where 1 but in order to get global existence for 1 and the decay property we need to find a uniform bound for the approximate solutions in 15 we introduced a new norm and a functional integral approach to prove a uniform bound for a model problem of euler equation in r 3 with spherical symmetry in order to overcome the geometric effects of spherical symmetry which leads to a non integrable source term we considered an infinite reflection problem and solved it by considering the cancellations between reflections of different orders in this paper we consider a system which describes the isentropic and spherically symmetric motion of gas flow surrounding a solid star with radius 1 and mass m it is interesting to note that the wave curves for this problem are no longer continuous and there is an extra term in the wave interaction estimates by introducing a new norm we prove a similar result as 15 in the appendix we present a local existence theorem for 1 which was also obtained by makino 9 by different method and we extend the results to the cases with different boundary conditions

master function approach to quantum solvable models on sl 2 c and sl 2 c gl 1 c manifolds by introducing a new parameter as a second associated index for special functions we construct the three dimensional differential generators of gl 2 c lie algebra together with the corresponding contracted form h4 non casimir quadratic as well as the casimir of gl 2 c and h4 generators can be considered as quantum solvable models on group manifold sl 2 c then by appropriate parametrization of group manifold sl 2 c and eliminating one of the coordinates we obtain quantum solvable hamiltonians on homogeneous manifold sl 2 c gl 1 c with the metric described by master function we show that two dimensional hamiltonian on sl 2 c gl 1 c derived from the reduction of casimir operator so 4 c lie algebra as a three dimensional hamiltonian on group manifold sl 2 c possesses the degeneracy sl 2 c group and also the shape invariance property where both have para supersymmetry representations of arbitrary order

convergence of shock capturing schemes for the compressible euler poisson equations we are concerned with approximate methods to construct global solutions with geometrical structure to the compressible euler poisson equations in several space variables a shock capturing numerical scheme is introduced to overcome the new difficulties from the nonlinear resonance of the system and the nonlocal behavior of the source terms the convergence and consistency of the shock capturing scheme for the equations is proved with the aid of the compensated compactness method then new existence results of the global solutions with geometrical structure are obtained the traces of the weak solutions are defined and then the weak solutions are proved to satisfy the boundary conditions the initial data are arbitrarily large withl bounds

supersymmetry operator transformations and exactly solvable potentials a large class of potentials can be solved algebraically by using supersymmetry and shape invariance the authors apply operator transformations f transformations to these algebraically solvable problems to obtain a larger class of solvable potentials the natanzon class of potentials which are not shape invariant the important condition which is independent of supersymmetry for finding new potentials from old ones using operator transformations is that the resulting schrodinger equation has a potential which does not depend on the state as a special case of the f transformation they rederive the previously known connection between the 3d harmonic oscillator the hydrogen atom and the morse potential they also discuss the lack of commutivity of susy and the f transformations

a master function approach for describing shape invariance parameters with respect to which quantum number they are shape invariant most of the one dimensional shape invariance potentials fall into two classes for each of which one of the quantum numbers describing the shape invariance is assigned on this basis the shape invariance parameters of every class are separately described in terms of a master function its weight function and an appropriate quantum number

differentiability and analyticity of topological entropy for anosov and geodesic flows in this paper we investigate the regularity of the topological entropyh top forc k perturbations of anosov flows we show that the topological entropy varies almost as smoothly as the perturbation the results in this paper along with several related results have been announced in kkpw

pathological foliations and removable zero exponents the ergodic theory of uniformly hyperbolic or axiom a diffeomorphisms has been studied extensively beginning with the pioneering work of anosov sinai ruelle and bowen an si ru bo while uniformly hyperbolic systems enjoy strong mixing properties they are not dense among c diffeomorphisms sm as using the concept of lyapunov exponents pesin introduced a weaker form of hyperbolicity which he termed nonuniform hyperbolicity nonuniformly hyperbolic diffeomorphisms share several mixing properties with uniformly hyperbolic ones our construction of the diffeomorphisms in this paper was motivated by the question of whether nonuniform hyperbolicity is dense among a large class of diffeomorphisms as a curious by product of our construction we prove that a pathological feature of central foliations the complete failure of absolute continuity can exist in a c open set of volume preserving diffeomorphisms we recall the definition of a nonuniformly hyperbolic diffeomorphism a real number is a lyapunov exponent of the diffeomorphism g m m if there exists a nonzero vector v tm such that

