convergence of relaxation schemes for initial boundary value problems for conservation laws abstract we consider a scalar conservation law with stiff source term in the quarter plan this equation is relaxed in a quasi linear hyperbolic system the relaxed system is approximated using an upwind scheme converging for fixed relaxation time and vanishing discretization mesh further we use chapman enskog expansion to obtain a viscous scheme for the equilibrium law and prove its convergence to the physical solution boundary information is carefully handled in an appropriate inequality linking the entropy numerical flux and the flux function an extension to general conservative schemes is also investigated

the riemann problem near a hyperbolic singularity ii this paper is interested in classifying the solutions of riemann problems for the 2 times 2 conservation laws that have homogeneous quadratic flux functions such flux functions approximate an arbitrary 2 times 2 system in a neighborhood of an isolated point where strict hyperbolicity fails here the solution for the symmetric systems in region iii of the four region classification of schaeffer and shearer is given the solution is based on the qualitative shape of the integral curves described by schaeffer and shearer and a numerical calculation of the hugoniot loci and their shock types

weak entropy boundary conditions for isentropic gas dynamics via kinetic relaxation abstract we consider a kinetic bgk model relaxing to isentropic gas dynamics previously introduced by the authors but with dirichlet boundary condition on the incoming velocities we pass to the limit as the relaxation parameter tends to zero by compensated compactness inside the domain and obtain that the limit satisfies entropy inequalities on the boundary involving weak traces of entropy fluxes our method is very general and could be applied to any entropy satisfying bgk model as soon as we have strong compactness of the macroscopic variables inside the domain

the classification of 2 2 systems of non strictly hyperbolic conservation laws with application to oil recovery on considere un systeme 2 2 a une variable d espace u f u x 0 0 ou u u x t r 2 et f r 2 r 2 on montre que pour des equations hyperboliques des points ombilicaux sont typiquement isoles et que en un point ombilical isole df est typiquement diagonalisable on donne une classification des equations hyperboliques a points ombilicaux isoles

divergence measure fields and hyperbolic conservation laws we analyze a class of l infty vector fields called divergence measure fields we establish the gauss green formula the normal traces over subsets of lipschitz boundaries and the product rule for this class of l infty fields then we apply this theory to analyze l infty entropy solutions of initial boundary value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data the examples of conservation laws include multidimensional scalar equations the system of nonlinear elasticity and a class of m x m systems with affine characteristic hypersurfaces the analysis in l infty also extends to l p

eigenvalue bifurcations in kac murdock szegő matrices with a complex parameter abstract for complex the spectral properties of the toeplitz matrix k n j k j k 1 n often called the kac murdock szeg matrix have been examined in detail in two recent papers the second paper in particular introduced the concept of borderline curves these are two closed curves in the complex plane that consist of all the for which k n possesses some eigenvalue whose magnitude equals the matrix dimension n the purpose of the present paper is to examine eigenvalue bifurcations in both a qualitative and a quantitative manner and to discuss connections between bifurcations and the borderline curves

singularities in multifractal turbulence dissipation networks and their degeneration abstract we suggest that large scale turbulence dissipation is concentrated along caustic networks that appear due to vortex sheet instability in three dimensional space leading to an effective fractal dimension d eff 5 3 of the networl backbone without caustic singularities and a turbulence intermittency exponent 1 6 if there are singularities on these caustic networks then d eff 5 3 and 1 6 it is shown using the theory of caustic singularities that the strongest however stable on the backbone singularities lead to d eff 4 3 an elastic backbone and to 1 3 thus there is a restriction of the network fractal variability 4 3 eff 5 3 and consequently 1 6 1 3 degeneration of these networks into a system of smooth vortex filaments deff 1 leads to 1 2 after degeneration the strongest singularities of the dissipation field e lose their power law form while the smoother field in e takes it it is shown using the method of multifractal asymptotics that the probability distribution of the dissipation changes its form from exponential like to log normal like with this degeneration and that the multifractal asymptote of the field in e is related to the multifractal asymptote of the energy field finally a phenomenon of acceleration of large scale turbulent diffusion of passive scalar by the singularities is briefly discussed all results are supported by comparison with experimental data obtained by different authors

