the riemann problem for the two dimensional zero pressure euler equations abstract the riemann problem for the two dimensional zero pressure euler equations is considered the initial data are constant values in each quadrant which satisfy an assumption that each initial discontinuity projects only one two dimensional wave the phenomenon of two dimensional delta shock wave with a dirac delta function in both density and internal energy is identified both generalized rankine hugoniot relation and entropy condition for this type of two dimensional delta shock wave are proposed the qualitative behavior of entropy solutions to this relation with certain special initial data is established based on these preparations we obtain twenty three explicit solutions and their corresponding criteria in particular the mach reflection like patterns arise in the exact solutions

existence and asymptotic behavior of measure valued solutions for degenerate conservation laws abstract we consider two classes of typical degenerate hyperbolic systems of conservation laws to provide a general approach for solving the existence and large time asymptotic behavior of measure valued solutions for initial boundary value problems some existence theorems of the measure valued solutions are established the convergence of large time averages of the measure valued solutions to a dirac mass concentrated at the input state on the boundary is proved for almost each fixed space variable although the measure valued solutions of the initial boundary problems may not be unique in general our results indicate that the asymptotic equilibrium of these measure valued solutions is unique

riemann problem for the relativistic chaplygin euler equations abstract the relativistic euler equations for a chaplygin gas are studied the riemann problem is solved constructively there are five kinds of riemann solutions in which four only contain different contact discontinuities and the other involves delta shock waves under suitable generalized rankine hugoniot relation and entropy condition the existence and uniqueness of delta shock solutions are established

towards a compact high order method for non linear hyperbolic systems ii the hermite hllc scheme in a finite volume framework we develop an approximate hll riemann solver specific to weakly hyperbolic systems those systems are obtained by considering not only the variable but also its first spatial derivative as unknowns to this aim we rely upon the theory of d shock waves newly developed in the scalar case first we demonstrate that the extended version of the hlle scheme to weakly hyperbolic systems is compatible with the existence of dirac measures in the solution then we develop a specific hermite least square hlsm interpolation that enables to generate a high order and compact scheme without creating spurious oscillations in the reconstruction of the variable or its first derivative extensive numerical experiments make it possible to validate the method and to check convergence to entropy solutions relying upon those results we construct a new hll riemann solver suited for the extended one dimensional euler equations for this purpose we introduce the contribution of a contact discontinuity inside the definition of the solver by using a formal analogy with the scalar study we demonstrate that this solver tolerates the existence of d shock waves in the solution numerical experiments that follow help to validate some of the assumptions made to generate this scheme

riemann problem and wave interactions for the one dimensional relativistic string equation in minkowski space abstract in this paper we consider the riemann problem for the one dimensional relativistic string equation in minkowski space it is shown that the delta shock wave will be constructed in the riemann solutions in some certain situations furthermore the interactions of elementary waves including contact discontinuity and delta shock wave are obtained

universal multifractal properties of circle maps from the point of view of critical phenomena ii analytical results the multifractal properties of maps of the circle exhibited in the preceding paper are analyzed from a simplified approach to the renormalization group of kadanoff this second renormalization group transformation whose formulation and interpretation are discussed here acts on the space of one time differentiable coordinate changes which associate a map on the critical manifold to the fixed point of the usual renormalization group while the dependence of the multifractal moments on the starting point can be described statistically and in particular through universal amplitude ratios as in paper i it is shown that fourier analysis is another possible approach for all multifractal moments the low frequency fourier coefficients have a universal self similar scaling behavior analogous to that found for the usual spectrum of circle maps in the case of the first moment it is demonstrated that the fourier coefficients are within constants equal to the usual spectrum the relation between amplitude ratios and fourier coefficients is established and it is demonstrated that the universal values of the ratios come from the universal low frequency fourier coefficients since for the universal ratios arising in the statistical description the scaling regime is much more easily accessible than for the spectrum the statistical approach described in paper i should be more convenient for experiments and could become an alternative to the usual spectral description the universal statistical description of the multifractal moments adopted here is possible because the choice of thea priori probability for the starting point is demonstrated to be irrelevant

two dimensional riemann problem for a hyperbolic system of nonlinear conservation laws i four j cases abstract the riemann problem for a two dimensional 2 2 hyperbolic system of nonlinear conservation laws has been solved analytically the solutions reveal a variety of geometric structures 57 exact solutions and the corresponding criteria have been obtained a kind of new nonlinear hyperbolic wave which we call delta shock appears in some solutions the delta shock is a discontinuity surface which is the support of a dirac function

