strong convergence to global solutions for multidimensional flows of compressible viscous fluids with polytropic equations of state and discontinuous initial data we extend to general polytropic pressures p k 1 the existence theory of 8 for isothermal 1 flows of navier stokes fluids in two and three space dimensions with fairly general initial data specifically we require that the initial density be close to a constant in l2 and l and that the initial velocity be small in l2 and bounded in l2n in two dimensions the l2 norms must be weighted slightly solutions are obtained as limits of approximate solutions corresponding to mollified initial data the key point is that the approximate densities are shown to converge strongly so that nonlinear pressures can be accommodated even in the absence of any uniform regularity information for the approximate densities

decay estimates of solutions for the equations of motion of compressible viscous and heat conductive gases in an exterior domain in ℝ3 we consider the equations of motion of compressible viscous and heat conductive gases in an exterior domain in 3 we give the l q l p estimates for solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem

on the outer pressure problem of the one dimensional compressible navier stokes equation with degenerate transport coefficients abstract this paper is concerned with the initial boundary value problem for the compressible navier stokes system with degenerate transport coefficients we prove the global existence of large solutions for the outer pressure problem of a viscous heat conducting one dimensional real gas the proof is based on several detailed analysis and key a priori estimates on the bounds on the density and temperature

asymptotic behavior of solutions of functional difference equations for linear functional difference equations we obtain some results on the asymptotic behavior of solutions which correspond to a perron type theorem for linear ordinary difference equations we also apply our results to volterra difference equations with infinite delay

rate of convergence for szasz mirakyan durrmeyer operators with derivatives of bounded variation abstract in the present paper we estimate the rate of convergence for szasz mirakyan durrmeyer operators with derivatives of bounded variation

the initial value problem for the compressible magnetohydrodynamic equations without heat conductivity abstract the cauchy problem for the three dimensional compressible magnetohydrodynamic equations without heat conductivity is considered in this paper the global existence and uniqueness of the strong solutions are established by the delicate energy method under the condition that the initial data is small perturbations of some given constant state in h 2 framework furthermore the optimal convergence rates of the solutions in l q norm with 2 q 6 and optimal convergence rates of their spatial derivatives in l 2 norm are obtained if additionally the initial data belongs to l p with 1 p 6 5

on spherically symmetric solutions of the compressible isentropic navier stokes equations we prove the global existence of weak solutions to the cauchy problem for the compressible isentropic navier stokes equations in n n 2 3 when the cauchy data are spherically symmetric the proof is based on the exploitation of the one dimensional feature of symmetric solutions and use of a new multidimensional property induced by the viscous flux the present paper extends lions existence theorem 15 to the case 1 n for spherically symmetric initial data where is the specific heat ratio in the pressure n 3 2 for n 2 and n 9 5 for n 3 dedicated to professor rolf leis on the occasion of his 70th birthday

the asymptotic estimates of solutions of difference equations abstract the asymptotic estimate of the solution y t of linear difference equations with almost constant coefficients and the condition of asymptotic equivalence between y t and y 0 t are given where y 0 t is the corresponding solution of linear equations with constant coefficients a brief and elementary proof of the results is presented the method of proof carries over to differential equations in which some similar results are stated

approximation properties of bernstein durrmeyer type operators abstract this paper deals with the approximation of continuous functions by sequences of some modified bernstein durrmeyer type operators that reproduce certain test functions the orders of approximation of the new versions turn to be at least as good as the one of the genuine bernstein durrmeyer operators moreover by extrapolating techniques recently applied to the classical bernstein operators we present a one parameter family of modified sequences of operators that reproduce certain polynomials and possess that popular genuine sequence as a limit case comparisons and some illustrative graphics are also presented

on blowup of classical solutions to the compressible navier stokes equations in this paper we study the finite time blow up of smooth solutions to the compressible navier stokes system when the initial data contain vacuums we prove that any classical solutions of viscous compressible fluids without heat conduction will blow up in finite time as long as the initial data has an isolated mass group see definition 2 2 the results hold regardless of either the size of the initial data or the far fields being vacuum or not this improves the blowup results of xin comm pure appl math 51 229 240 1998 by removing the crucial assumptions that the initial density has compact support and the smooth solution has finite total energy furthermore the analysis here also yields that any classical solutions of viscous compressible fluids without heat conduction in bounded domains or periodic domains will blow up in finite time if the initial data have an isolated mass group satisfying some suitable conditions

the pointwise estimates of diffusion wave for the navier stokes systems in odd multi dimensions we study the dissipation of solutions of the isentropic navier stokes equations in odd multi dimensions pointwise estimates of the time asymptotic shape of the solutions are obtained and shown to exhibit the generalized huygen s principle our approach is based on the detailed analysis of the green function of the linearized system this is used to study the coupling of nonlinear diffusion waves

