vanishing of vacuum states and blow up phenomena of the compressible navier stokes equations the navier stokes systems for compressible fluids with density dependent viscosities are considered in the present paper these equations in particular include the ones which are rigorously derived recently as the saint venant system for the motion of shallow water from the navier stokes system for incompressible flows with a moving free surface 14 these compressible systems are degenerate when vacuum state appears we study initial boundary value problems for such systems for both bounded spatial domains or periodic domains the dynamics of weak solutions and vacuum states are investigated rigorously

asymptotic constraint qualifications and global error bounds for convex inequalities in this paper we study various asymptotic constraint qualifications for the existence of global error bounds for approximate solutions of convex inequalities many known conditions that ensure the existence of such a global error bound are shown to be equivalent to one of the following three conditions i the bounded excess condition ii slater condition together with the asymptotic constraint qualification defined by auslender and crouzeix 1 and iii positivity of normal directional derivatives of the maximum of the constraint functions introduced by lewis and pang 12

error bounds for 2 regular mappings with lipschitzian derivatives and their applications we obtain local estimates of the distance to a set defined by equality constraints under assumptions which are weaker than those previously used in the literature specifically we assume that the constraints mapping has a lipschitzian derivative and satisfies a certain 2 regularity condition at the point under consideration this setting directly subsumes the classical regular case and the twice differentiable 2 regular case for which error bounds are known but it is significantly richer than either of these two cases when applied to a certain equation based reformulation of the nonlinear complementarity problem our results yield an error bound under an assumption more general than b regularity the latter appears to be the weakest assumption under which a local error bound for complementarity problems was previously available we also discuss an application of our results to the convergence rate analysis of the exterior penalty method for solving irregular problems

global regularity theorems for a system of nondifferentiable convex inequalities and linear equalities the best bound is given for the quotient of the distance of an infeasible point and the norm of the residual vector applications to stability theory are then obtained

error bounds for degenerate cone inclusion problems error bounds for cone inclusion problems in banach spaces are established under conditions weaker than robinson s constraint qualification the results allow the cone to be more general than the origin and therefore also generalize a classical error bound result concerning equality constrained sets in optimization applications in finite dimensional differentiable inequalities and in tangent cones are discussed

strong solutions of the navier stokes equations for isentropic compressible fluids we study strong solutions of the isentropic compressible navier stokes equations in a domain r3 we first prove the local existence of unique strong solutions provided that the initial data 0 and u0 satisfy a natural compatibility condition the important point in this paper is that we allow the initial vacuum the initial density may vanish in an open subset of we then prove a new uniqueness result and stability result our results are valid for unbounded domains as well as bounded ones

perturbation analysis of a condition number for convex inequality systems and global error bounds for analytic systems in this paper several types of perturbations on a convex inequality system are considered and conditions are obtained for the system to be well conditioned under these types of perturbations where the well conditionedness of a convex inequality system is defined in terms of the uniform boundedness of condition numbers under a set of perturbations it is shown that certain types of perturbations can be used to characterize the well conditionedness of a convex inequality system in which either the system has a bounded solution set and satisfies the slater condition or an associated convex inequality system which defines the recession cone of the solution set for the system satisfies the slater condition finally sufficient conditions are given for the existence of a global error bound for an analytic system it is shown that such a global error bound always holds for any inequality system defined by finitely many convex analytic functions when the zero vector is in the relative interior of the domain of an associated convex conjugate function

global error bounds with fractional exponents using the partial order induced by a proper weakly lower semicontinuous function on a reflexive banach space x we give a sufficient condition for f to have error bounds with fractional exponents application is given to identify the set of such exponents for quadratic functions

global error bounds for convex inequality systems in banach spaces we study conditions under which a global error bound in terms of a natural residual exists for a convex inequality system specifically we obtain an error bound result which unifies many existing results assuming a slater condition we also derive two characterizations for a convex inequality system to possess a global error bound one is in terms of metric regularity and the other is in terms of an associated convex inequality system as a consequence we show that in font bbbabstract msbm10 at 8pt mbox bbbabstract r n a global error bound holds for such a system under the assumption of the zero vector in the relative interior of the domain of an associated conjugate function along with metric regularity at every point of the feasible set defined by the system finally we discuss some applications of these results to convex programs