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

nonlinear resonance for quasilinear hyperbolic equation the purpose of this paper is to study the wave behavior of hyperbolic conservation laws with a moving source resonance occurs when the speed of the source is too close to one of the characteristic speeds of the system for the nonlinear system characteristic speeds depend on the basic dependence variables and resonance gives rise to nonlinear interactions which lead to rich wave phenomena motivated by physical examples a scalar model is proposed and analyzed to describe the qualitative behavior of waves for a general system in resonance with the source analytical understanding is used to design a numerical scheme based on the random choice method an important physical example is transonic gas flow through a nozzle this analysis provides a transparent and revealing qualitative understanding of wave behavior of gas flow including such phenomena as nonlinear stability instability and changing types of waves

on the two dimensional gas expansion for compressible euler equations we investigate the problem of two dimensional unsteady expansion of an inviscid polytropic gas which can be interpreted as the collapse of a wedge shaped dam containing water initially with a uniform velocity we model this problem by isentropic euler equations the flow is quasi stationary and using hodograph transform we describe it by a partial differential equation of second order in the state space if it is irrotational initially furthermore this equation is reduced to a linearly degenerate system of three partial differential equations with inhomogeneous source terms these properties are used to prove that the flow is globally smooth when a wedge of gas expands into a vacuum and to analyze that shocks may appear in the interaction of four planar rarefaction waves

the riemann problem for the euler equations with nonconvex and nonsmooth equation of state construction of wave curves the riemann problem is considered for the investigation of nonclassical wave phenomena in bethe zel dovich thompson bzt fluids and fluids which undergo a phase transition i e the isentropes suffer kinks at the saturation boundaries these fluids are modeled by a general equation of state that also takes into account regions of negative nonlinearity where the isentropes are nonconvex in the pressure volume plane here the physical model is based on thermodynamical equilibrium both nonsmoothness as well as nonconvexity significantly influence the composition of the wave curves which are crucial for determining the solution for the riemann problem for the euler equations this work gives explicit criteria for the composition of these wave curves by elementary wave types such as shocks simple waves and composite waves numerous examples for different wave curve compositions are presented

accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1d center bundle we prove that stable ergodicity is cr open and dense among conservative partially hyperbolic diffeomorphisms with one dimensional center bundle for all r 2

new ladder operators for the monopole harmonics using the ladder operators shifting the index m of the associated jacobi functions for a given n the monopole harmonics and their corresponding angular momentum operators are respectively extracted as the irreducible representation space and generators of su 2 lie algebra the indices n and m play the role of principal and azimuthal quantum numbers by introducing the ladder operators shifting the index n of the same associated jacobi functions we also get a new type of the raising and lowering relations which are realized by the operators shifting only the index n of the monopole harmonics moreover other symmetries including the transformation of the irreducible representation spaces into each other are derived based on the operators that shift the indices n and m of the monopole harmonics simultaneously and agreeably as well as simultaneously and inversely our results are reduced to spherical harmonics by eliminating magnetic charge of the monopole

global solution of an initial value problem for two dimensional compressible euler equations abstract this paper is concerned with the existence of global continuous solutions of the expansion of a wedge of gas into a vacuum for compressible euler equations by hodograph transformation we first prove that the flow is governed by a partial differential equation of second order which is further reduced to a system of two nonhomogeneous linearly degenerate equations in the phase space under an irrotationality condition then this conclusion is applied to solving the problem that a wedge of gas expands into a vacuum which is actually a goursat type problem for these two equations in the supersonic domain