δ shock waves as a new type of solutions to systems of conservation laws abstract a concept of a new type of singular solutions to systems of conservation laws is introduced it is so called n shock wave where n is n th derivative of the dirac delta function n 1 2 in this paper the case n 1 is studied in details we introduce a definition of shock wave type solution for the system u t f u x 0 v t f u v x 0 w t f u v 2 f u w x 0 within the framework of this definition the rankine hugoniot conditions for shock are derived and analyzed from geometrical point of view we prove shock balance relations connected with area transportation finally a solitary shock wave type solution to the cauchy problem of the system of conservation laws u t u 2 x 0 v t 2 u v x 0 w t 2 v 2 u w x 0 with piecewise continuous initial data is constructed these results first show that solutions of systems of conservation laws can develop not only dirac measures as in the case of shocks but their derivatives as well

new developments of delta shock waves and its applications in systems of conservation laws a theory of delta shock waves with dirac delta functions developing in both state variables for a class of nonstrictly hyperbolic systems of conservation laws is established in this paper we solve constructively the riemann problems for the system under consideration in solutions we find another kind of delta shock waves on which both state variables simultaneously contain the dirac delta functions it is quite different from the previous ones on which only one state variable contains the dirac delta function we strictly prove it satisfies the system in the sense of distributions the generalized rankine hugoniot relation and entropy condition are proposed to solve the delta shock waves furthermore we show all of the existence and stability of solutions including the delta shock waves to reasonable viscous perturbations the generalized rankine hugoniot relation is also confirmed in particular our theory on the delta shock waves possesses the generality and practicability which can be conveniently and successfully applied to those systems investigated by korchinski 1977 tan zhang and zheng 1991 ercole 2000 cheng and yang 2011 etc and we also give a simplified approach to solve a 2 d riemann problem for the system studied by tan and zhang 1990 for the case 2j 2j and obtain the explicit formulae of the delta shock waves finally the numerical simulations completely coinciding with theoretical analysis are presented

eulerian droplet model delta shock waves and solution of the riemann problem abstract we study an eulerian droplet model which can be seen as the pressureless gas system with a source term a subsystem of this model and the inviscid burgers equation with source term the condition for loss of regularity of a solution to burgers equation with source term is established the same condition applies to the eulerian droplet model and its subsystem the riemann problem for the eulerian droplet model is constructively solved by going through the solution of the riemann problems for the inviscid burgers equation with a source term and the subsystem respectively under suitable generalized rankine hugoniot relations and entropy condition the existence of delta shock solution is established the existence of a solution to the generalized rankine hugoniot conditions is proven some numerical illustrations are presented

two dimensional riemann problem for a hyperbolic system of conservation laws in three pieces abstract this paper studies the two dimensional riemann problem for a hyperbolic system of conservation laws with three constant initial data separated by x positive y positive and x negative axes using the generalized characteristic analysis method by considering the interactions of rarefaction waves shock waves and contact discontinuities 13 exact solutions and the corresponding criteria are obtained delta shock waves appear in some solutions moreover the numerical illustrations agree completely with the geometric structures of the exact solutions

two dimensional riemann problems for zero pressure gas dynamics with three constant states abstract the riemann problems for two dimensional zero pressure gas dynamics are solved completely when the initial data take three constant states having discontinuities on x y positive and x negative axes with the help of characteristic analysis by studying interactions among delta shocks vacuums and contact discontinuities the riemann solutions constructed exhibit nine different explicit configurations the mach reflection like configurations appear in some solutions

riemann problems for nonstrictly hyperbolic 2 2 systems of conservation laws on resout le probleme de riemann pour des systemes 2 2 de lois de conservation hyperboliques ayant des fonctions flux quadratiques

the riemann problem with delta initial data for a class of coupled hyperbolic systems of conservation laws abstract in this paper we solve the riemann problem with the initial data containing dirac delta functions for a class of coupled hyperbolic systems of conservation laws under suitably generalized rankine hugoniot relation and entropy condition the existence and uniqueness of solutions involving delta shock waves are proved further four kinds of different structure for solutions are established uniquely