existence and uniqueness of solutions for a class of non newtonian fluids with vacuum and damping abstract in this paper we proved local existence and uniqueness of solutions for a class of non newtonian fluids with vacuum and damping in one dimensional bounded intervals the main difficulty is due to the strong nonlinearity of the system and initial vacuum

on the quadrature methods for the numerical solution of singular integral equations abstract a cauchy type singular integral equation can be numerically solved by the use of an appropriate numerical integration rule and the reduction of this equation to a system of linear algebraic equations either directly or after the reduction of the cauchy type singular integral equation to an equivalent fredholm integral equation of the second kind in this paper two fundamental theorems on the equivalence under appropriate conditions of the aforementioned methods of numerical solution of cauchy type singular integral equations are proved in sufficiently general cases of cauchy type singular integral equations of the second kind

on the uniform convergence of gaussian quadrature rules for cauchy principal value integrals and their derivatives the convergence of the aforementioned quadrature rules for integrands possessing h6lder continuous derivatives of an appropriate order is proved to be uniform and not only pointwise the rate of convergence is also established and an application to the numerical solution of singular integral equations is made

existence and uniqueness of solutions for a class of non newtonian fluids with singularity and vacuum abstract the aims of this paper are to discuss existence and uniqueness of local solutions for a class of non newtonian fluids with singularity and vacuum in one dimensional bounded intervals there are two important points in this paper one is that we allow the initial vacuum another one is that the viscosity term of momentum equation is with singularity and fully nonlinearity

extension of hoffman s error bound to polynomial systems given any linear system defined by linear inequalities equalities in re n hoffman s error bound says that the distance from any point x in re n to the solution set of the linear system is bounded by a constant independent of x times a certain residual function evaluated at x this paper considers arbitrary possibly nonconvex polynomial systems defined by polynomial inequalities equalities it is shown that the solution sets of such systems are in general holder continuous as the right hand side changes in contrast to the lipschitzian continuity for the linear case also for any convex quadratic inequality system possessing an interior solution the authors show that its solution set behaves in a lipschitzian manner as the right hand side varies this result sharpens the earlier error bound results by mangasarian and robinson for general convex differentiable inequality systems which in addition to the interiority assumption require either an asymptotic constraint qualification con

the classical collocation method for singular integral equations first of all a method of collocation which we call classical collocation is described for the approximate solution of complete singular integral equations with cauchy kernel taken over the arc 1 1 secondly we demonstrate that under reasonable conditions the approximate solutions converge to the solution of the original equation

asymptotic behavior of the solutions to the damped compressible euler equations with vacuum abstract the asymptotic behavior of solutions of the damped compressible euler equations is conjectured to obey to the famous porous media equations pmes the previous works on this topic concern the case away from vacuum where the system is strictly hyperbolic in present paper we prove that the l entropy weak solution with vacuum obtained by the compensated compactness theory converges strongly in l loc p 1 p space to the unique similarity solution of the related pme as time goes to infinity

a new interpretation of cauchy type singular integrals with an application to singular integral equations abstract cauchy type integrals were given the interpretation of the principal value for points inside the integration interval here this interpretation is modified and generalized in a very simple manner the new interpretation in general is not equivalent to the classical one the relationship between the new interpretation and the classical one is investigated and various applications of the new interpretation to the plemelj formulas the riemann hilbert boundary value problem singular integral equations the inversion formula quadrature rules and interface crack problems are presented

on the transformation of discontinuity waves from spatial to pressure coordinates for the primitive system of geophysical fluid dynamics abstract we give a complete analysis of the transformation of discontinuity waves from spatial coordinates to pressure coordinates for the primitive equations of geophysical fluid dynamics these waves are moving surfaces in physical space across which some or all of the fluid variables are discontinuous and therefore correspond to regions of steep gradient in solutions of related models which incorporate frictional forces or viscosity discontinuity waves consequently represent the most easily discernable and frequently the most interesting and important features of the flow our results show that the primitive equations in either coordinate system inherit from the underlying euler equations all the physically correct shear waves that the acoustic waves are completely eliminated in pressure coordinates but not in spatial coordinates and that unphysical anomolous waves occur in both coordinate systems