analysis of a singular hyperbolic system of conservation laws abstract we solve the riemann and cauchy problems globally for a singular system of n hyperbolic conservation laws the system which arises in the study of oil reservoir simulation has only two wave speeds and these coincide on a surface of codimension one in state space the analysis uses the random choice method of glimm comm pure appl math 18 1965 697 715

nonuniqueness for the vanishing viscosity solution with fixed initial condition in a nonstrictly hyperbolic system of conservation laws we consider the riemann problem for a system of two decoupled nonstrictly hyperbolic burgers like conservation equations with added artificial viscosity we analytically establish two different vanishing viscosity limits for the solution of this system which correspond to the two cases where one of the viscosities vanishes much faster than the other this is done without altering the initial condition as is necessary with travelling wave methods numerical evidence is then provided to show that when the two viscosities vanish at the same rate the solution converges to a limit that lies strictly between the two previously established limits finally we use control theory to explain the mechanism behind this nonuniqueness behavior which indicates other systems of nonstrictly hyperbolic conservation laws where nonuniqueness will occur

axisymmetric solutions of the euler equations for polytropic gases we construct rigorously a three parameter family of self similar globally bounded and continuous weak solutions in two space dimensions for all positive time to the euler equations with axisymmetry for polytropic gases with a quadratic pressure density law we use the axisymmetry and self similarity assumptions to reduce the equations to a system of three ordinary differential equations from which we obtain detailed structures of solutions besides their existence these solutions exhibit familiar structures seen in hurricanes and tornadoes they all have finite local energy and vorticity with well defined initial and boundary values these solutions include the one parameter family of explicit solutions reported in a recent article of ours

existence of global solutions for unsteady isentropic gas flow in a laval nozzle in this paper we study the motion of isentropic gas in the laval nozzle the laval nozzle is the most important type of nozzle utilized in some turbines in particular we consider unsteady flows including transonic gas flows and prove the existence of global solutions for the cauchy problem in spite of its importance this problem has received little attention until now the most difficult point is to obtain bounded estimates for approximate solutions to overcome this we introduce a modified godunov scheme the corresponding approximate solutions consist of piecewise steady state solutions of an auxiliary equation and yield a sharper bounded estimate as a result we find an invariant region for our solutions finally in order to prove their convergence we use the compensated compactness framework

well balanced schemes for conservation laws with source terms based on a local discontinuous flux formulation we propose and analyze a finite volume scheme of the godunov type for conservation laws with source terms that preserve discrete steady states the scheme works in the resonant regime as well as for problems with discontinuous flux moreover an additional modification of the scheme is not required to resolve transients and solutions of non linear algebraic equations are not involved our well balanced scheme is based on modifying the flux function locally to account for the source term and to use a numerical scheme especially designed for conservation laws with discontinuous flux due to the difficulty of obtaining bv estimates we use the compensated compactness method to prove that the scheme converges to the unique entropy solution as the discretization parameter tends to zero we include numerical experiments in order to show the features of the scheme and how it compares with a wellbalanced scheme from the literature

stationary solutions to euler equations with spherical symmetry abstract in this paper as a first step to study the large time behavior of solutions to the compressible euler equations on a half line we consider the two classes of compressible euler equations and prove the existence of stationary solutions with spherical symmetry form the aim is to concentrate on understanding the influence of boundary conditions

dimension reduction of compressible fluid models over product manifolds abstract in this paper we study the dimension reduction limits of the compressible navier stokes equations over product riemannian manifolds o m f such that dim m n and dim f d are arbitrary using the method of relative entropies we establish the convergence of the suitable weak solutions of the navier stokes equations on o to the classical solution of the limiting equations on m as 0 provided the latter exists in addition we also deduce the vanishing viscosity limit the limiting equations identified through our analysis contain the weight function a m r as a parameter where a x area of the fibre f x our work is based on and generalises the results in bella et al 2016 3 bella et al 2014 4 and maltese and novotn 2014 19 and it contains as special cases the physical models of circular nozzles thin plate limits and finite length longitudinal nozzles

partial differentiability of invariant splittings a key feature of a general nonlinear partially hyperbolic dynamical system is the absence of differentiability of its invariant splitting in this paper we show that often partial derivatives of the splitting exist and the splitting depends smoothly on the dynamical system itself