a new canonical form for complex symmetric matrices it is well known that every real symmetric matrix and every complex hermitian matrix is diagonalizable i e orthogonally similar to a diagonal matrix however a complex symmetric matrix with repeated eigenvalues may fail to be diagonalizable we present a block diagonal canonical form in which each block is quasi diagonal to which every complex symmetric matrix is orthogonally similar as far as applications are concerned complex symmetric matrices as opposed to hermitian matrices play an important role in theories of wave propagation in continuous media e g elasticity thermoelasticity

dynamic green s functions in anisotropic piezoelectric thermoelastic and poroelastic solids a procedure is described to generate fundamental solutions or green s functions for time harmonic point forces and sources the linearity of the field equations permits the green s function to be represented as an integral over the surface of a unit sphere where the integrand is the solution of a one dimensional impulse response problem the method is demonstrated for the theories of piezoelectricity thermoelasticity and poroelasticity time domain analogues are discussed and compared with known expressions for anisotropic elasticity

spectral properties of kac murdock szego matrices with a complex parameter abstract when 0 1 the kac murdock szego matrix k n j k j k 1 n is a toeplitz correlation matrix with many applications and very well known spectral properties we study the eigenvalues and eigenvectors of k n for the general case where is complex pointing out similarities and differences to the case 0 1 we then specialize our results to real with 1 emphasizing the continuity of the eigenvalues as functions of for 1 we develop simple approximate formulas for the eigenvalues and pinpoint all eigenvalues locations our study starts from a certain polynomial whose zeros are connected to the eigenvalues by elementary formulas we discuss relations of our results to earlier results of w f trench

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

self organization and fractal dynamics in turbulence abstract results of analysis of the field of helicity obtained in three different turbulent laboratory flows grid flow boundary layer and jet and a simple helical fracton model has been used in order to provide a quantitative explanation of anomalous turbulent diffusion in the troposphere and in the ocean it is shown that kolmogorov turbulence is critical in respect to the localization effects of subregions with large helicity helical fractons and it breaks up into helical fractons under the condition df 2 where df 2d dw is the so called fracton dimension d is the fractal dimension of the turbulent fractal and dw is the dimension of random walks on this fractal for strictly kolmogorov turbulence d1 2 we study the internal structure of helical fractons and demonstrate that they are characterized by d f 4 3 finally we look at the influence of helical fractons on diffusion of a passive scalar in turbulence it is shown that their influence is manifested in the scaling law for the turbulent diffusivity in the form k l 8 7 in both three dimensional and quasi two dimensional situations this anomalous law is in a very good agreement with a large number of experimental data of different authors in the troposphere and in the upper ocean

kinetic relaxation to entropy based coupling conditions for isentropic flow on networks abstract we consider networks for isentropic gas and prove existence of weak solutions for a large class of coupling conditions first we construct approximate solutions by a vector valued bgk model with a kinetic coupling function introducing so called kinetic invariant domains and using the method of compensated compactness justifies the relaxation towards the isentropic gas equations we will prove that certain entropy flux inequalities for the kinetic coupling function remain true for the traces of the macroscopic solution these inequalities define the macroscopic coupling condition our techniques are also applicable to networks with arbitrary many junctions which may possibly contain circles we give several examples for coupling functions and prove corresponding entropy flux inequalities we prove also new existence results for solid wall boundary conditions and pipelines with discontinuous cross sectional area

the numerical interface coupling of nonlinear hyperbolic systems of conservation laws i the scalar case we study the theoretical and numerical coupling of two general hyperbolic conservation laws the coupling preserves in a weak sense the continuity of the solution at the interface without imposing the overall conservativity of the coupled model in order to analyze the convergence of the coupled numerical scheme we first revisit the approximation of the boundary value problems we then prove the convergence and characterize the limit solution of the coupled schemes in a few simple but significative coupling situations the general coupling problem is analyzed for riemann initial data and illustrated by numerical simulations resume nous nous interessons a une nouvelle forme de couplage de deux systemes hyperboliques de lois de conservation ce couplage assure de facon faible la continuite de la solution a l interface sans imposer la conservativite du modele couple pour etudier la convergence du schema d approximation numerique nous commencons par reprendre les resultats concernant l approximation du probleme aux limites nous demontrons ensuite la convergence du schema couple dans un certain nombre de cas interessants le cas general du couplage est etudie et illustre numeriquement pour une donnee initiale de riemann

