singular behavior of vacuum states for compressible fluids in this survey paper we will present the recent work on the study of the compressible fluids with vacuum states by illustrating its interesting and singular behavior through some systems of fluid dynamics that is euler equations euler poisson equations and navier stokes equations the main concern is the well posedness of the problem when vacuum presents and the singular behavior of the solution near the interface separating the vacuum and the gas furthermore the relation of the solutions for the gas dynamics with vacuum to those of the boltzmann equation will also be discussed in fact the results obtained so far for vacuum states are far from being complete and satisfactory therefore this paper can only be served as an introduction to this interesting field which has many open and challenging mathematical problems moreover the problems considered here are limited to the author s interest and knowledge in this area

temps de vie des solutions regulieres des equations d euler compressibles axisymetriques en dimension deux nous considerons le systeme des equations d euler isentropiques en dimension deux pour des donnees initiales invariantes par rotation et perturbations de taille e d un etat de repos on etablit un equivalent du temps de viet e de la solution classique lim 2 t 2 de plus on donne pour t leqq frac a 2 varepsilon 2 a tau une estimation de la vraie solution en calculant la taille de son ecart a une solution approchee construite dans un precedent travail

well posedness in smooth function spaces for moving boundary 1 d compressible euler equations in physical vacuum we prove well posedness for the 3 d compressible euler equations with moving physical vacuum boundary with an equation of state given by the so called gamma gas law for gamma 1 the physical vacuum singularity requires the sound speed c to go to zero as the square root of the distance to the moving boundary and thus creates a degenerate and characteristic hyperbolic free boundary system wherein the density vanishes on the free boundary the uniform kreiss lopatinskii condition is violated and manifest derivative loss ensues nevertheless we are able to establish the existence of unique solutions to this system on a short time interval which are smooth in sobolev spaces all the way to the moving boundary and our estimates have no derivative loss with respect to initial data our proof is founded on an approximation of the euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term chosen to preserve as much of the geometric structure of the euler equations as possible we first construct solutions to this degenerate parabolic regularization using a new higher order hardy type inequality we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter solutions to the compressible euler equations are found in the limit as the artificial viscosity tends to zero our regular solutions can be viewed as degenerate viscosity solutions out methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws

wave equations with time dependent dissipation ii effective dissipation abstract this article is intended to present a construction of structural representations of solutions to the cauchy problem for wave equations with time dependent dissipation above scaling these representations are used to give estimates of the solution and its derivatives based on l q r n q 2 the article represents the second part within a series in jens wirth wave equations with time dependent dissipation i non effective dissipation j differential equations 222 2 2006 487 514 weak dissipations below scaling were discussed

on aggregate farrell efficiencies in this paper we establish the fact that an industry maximal revenue is the sum of its firms maximal revenues this fact enables us to discover conditions for aggregation of farrell efficiencies

on global multidimensional supersonic flows with vacuum states at infinity in this paper we are concerned with the global existence and stability of a smooth supersonic flow with vacuum state at infinity in a three dimensional infinitely long divergent nozzle the flow is described by a three dimensional steady potential equation which is multi dimensional quasilinear hyperbolic but degenerate at infinity with respect to the supersonic direction and whose linearized part admits the form partial t 2 frac 1 1 t 2 gamma 1 partial 1 2 partial 2 2 frac 2 gamma 1 1 t partial t for 1 gamma 2 from the physical point of view due to the expansive geometric property of the divergent nozzle and the mass conservation of gases the moving gases in the nozzle will gradually become rarefactive and tend to vacuum states at infinity which implies that such a smooth supersonic flow should be globally stable for small perturbations since there are no strong resulting compressions in the motion of the flow we will confirm such a global stability phenomenon by rigorous mathematical proofs and further show that there do not exist vacuum domains in any finite parts of the nozzle