on hyperbolic dissipative systems of composite type abstract the shizuta kawashima condition plays the fundamental role in guaranteeing global stability for systems of hyperbolic parabolic hyperbolic with relaxation however there are many important physical systems not satisfying this coupling condition which are of composite type with regard to dissipation the compressible navier stokes equations with zero heat conductivity and euler equations of adiabatic flow through porous media are two typical examples in this paper we construct the global unique solution near constant equilibria to these two systems in three dimensions for the small h l l 3 initial data our proof is based on a reformation of the systems in terms of the pressure velocity and entropy a scaled energy estimates with minimal fractional derivative counts in conjunction with the linear l 2 l 2 decay estimates to extract a fast enough decay of velocity gradient which is used to close the energy estimates for the non dissipative entropy we also include an application to certain two phase models

optimal lp lq convergence rates for the compressible navier stokes equations with potential force in this paper we are concerned with the optimal lp lq convergence rates for the compressible navier stokes equations with a potential external force in the whole space under the smallness assumption on both the initial perturbation and the external force in some sobolev spaces the optimal convergence rates of the solution in lq norm with 2 q 6 and its first order derivative in l2 norm are obtained when the initial perturbation is bounded in lp with 1 p 6 5 the proof is based on the energy estimates on the solution to the nonlinear problem and some lp lq estimates on the semigroup generated by the corresponding linearized operator

global well posedness of classical solutions with large oscillations and vacuum to the three dimensional isentropic compressible navier stokes equations we establish the global existence and uniqueness of classical solutions to the cauchy problem for the isentropic compressible navier stokes equations in three spatial dimensions with smooth initial data that are of small energy but possibly large oscillations with constant state as far field which could be either vacuum or nonvacuum the initial density is allowed to vanish and the spatial measure of the set of vacuum can be arbitrarily large in particular the initial density can even have compact support these results generalize previous results on classical solutions for initial densities being strictly away from vacuum and are the first for global classical solutions that may have large oscillations and can contain vacuum states 2012 wiley periodicals inc

rate of convergence of bernstein polynomials for functions with derivatives of bounded variation abstract for any x 0 1 we first prove that if x t t x on 0 1 then the bernstein polynomials of x satisfy the asymptotic relation k 0 n k n x k n x k 1 x n k 2x 1 x 1 2 1 n o 1 n this asymptotic relation is then used to study the rate of convergence of bernstein polynomials for functions with derivatives of bounded variation an estimate of the rate of convergence is given this estimate is asymptotically the best possible at points where is continuous

on asymptotic equivalence of perturbed linear systems of differential and difference equations abstract standard results on asymptotic integration of systems of linear differential equations give sufficient conditions which imply that a system is strongly asymptotically equivalent to its principal diagonal part these involve certain dichotomy conditions on the diagonal part as well as growth conditions on the off diagonal perturbation terms here we study perturbations with a triangularly induced structure and see that growth conditions can be substantially weakened in addition we give results for not necessarily triangular perturbations which in some sense interpolate between the classical theorems of levinson and hartman wintner some analogous results for systems of linear difference equations are also given

global existence in critical spaces for flows of compressible viscous and heat conductive gases we are concerned with global existence and uniqueness of strong solutions for a general model of viscous and heat conductive gases the initial data are supposed to be close to a stable equilibrium with constant density and temperature using uniform estimates for the linearized system with a convection term we get global well posedness in a functional setting invariant with respect to the scaling of the associated equations in space dimension n 3 we also show a smoothing effect on the velocity and the temperature and a decay on the difference between the density and the constant reference state these results extend a previous paper devoted to the barotropic case see 5

asymptotic behavior results for solutions to some nonlinear difference equations abstract we give asymptotic results for convergent solutions x n of real or complex difference equations x n 1 j x n f n x n where x n is an m vector j is a constant m m matrix and f n y is a vector valued function which is continuous in y for fixed n and where f n y is small in a sense in addition we obtain asymptotic results for solutions x n of the poincare difference equation x n 1 a b n x n where b n satisfies b n o n with 0 1 an application illustrates the results

convergence of the q analogue of szasz beta operators in the present paper we introduce the q analogue of the well known szasz beta operators 11 we also establish the approximation properties of these operators and estimate convergence results in the end we propose an open problem

global solutions of the navier stokes equations for multidimensional compressible flow with discontinuous initial data abstract we prove the global existence of weak solutions of the navier stokes equations for compressible isothermal flow in two and three space dimensions when the initial density is close to a constant in l 2 and l and the initial velocity is small in l 2 and bounded in l 2 n in two dimensions the l 2 norms must be weighted slightly a great deal of qualitative information about the solution is obtained for example we show that the velocity and vorticity are relatively smooth in positive time as is the effective viscous flux f which is the divergence of the velocity minus a certain multiple of the pressure we find that f plays a crucial role in the entire analysis particularly in closing the required energy estimates understanding rates of regularization near the initial layer and most important obtaining time independent pointwise bounds for the density