non equilibrium critical behaviour on fractal lattices a simple one component surface reaction model exhibiting a critical point has been studied on deterministic fractal lattices with fractal dimension 1 df 2 results from time dependent simulations show a change in the value of the critical exponents with df two of the exponents interpolate very nicely between the values in one and two dimensions

compressible euler equations with general pressure law we study the hyperbolic system of euler equations for an isentropic compressible fluid governed by a general pressure law the existence and regularity of the entropy kernel that generates the family of weak entropies is established by solving a new euler poisson darboux equation which is highly singular when the density of the fluid vanishes new properties of cancellation of singularities in combinations of the entropy kernel and the associated entropy flux kernel are found

existence of global weak solutions to a special system of euler equation with a source ii general case abstract in this paper we apply the maximum principle and the theory of compensated compactness to establish an existence theorem for global weak solutions to the cauchy problem of the non strictly hyperbolic system a special system of euler equation with a general source term

on a weak solution for a transonic flow problem on essaie d etablir l existence de solutions faibles pour un ecoulement stationnaire bidimensionnel irrotationnel compressible non visqueux dans un canal de longueur infinie ou autour d un profil aerodynamique la vitesse de l ecoulement etant donnee a l infini

multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type we are concerned with the existence and stability of multidimensional transonic shocks for the euler equations for steady potential compressible fluids the euler equations consisting of the conservation law of mass and the bernoulli law for the velocity can be written as the following second order nonlinear equation of mixed elliptic hyperbolic type for the velocity potential r r 1 1 div d d 0 where the density function q is

differentiation of srb states abstract let f be a diffeomorphism of a manifold m and f a generalized srb state for f if supp f is a hyperbolic compact set we show that the map is differentiable in a suitable functional setup and we compute the derivative when supp f is an attractor the derivative is given by where x is the vector field this formula can be extended to time dependent situations and also at least formally to nonuniformly hyperbolic situations the above results will find their use in the study of the onsager reciprocity relations and the fluctuation dissipation formula of nonequilibrium statistical mechanics

extended supersymmetry for the bound states of the generalized hulthen potential hierarchy using the associated hypergeometric differential equation we analytically solve the bound states corresponding to a hierarchy of the radial potential v0 e r 1 e r c e r 1 e r 2 as a generalization of the hulth n potential then an analytic solution corresponding to a special case for which the parameter c is expected to be in terms of l l 1 is also derived meanwhile without introducing a superpotential and in the framework of supersymmetric quantum mechanics it is shown that these bound states can be calculated by two different algebraic methods based on these two approaches it is noted that the bound states realize an extended supersymmetry structure

axisymmetric solutions of the pressure gradient system a family of self similar and global bounded weak solutions are constructed rigorously for all positive time to the two dimensional pressure gradient system in this paper by using the axisymmetric and self similar assumptions the equations are reduced to a system of ordinary differential equations from which we obtain detailed structures of solutions as well as their existence on this basis we explore the behavior of characteristics near the sonic boundary

blowup phenomena of solutions to the euler equations for compressible fluid flow abstract the blowup phenomena of solutions is investigated for the euler equations of compressible fluid flow the approach is to construct special explicit solutions with spherical symmetry to study certain blowup behavior of multi dimensional solutions in particular the special solutions with velocity of the form c t x are constructed to show the expanding and blowup properties the solution with velocity of the form a t x a t for 1 and for any space dimensions is obtained as a corollary another conclusion is that there is only trivial solution with velocity of the form c t x 1 x for 1 and multi space dimensions