estimation of returns to scale using data envelopment analysis abstract generalization of the measure of returns to scale from a single number to an interval permits extension of the concept to dea data domains with multiple inputs and multiple outputs the key new approach is a partition of the optimal frontier into three parts corresponding respectively to increasing constant and decreasing returns to scale these parts are characterized in terms of optimal primal solutions and optimal dual solutions for both the original charnes cooper rhodes model 1978 and the later banker charnes cooper model 1984 and relying on concepts developed by r d banker 1984 and r m thrall 1988

well posedness in smooth function spaces for the moving boundary three dimensional compressible euler equations in physical vacuum we prove well posedness for the three dimensional compressible euler equations with moving physical vacuum boundary with an equation of state given by p c for 1 the physical vacuum singularity requires the sound speed c to go to zero as the square root of the distance to the moving boundary and thus creates a degenerate and characteristic hyperbolic free boundary system wherein the density vanishes on the free boundary the uniform kreiss lopatinskii condition is violated and manifest derivative loss ensues nevertheless we are able to establish the existence of unique solutions to this system on a short time interval which are smooth in sobolev spaces all the way to the moving boundary and our estimates have no derivative loss with respect to initial data our proof is founded on an approximation of the euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term chosen to preserve as much of the geometric structure of the euler equations as possible we first construct solutions to this degenerate parabolic regularization using a higher order version of hardy s inequality we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter solutions to the compressible euler equations are found in the limit as the artificial viscosity tends to zero our regular solutions can be viewed as degenerate viscosity solutions our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws

well posedness for compressible euler equations with physical vacuum singularity an important problem in the theory of compressible gas flows is to understand the singular behavior of vacuum states the main difficulty lies in the fact that the system becomes degenerate at the vacuum boundary where the characteristic speeds u c coincide and have unbounded spatial derivative since c behaves like x 1 2 close to the boundary in this paper we overcome this difficulty by presenting a new formulation and new energy spaces we establish the local in time well posedness of one dimensional compressible euler equations for isentropic flows with the physical vacuum singularity in some spaces adapted to the singularity 2009 wiley periodicals inc

the denominator rule for share weighting aggregation abstract in this paper we prove that the denominator rule namely that weighting shares should be defined in term of the denominator variable of the relevant index is a necessary and sufficient condition for consistent arithmetic aggregation of any ratio type performance measures including efficiency indices we then illustrate the applicability of the denominator rule in the aggregation of scale efficiency measures of the effect of input congestion and of capacity utilization indices

the lifespan of a class of smooth spherically symmetric solutions of the compressible euler equations with variable entropy in three space dimensions we consider smooth three dimensional spherically symmetric eulerian flows of ideal polytropic gases with variable entropy whose initial data are obtained by adding a small smooth perturbation with compact support to a constant state under a natural assumption we obtain precise information on the asymptotic behavior of their lifespan when the size of the initial perturbation tends to 0 this is achieved by the construction and estimate of a suitable approximate flow

modeling stochastic frontier based on vine copulas abstract this article models a production function and analyzes the technical efficiency of listed companies in the united states germany and england between 2005 and 2012 based on the vine copula approach traditional estimates of the stochastic frontier assume that data is multivariate normally distributed and there is no source of asymmetry the proposed method based on vine copulas allow us to explore different types of asymmetry and multivariate distribution using data on product capital and labor we measure the relative efficiency of the vine production function and estimate the coefficient used in the stochastic frontier literature for comparison purposes this production vine copula predicts the value added by firms with given capital and labor in a probabilistic way it thereby stands in sharp contrast to the production function where the output of firms is completely deterministic the results show that on average s p500 companies are more efficient than companies listed in england and germany which presented similar average efficiency coefficients for comparative purposes the traditional stochastic frontier was estimated and the results showed discrepancies between the coefficients obtained by the application of the two methods traditional and frontier vine opening new paths of non linear research

extending fare and zelenyuk 2003 abstract in our recent work fare and zelenyuk 2003 on aggregate farrell efficiency scores european journal of operational research 146 3 615 620 we have proposed a way of aggregating farrell type efficiency scores with weights and aggregation function derived from economic type optimization behaviour in this comment we correct a mathematical error present in that work as well as generalize the price independent weights we have proposed earlier