the initial value problem for the compressible navier stokes equations without heat conductivity abstract in this paper we are concerned with the global existence and convergence rates of strong solutions for the compressible navier stokes equations without heat conductivity in r 3 the global existence and uniqueness of strong solutions are established by the delicate energy method under the condition that the initial data are close to the constant equilibrium state in h 2 framework furthermore if additionally the initial data belong to l 1 the optimal convergence rates of the solutions in l 2 norm and convergence rates of their spatial derivatives in l 2 norm are obtained

on asymptotic summation of potentially oscillatory difference systems a new technique for the asymptotic summation of linear systems of difference equations y t 1 d t r t y t is derived a fundamental solution y t t i p t is constructed in terms of a product of two matrix functions the first function t is a product of the diagonal part d t the second matrix i p t is a perturbation of the identity matrix i conditions are given on the matrix d t r t that allow us to represent i p t as an absolutely convergent resolvent series without imposing stringent conditions on r t our method could be applied to discretized version of singularly perturbed differential equations y t a t y t that fit the setting of quantum mechanics

a global existence result for the compressible navier stokes equations in the critical l p framework the present paper is dedicated to the global well posedness issue for the barotropic compressible navier stokes system in the whole space mathbb r d with d 2 we aim at extending the work by danchin inventiones mathematicae 141 3 579 614 2000 to a critical framework which is not related to the energy space for small perturbations of a stable equilibrium state in the sense of suitable lp type besov norms we establish the global existence as a consequence like for incompressible flows one may exhibit a class of large highly oscillating initial velocity fields for which global existence and uniqueness holds true in passing we obtain new estimates for the linearized and the paralinearized systems which may be of interest for future works on compressible flows

global existence and asymptotic behavior for the 3d compressible navier stokes equations without heat conductivity in a bounded domain abstract in this paper we investigate the global existence and uniqueness of strong solutions to the initial boundary value problem for the 3d compressible navier stokes equations without heat conductivity in a bounded domain with slip boundary the global existence and uniqueness of strong solutions are obtained when the initial data is near its equilibrium in h 2 furthermore the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods

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

asymptotic properties of solutions to nonautonomous volterra difference systems with infinite delay abstract we show that under certain conditions there is a one to one correspondence between weighted bounded solutions of a linear nonautonomous volterra difference system with infinite delay and its perturbed system under additional assumptions this correspondence is elevated to asymptotic equivalence on the other side we prove the existence of weighted convergent solutions

on asymptotic summations of almost diagonal difference systems and their applications in this paper we derive some new results of asymptotic summations of left and right almost diagonal difference systems in particular we study a special case where the diagonal elements are constants with magnitude 1 conditions on the perturbation matrices are given under which the system is left and right almost diagonal an application of this result in a discrete adiabatic oscillator equation is discussed we also introduce a result for the case when the diagonal matrix is approaching the identity matrix i an example is given to illustrate that the new result can be applied to some cases that do not satisfy the growth condition

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

asymptotic solutions and error estimates for linear systems of difference and differential equations abstract classical results concerning the asymptotic behavior solutions of systems of linear differential or difference equations lead to formulas containing factors that are asymptotically constant i e k o 1 as t tends to infinity here we are interested in more precise information about the o 1 terms specifically how they depend precisely on certain perturbation terms in the equation results along these lines were given by gel fond and kubenskaya for scalar difference equations and we will both extend and generalize one of them as well as provide some corresponding results for differential equations

direct results for a certain family of summation integral type operators abstract in the present paper we establish some local and global direct results for a certain family of summation integral operators b n we also estimate the rate of convergence of b n f for a function f having derivatives of bounded variation

global existence and optimal decay rate for the strong solutions in h2 to the 3 d compressible navier stokes equations without heat conductivity abstract in this paper we consider the 3 d compressible navier stokes equations without heat conductivity which form a hyperbolic parabolic system we prove the global existence of a strong solution when the initial perturbation is small in h 2 and its l 1 norm is bounded moreover we can obtain the optimal decay rates for such a solution

global existence in critical spaces for compressible navier stokes equations we investigate global strong solutions for isentropic compressible fluids with initial data close to a stable equilibrium we obtain the existence and uniqueness of a solution in a functional setting invariant by the scaling of the associated equations more precisely the initial velocity has the same critical regularity index as for the incompressible homogeneous navier stokes equations and one more derivative is needed for the density we point out a smoothing effect on the velocity and a l 1 decay on the difference between the density and the constant reference state the proof lies on uniform estimates for a mixed hyperbolic parabolic linear system with a convection term

boundedness of solutions for singular hamiltonian differential systems with applications to deficiency indices abstract the paper gives boundedness estimation of solutions for singular hamiltonian differential systems as corollaries limit circle criteria are given and improve some previous results

compressible navier stokes equations with zero heat conductivity abstract we study the large time behavior of solutions of the compressible navier stokes equations without heat conductivity acoustic waves are dissipative due to the viscosity entropy waves are nondissipative due to lack of heat conductivity thus the system is of composite type the simplest prototype of this class of nonlinear partial differential equations we study the nonlinear interactions of waves using a combination of energy method weighted energy method and pointwise estimates