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

vanishing viscosity solutions of the compressible euler equations with spherical symmetry and large initial data we are concerned with spherically symmetric solutions of the euler equations for multidimensional compressible fluids which are motivated by many important physical situations various evidences indicate that spherically symmetric solutions of the compressible euler equations may blow up near the origin at a certain time under some circumstance the central feature is the strengthening of waves as they move radially inward a longstanding open fundamental problem is whether concentration could be formed at the origin in this paper we develop a method of vanishing viscosity and related estimate techniques for viscosity approximate solutions and establish the convergence of the approximate solutions to a global finite energy entropy solution of the isentropic euler equations with spherical symmetry and large initial data this indicates that concentration is not formed in the vanishing viscosity limit even though the density may blow up at a certain time to achieve this we first construct global smooth solutions of appropriate initial boundary value problems for the euler equations with designed viscosity terms approximate pressure function and boundary conditions and then we establish the strong convergence of the viscosity approximate solutions to a finite energy entropy solution of the euler equations

hyperbolic conservation laws with umbilic degeneracy i in this paper a compactness framework for approximate solutions to nonlinear hyperbolic systems with umbilic degeneracy is established by combining techniques of compensated compactness with some classical methods and by a detailed analysis of a highly singular equation of euler poisson darboux type then this framework is successfully applied to prove the convergence of the viscosity method and to prove the existence of global entropy solutions for the cauchy problem with large initial data for a canonical class of the systems with quadratic flux form

existence of global smooth solution to the relativistic euler equations abstract in this paper we prove an existence theorem of global smooth solutions to the cauchy problem for the one dimensional relativistic euler equations the analysis is based on a priori estimates which are obtained by the characteristic method and maximum principle

kinetic formulation of the isentropic gas dynamics and p systems we consider the 2 x 2 hyperbolic system of isentropic gas dynamics in both eulerian or lagrangian variables also called the p system we show that they can be reformulated as a kinetic equation using an additional kinetic variable such a formulation was first obtained by the authors in the case of multidimensio nal scalar conservation laws a new phenomenon occurs here namely that the advection velocity is now a combination of the macroscopic and kinetic velocities various applications are given we recover the invariant regions deduce new l estimates using moments lemma and prove l w stability for 7 3

dirac equation for a spin charged particle on the 2d sphere s2 and the hyperbolic plane h2 using the idea of shape invariance with respect to the main quantum number n we represent lie algebras u 2 and u 1 1 the induced metric by the casimir operator of lie algebras u 2 and u 1 1 leads us to obtain new solutions of the dirac equation corresponding to a spin charged particle on the 2d sphere s2 and the hyperbolic plane h2 in the presence of a magnetic monopole it is shown that the related new spinors represent the supersymmetry algebra and that they satisfy shape invariance equations with respect to n

the riemann problem near a hyperbolic singularity the classification of solutions of quadratic riemann problems 1 the purpose of this paper is to classify the solutions of riemann problems near a hyperbolic singularity in a nonlinear system of conservation laws hyperbolic singularities play the role in the theory of riemann problems that rest points play in the theory of ordinary differential equations indeed generically only a finite number of structures can appear in a neighborhood of such a singularity in this the first of three papers the program of classification is discussed in general and the simplest structure that occurs is characterized

the generalized riemann problem method for the shallow water equations with bottom topography this paper extends the generalized riemann problem method grp to the system of shallow water equations with bottom topography the main contribution is that the generalized riemann problem method j comput phys 1984 55 1 1 32 is used to evaluate the midpoint values of solutions at each cell interface so that the bottom topography effect is included in numerical fluxes and at the same step the source term is discretized with an interface method in which only mid point values are plugged in this scheme is well balanced between the flux gradient and bottom topography when incorporating the surface gradient method sgm j comput phys 2001 168 1 1 25 into data reconstruction step and it is also suitable for both steady and unsteady flow simulations we illustrate the accuracy of this scheme by several 1 d and 2 d numerical experiments copyright 2005 john wiley sons ltd

global solutions to the compressible euler equations with geometrical structure we prove the existence of global solutions to the euler equations of compressible isentropic gas dynamics with geometrical structure including transonic nozzle flow and spherically symmetric flow due to the presence of the geometrical source terms the existence results themselves are new especially as they pertain to radial flow in an unbounded region vec x geqq 1 and to transonic nozzle flow arbitrary data withl bounds are allowed in these results a shock capturing numerical scheme is introduced to compute such flows and to construct approximate solutions the convergence and consistency of the approximate solutions generated from this scheme to the global solutions are proved with the aid of a compensated compactness framework