the riemann problem admitting δ δ shocks and vacuum states the vanishing viscosity approach in this paper using the vanishing viscosity method we construct a solution of the riemann problem for the system of conservation laws ut u2 x 0 vt 2 uv x 0 wt 2 v2 uw x 0 with the initial data u x 0 v x 0 w x 0 u v w x 0 this problem admits shock wave type solutions and vacuum states shock is a new type of singular solutions to systems of conservation laws first introduced in e yu panov v m shelkovich shock waves as a new type of solutions to systems of conservation laws j differential equations 228 2006 49 86 it is a distributional solution of the riemann problem such that for t 0 its second component v may contain dirac measures the third component w may contain a linear combination of dirac measures and their derivatives while the first component u has bounded variation using the above mentioned results we also solve the shock cauchy problem for the first two equations of the above system since shocks can be constructed by the vanishing viscosity method they are natural solutions to systems of conservation laws we describe the formation of the shocks and the vacuum states from smooth solutions of the parabolic problem the results of this paper as well as those of the above mentioned paper show that solutions of systems of conservation laws can develop not only dirac measures as in the case of shocks but their derivatives as well

generalized variational principles global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics we study systems of conservation laws arising in two models of adhesion particle dynamics the first is the system of free particles which stick under collision the second is a system of gravitationally interacting particles which also stick under collision in both cases mass and momentum are conserved at the collisions so the dynamics is described by 2 2 systems of conservations laws we show that for these systems global weak solutions can be constructed explicitly using the initial data by a procedure analogous to the lax oleinik variational principle for scalar conservation laws however this weak solution is not unique among weak solutions satisfying the standard entropy condition we also study a modified gravitational model in which instead of momentum some other weighted velocity is conserved at collisions for this model we prove both existence and uniqueness of global weak solutions we then study the qualitative behavior of the solutions with random initial data we show that for continuous but nowhere differentiable random initial velocities all masses immediately concentrate on points even though they were continuously distributed initially and the set of shock locations is dense

delta shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws abstract for simple models of hyperbolic systems of conservation laws we study a new type of nonlinear hyperbolic wave a delta shock wave which is a dirac delta function supported on a shock we prove that delta shock waves are w limits in l1 of solutions to some reasonable viscous perturbations as the viscosity vanishes further we solve the multiplication problem of a delta function with a discontinuous function to show that delta shock waves satisfy the equations in the sense of distributions under suitable generalized rankine hugoniot and entropy conditions we establish the existence and uniqueness of solutions involving delta shock waves for the riemann problems the existence of solutions to the cauchy problem is also investigated

sizing optimization of truss structures by method of centers and force formulation this paper describes a new approach to optimum weight design of truss structures the force method is incorporated in an optimization algorithm based on the method of center points design variables are the member cross sectional areas and the redundant forces evaluated for each independent loading condition acting on the structure the optimization method utilizes the largest hyperspheres inscribed within the feasible space the method of hyperspheres has been enhanced here to handle the compatibility equality constraints as well by including the analysis step in the optimization cycle there is no longer the need to perform separate structural analyses thus saving computation time the viability and efficiency of the proposed method are demonstrated for truss structures subject to multiple loading conditions and constraints on member stresses nodal displacement and minimum gage numerical results are compared with those reported in the literature

generalized plane delta shock waves for n dimensional zero pressure gas dynamics abstract with the help of a generalized plane wave solution we study a type of generalized plane delta shock wave for the n dimensional zero pressure gas dynamics and refine its generalized rankine hugoniot relation which is a system of ordinary equations this relation describes accurately the character of the generalized plane delta shock location propagation speed and weight under a suitable entropy condition four different explicit constructions of solutions for a kind of riemann problem with radon measure as initial data are established uniquely the overtaking of two plane delta shocks forming a new generalized plane delta shock is also investigated finally the 2 d riemann problem with four pieces of initial data is solved in a simplified situation

riemann problems for a class of coupled hyperbolic systems of conservation laws abstract we study a class of coupled hyperbolic systems of conservation laws which contain the one dimensional zero pressure gas dynamics as a prototypical example the riemann problems are solved constructively the riemann solutions exactly include two kinds delta shock wave solutions and vacuum solutions under the generalized rankine hugoniot relation and entropy condition all of the existence uniqueness and stability of solutions to viscous perturbations are proved two typical examples are presented finally

delta shock waves in chromatography equations abstract the previous investigations on delta shock waves were mostly focused on those with dirac delta function in only one state variable in this paper we obtain another kind from the nonlinear chromatography equations in which the dirac delta functions develop simultaneously in both state variables it is strictly proved to satisfy the system in the sense of distributions the generalized rankine hugoniot relation and entropy condition are clarified the numerical results completely coinciding with the theoretical analysis are presented