initial boundary value problem for conservation laws this paper concerns the initial boundary value problems for some systems of quasilinear hyperbolic conservation laws in the space of bounded measurable functions the main assumption is that the system under study admits a convex entropy extension it is proved that then any twicely differentiable entropy fluxes have traces on the boundary if the bounded solutions are generated by either godunov schemes or by suitable viscous approximations furthermore in the case that the weak interior solutions are generated by godunov schemes any lipschitz continuous entropy fluxes corresponding to convex entropies have traces on the boundary and the traces are bounded above by computable numerical boundary values this in particular gives a trace formula for the flux functions in terms of the numerical boundary data we also investigate the formulation of boundary conditions for systems of hyperbolic conservation laws it is shown that the set of expected boundary values derived from the viscous approximation contains the one derived in terms of the boundary riemann problems and the converse is not true in general the general theory is then applied to some specific examples first several new facts are obtained for convex scalar conservation laws for example we give example which show that godunov schemes produce numerical boundary layers it is shown that any continuous functions of density have traces on the boundary instead of only entropy fluxes we also obtain interior and boundary regularity of the weak solutions for bounded measurable initial and boundary data a generalized oleinik entropy condition is also obtained next we prove the existence of a weak solution to the initial boundary value problem for a family of quadratic system with a uniformly characteristic boundary condition

exceptional points of non hermitian operators exceptional points associated with non hermitian operators i e operators being non hermitian for real parameter values are investigated the specific characteristics of the eigenfunctions at the exceptional point are worked out within the domain of real parameters the exceptional points are the points where eigenvalues switch from real to complex values these and other results are exemplified by a classical problem leading to exceptional points of a non hermitian matrix

a theorem in thermoelasticity and its application to linear stability equations are written down governing the propagation of plane sinusoidal waves of small amplitude through a homogeneously prestrained equilibrium state of a materially homogeneous thermoelastic body of arbitrary elastic and thermal symmetry the symmetric isothermal and isentropic acoustic tensors are defined in the usual way and it is assumed that the former is positive definite so that it has three real and positive eigenvalues it is shown under the usual assumption that the specific heat at constant deformation is positive that the three real and positive eigenvalues of the isentropic acoustic tensor are interlaced with those of the isothermal acoustic tensor the smallest eigenvalue belonging to the isothermal and the largest to the isentropic acoustic tensor under the additional assumption that the symmetrized thermal conductivity tensor is positive definite it is further shown that this result on the interlacing of the eigenvalues is sufficient to guarantee for all positive values of the frequency of the sinusoidal waves that the material is linearly stable in the sense that sinusoidal waves may not increase without bound in the direction of propagation in the final section the wide diversity in behaviour of the complex squared wave speed as a function of frequency is illustrated graphically the stability result is extended to negative frequencies as these would be required in any fourier synthesis of the sinusoidal wave solutions a connection with whitham s wave hierarchy approach is mentioned

initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws we consider initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws through the scalar case the entropy solutions we address assume their initial data only in the sense of weak star in l as t 0 and satisfy the entropy inequality in the sense of distributions for t 0 we prove that if the flux function has weakly genuine nonlinearity then the entropy solutions are always unique and the initial layers do not appear we also discuss applications to the zero relaxation limit for hyperbolic systems of conservation laws with relaxation

double borderline and extraordinary eigenvalues of kac murdock szegő matrices with a complex parameter abstract for all sufficiently large complex and for arbitrary matrix dimension n it is shown that the kac murdock szeg matrix k n j k j k 1 n possesses exactly two eigenvalues whose magnitude is larger than n we discuss a number of properties of the two extraordinary eigenvalues conditions are developed that given n allow us without actually computing eigenvalues to find all values that give rise to eigenvalues of magnitude n termed borderline eigenvalues the aforementioned values of form two closed curves in the complex plane we describe these curves which are n dependent in detail an interesting borderline case arises when an eigenvalue of k n equals n apart from certain exceptional cases this occurs if and only if the eigenvalue is a double one and if and only if the point is a cusp like singularity of one of the two closed curves