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

sufficient conditions for error bounds for a lower semicontinuous l s c inequality system on a banach space it is shown that error bounds hold provided every element in an abstract subdifferential of the constraint function at each point outside the solution set is norm bounded away from zero a sufficient condition for a global error bound to exist is also given for an l s c inequality system on a real normed linear space it turns out that a global error bound closely relates to metric regularity which is useful for presenting sufficient conditions for an l s c system to be regular at sets under the generalized slater condition a continuous convex system on rn is proved to be metrically regular at bounded sets

extended newton methods for conic inequalities approximate solutions and the extended smale α theory abstract using the convex process theory and the majorizing technique we study the convergence issues of the iterative sequences generated by the extended newton method for solving the conic inequality system f c 0 defined by a cone c and a frechet differentiable function f satisfying the extended weak condition convergence criterion of the extended newton method is presented as an application we establish when f is analytic an extended theory similar to smale s theory for nonlinear analytic equations

error bounds for nondifferentiable convex inequalities under a strong slater constraint qualification a global error bound is given on the distance between an arbitrary point in then dimensional real spacern and its projection on a nonempty convex set determined bym convex possibly nondifferentiable inequalities the bound is in terms of a natural residual that measures the violations of the inequalities multiplied by a new simple condition constant that embodies a single strong slater constraint qualification cq which implies the ordinary slater cq a very simple bound on the distance to the projection relative to the distance to a point satisfying the ordinary slater cq is given first and then used to derive the principal global error bound

application of quadrature rules for cauchy type integrals to the generalized poincare bertrand formula the classical poincare bertrand transposition formula for the inversion of the order of integration in repeated cauchy type integrals is generalized in accordance with a new interpretation of cauchy type integrals next the gauss jacobi quadrature rule is applied in a particular case of the generalized poincare bertrand formula to both members of this formula and it is proved that this formula still remains valid after the approximation of the integrals by quadrature sums two simple applications of this result one concerning the convergence of a quadrature rule for repeated cauchy type integrals and the other the numerical solution of singular integral equations are made further generalizations and applications of the present results follow easily

error bounds for lower semicontinuous functions in normed spaces without the convexity or analyticity assumption we study error bounds for an inequality system defined by a general lower semicontinuous function and establish sufficient necessary conditions on the existence of error bounds in infinite dimensional normed spaces some characterizations for a convex inequality system to possess an error bound in a reflexive banach space are also given as applications in dealing with the hoffman error bound result in normed spaces we give a computable lipschitz bound constant which is better than previous lipschitz bound constants in some examples we also consider error bounds for quadratic functions on rn

a remark on the application of interpolatory quadrature rules to the numerical solution of singular integral equations abstract a cauchy type singular integral equation of the first or the second kind can be numerically solved either directly or after its reduction by the usual regularization procedure to an equivalent fredholm integral equation of the second kind the equivalence of these two methods that is the equivalence both of the systems of linear algebraic equations to which the singular integral equation is reduced and of the natural interpolation formulae is proved in this paper for a class of cauchy type singular integral equations of the first kind and of the second kind but with constant coefficients for general interpolatory quadrature rules under sufficiently mild assumptions the present results constitute an extension of a series of previous results concerning only gaussian quadrature rules based on the corresponding orthogonal polynomials and their properties

a unified analysis of hoffman s bound via fenchel duality in 1952 hoffman showed that the distance from any point to the solution set of a linear system is bounded above by a constant times the norm of the residual subsequently this bound has been studied extensively and has found many practical uses in this paper we consider an extension of hoffman s bound to a partially infinite dimensional setting in which the norm defining the distance is replaced by a positively homogeneous convex function and the linear system is replaced by a convex inclusion of the form ax a in k where a is a continuous linear operator from some normed linear space into mathbf r m a is an element of the range space of a and k is a nonempty closed convex cone in mathbf r m when specialized to the finite dimensional case we unify and extend many existing results on hoffman s bound our analysis is based on the use of fenchel duality to express the distance as the supremum of a certain concave function over a bounded subset of the polar of k much of our analysis als