analysis and approximation of conservation laws with source terms we consider a conservation law of the form cl i ut f u x ax where a cdot is a bounded piecewise smooth source term and f an even convex function we first characterize the solution to the riemann problem through a new lax type formula then we prove that for a cdot fixed the semigroup associated with cl is an l1 contraction and we obtain an existence theorem for weak solutions to cl we conclude by constructing godunov type difference schemes and proving that these schemes are l infty stable and have stable steady solutions similar in structure to those of cl some numerical tests are reported

first and second order finite volume methods for the one dimensional nonconservative euler system gas flow in porous media with a nonconstant porosity function provides a nonconservative euler system we propose a new class of schemes for such a system for the one dimensional situations based on nonconservative fluxes preserving the steady state solutions we derive a second order scheme using a splitting of the porosity function into a discontinuous and a regular part where the regular part is treated as a source term while the discontinuous part is treated with the nonconservative fluxes we then present a classification of all the configurations for the riemann problem solutions in particularly we carefully study the resonant situations when two eigenvalues are superposed based on the classification we deal with the inverse riemann problem and present algorithms to compute the exact solutions we finally propose new sod problems to test the schemes for the resonant situations where numerical simulations are performed to compare with the theoretical solutions the schemes accuracy first and second order is evaluated comparing with a nontrivial steady state solution with the numerical approximation and convergence curves are established

riemann problem with nonlinear resonance effects and well balanced godunov scheme for shallow fluid flow past an obstacle we are interested in characterizing the resonant behavior that occurs in a shallow water system with an arbitrary bottom topography the inhomogeneity introduced by the bottom topography elevation term in the system of equations is modeled by a linearly degenerate field strict hyperbolicity fails when one of the different nonlinear waves has a zero speed a proliferation of reflected waves can occur by interactions and then resonance effects are important for a transonic flow classical nonlinear waves interact with standing waves and exhibit overcompressive undercompressive and marginal compressive waves analytical understanding is used to introduce a new kind of numerical processing of the source term the riemann problem with nonlinear resonance is solved the so called well balanced godunov scheme is constructed which yields more accurate asymptotic states and preserves the balance between the source terms and flux terms the numerical flux and source terms are simultaneously solved in both space and time to overcome resonance

on the compressible euler dynamics equations in transonic flow abstract this paper is concerned with the transonic flow in a variable area duct for the full compressible euler system we developed a generalized glimm method to confirm the existence of a solution for the initial boundary value problem the stability is resulted from the estimates on the interactions among the classical elementary waves and the perturbations caused by the operator splitting when the method is applied to the equations the limit of approximate solutions serves as a bv entropy solution in the cases where the duct has the positive material derivative or small l 1 norms of derivatives in particular both situations are consistent when the duct is expanded

a comparison of convergence rates for godunov s method and glimm s method in resonant nonlinear systems of conservation laws we obtain time independent bounds on derivatives prove convergence and establish a rate of convergence for godunov s numerical method as applied to the initial value problem for a resonant inhomogeneous conservation law which is treated as a 2 times 2 nonstrictly hyperbolic system we compare the results with a corresponding analysis of glimm s method and see that our analysis gives equivalent sharp convergence rates in the strictly hyperbolic setting but an improvement is seen in godunov s method over glimm s method in the nonstrictly hyperbolic resonant regime the 2 times 2 glimm and godunov methods are the only methods for which we can obtain time independent bounds on derivatives these bounds represent a purely nonlinear phenomenon because there are no corresponding time independent bounds for the linearized equations which blow up at a linear rate in time

supersymmetry factorisation of the schrodinger equation and a hamiltonian hierarchy the author presents a systematic procedure for constructing a hierarchy of non relativistic hamiltonians with the property that the adjacent members of the hierarchy are supersymmetric partners i e they share the same eigenvalue spectrum except for the missing ground state and the eigenvectors are simply related