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

probability densities of homogeneous functions explicit approximation and applications to percolating networks the author treats the probability density of homogeneous functions of independent variables assuming these variables are narrowly distributed he finds an explicit approximate form for the laplace transform of the function s density this expression is used to show that if the variables are all gaussian then so is the global function the latter result is also claimed to hold for the total resistance of any random resistor network between two arbitrary terminals he further utilises the result to explicitly find the laplace transform of the conductance density of i a hierarchical structure and ii a random network in 1 in dimensions using the migdal kadanoff procedure in the last case he also finds the scaling of all the cumulants and finds that they exhibit a constant gap

eigenvectors of hermitian toeplitz matrices with smooth simple loop symbols abstract the paper is devoted to the structure and the asymptotics of the eigenvector matrix of hermitian toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line the results extend existing results on banded toeplitz matrices to full toeplitz matrices with temperate decay of the entries in the first row and column we establish formulas for both the exact and the asymptotic computation of arbitrarily prescribed individual components of arbitrarily prescribed individual eigenvectors these formulas are in terms of the wiener hopf factorization of a function which depends solely on the symbol and the corresponding eigenvalue or even only on an approximation to this eigenvalue the paper does not aim at providing numerical algorithms its main purpose is rather to reveal the structure underneath the eigenvectors which are shown to be the sum of two harmonics and certain edge corrections

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

kinetic approximation of a boundary value problem for conservation laws we design numerical schemes for systems of conservation laws with boundary conditions these schemes are based on relaxation approximations taking the form of discrete bgk models with kinetic boundary conditions the resulting schemes are riemann solver free and easily extendable to higher order in time or in space for scalar equations convergence is proved we show numerical examples including solutions of euler equations

a theorem on isotropic null vectors and its application to thermoelasticity let a be a symmetric n times n matrix over an arbitrary field we show that if a possesses a non zero null vector x which is also isotropic i e such that x cdot x 0 then both the trace of the adjugate and the determinant of a vanish for the complex field the converse is also true in purely mechanical elasticity it is known that an isotropic wave amplitude vector which corresponds to a circularly polarized wave is necessarily associated with a double root of the characteristic equation of wave propagation whose roots give essentially the squared wave speeds we apply the preceding results to obtain concise conditions for the occurrence both of isotropic wave amplitudes and of double roots in the theory of thermoelastic wave propagation in sharp contrast with the purely mechanical theory we find that these two phenomena generally occur separately in thermoelasticity

boundary conditions for nonlinear hyperbolic systems of conservation laws we propose two formulations of the boundary conditions for nonlinear hyperbolic systems of conservation laws a first approach is based on the vanishing viscosity method and a second one is related to the riemann problem the equivalence between these two conditions is studied the latter formulation is extended to treat numerically physically relevant boundary conditions monodimensional experiments are presented

coupling of eigenvalues of complex matrices at diabolic and exceptional points the paper presents a general theory of coupling of eigenvalues of complex matrices of an arbitrary dimension depending on real parameters the cases of weak and strong coupling are distinguished and their geometric interpretation in two and three dimensional spaces is given general asymptotic formulae for eigenvalue surfaces near diabolic and exceptional points are presented demonstrating crossing and avoided crossing scenarios two physical examples illustrate effectiveness and accuracy of the presented theory

interaction of eigenvalues in multi parameter problems the paper presents a new general theory of interaction of eigenvalues of matrix operators depending on parameters both complex and real eigenvalues are considered strong and weak interactions are distinguished and their geometric interpretation on the complex plane is given mechanical examples are presented and discussed in detail