aggregation of scale efficiency abstract in this article we generalize the aggregation theory in efficiency and productivity analysis by deriving solutions to the problem of aggregation of individual scale efficiency measures primal and dual into aggregate primal and dual scale efficiency measures of a group e g industry the new aggregation result is coherent with aggregation framework and solutions that were earlier derived for other related efficiency measures and can be used in practice for estimation of scale efficiency of an industry or other groups of firms within it

transonic shocks for the full compressible euler system in a general two dimensional de laval nozzle in this paper we study the transonic shock problem for the full compressible euler system in a general two dimensional de laval nozzle as proposed in courant and friedrichs supersonic flow and shock waves interscience new york 1948 given the appropriately large exit pressure pe x if the upstream flow is still supersonic behind the throat of the nozzle then at a certain place in the diverging part of the nozzle a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes pe x we solve this problem completely for a general class of de laval nozzles whose divergent parts are small and arbitrary perturbations of divergent angular domains for the full steady compressible euler system the problem can be reduced to solve a nonlinear free boundary value problem for a mixed hyperbolic elliptic system one of the key ingredients in the analysis is to solve a nonlinear free boundary value problem in a weighted holder space with low regularities for a second order quasilinear elliptic equation with a free parameter the position of the shock curve at one wall of the nozzle and non local terms involving the trace on the shock of the first order derivatives of the unknown function

blowup of smooth solutions to the compressible navier stokes equation with compact density we present a sufficient condition on the blowup of smooth solutions to the compressible navier stokes equations in arbitrary space dimensions with initial density of compact support as an immediate application it is shown that any smooth solutions to the compressible navier stokes equations for polytropic fluids in the absence of heat conduction will blow up in finite time as long as the initial densities have compact support and an upper bound which depends only on the initial data on the blowup time follows from our elementary analysis immediately another implication is that there is no global small decay in time or even bounded in the case that all the viscosity coefficients are positive smooth solutions to the compressible navier stokes equations for polytropic fluids no matter how small the initial data are as long as the initial density is of compact support this is in contrast to the classical theory of global existence of small solutions to the same system with initial data being a small perturbation of a constant state that is not a vacuum the blowup of smooth solutions to the compressible euler system with initial density and velocity of compact support is a simple consequence of our argument 1998 john wiley sons inc

determining merged relative scores the starting point in this paper is the following type of situation performances of people or technologies are measured by their scores with respect to different benchmarks relative to those of some base performance in order to arrive at a decision one wants to merge average aggregate the scores of each individual or technology into a single number which can then be compared with that of another individual or technology the situation is similar at least technically to price indices where present prices of different products are considered relative to prices of these products in a base year and we again want to merge these into one number in this paper one solves most of these problems one also states a few problems which are still open

global solutions of nonlinear hyperbolic equations for small initial data on considere des systemes quasilineaires d equations hyperboliques d ordre 2 qui sont des deformations non lineaires de l equation d onde on etudie l existence globale et le comportement asymptotique des solutions du probleme de cauchy quand la donnee initiale est suffisamment petite

local existence with physical vacuum boundary condition to euler equations with damping abstract in this paper we consider the local existence of solutions to euler equations with linear damping under the assumption of physical vacuum boundary condition by using the transformation introduced in lin and yang methods appl anal 7 3 2000 495 to capture the singularity of the boundary we prove a local existence theorem on a perturbation of a planar wave solution by using littlewood paley theory and justifies the transformation introduced in liu and yang 2000 in a rigorous setting

a projection based approach to general form tikhonov regularization we present a projection based iterative algorithm for computing general form tikhonov regularized solutions to the problem min x ax b 2 2 lambda 2 lx 2 2 where the regularization matrix l is not the identity our algorithm is designed for the common case where lambda is not known a priori it is based on a joint bidiagonalization algorithm and is appropriate for large scale problems when it is computationally infeasible to transform the regularized problem to standard form by considering the projected problem we show how estimates of the corresponding optimal regularization parameter can be efficiently obtained numerical results illustrate the promise of our projection based approach