hilbert s double series theorem extended to the case of non homogeneous kernels abstract sharp version of celebrated hilbert s double series theorem is given in the case of non homogeneous kernel the main mathematical tools are the integral representation of mathieu s a series the holder inequality and an extension of the double series theorem by yang

global strong solutions to 1 d vacuum free boundary problem for compressible navier stokes equations with variable viscosity and thermal conductivity abstract in this paper we investigate the vacuum free boundary problem of one dimensional heat conducting compressible navier stokes equations where the viscosity coefficient depends on the density and the heat conductivity coefficient depends on the temperature satisfying a physical assumption from the chapman enskog expansion of the boltzmann equation the fluid connects to the vacuum continuously thus the system is degenerate near the free boundary the global existence and uniqueness of strong solutions for the free boundary problem are established when the initial data are large the result is proved by using both the lagrangian mass coordinate and the lagrangian trajectory coordinate an key observation is that the jacobian between these coordinates are bounded from above and below by positive constants

asymptotic behavior of the solution to the system for a viscous reactive gas abstract in this paper we investigate the asymptotic behavior of the generalized solution to the compressible navier stokes equations which describes a thermal explosion in a viscous reactive perfect gas confined between two infinite parallel plates in combustion theory

some uniform estimates and large time behavior of solutions to one dimensional compressible navier stokes system in unbounded domains with large data we study the large time behavior of solutions to the initial and initial boundary value problems with large initial data for the compressible navier stokes system describing the one dimensional motion of a viscous heat conducting perfect polytropic gas in unbounded domains the temperature is proved to be bounded from below and above independent of both time and space moreover it is shown that the global solution is asymptotically stable as time tends to infinity note that the initial data can be arbitrarily large this result is proved by using elementary energy methods

the duffin schaeffer conjecture with extra divergence given a nonnegative function psi mathbb n to mathbb r let w denote the set of real numbers x such that nx a 0 a consequence of our main result is that w is of full lebesgue measure if there exists an epsilon 0 such that textstyle sum n in mathbb n left frac psi n n right 1 epsilon varphi n infty the duffin schaeffer conjecture is the corresponding statement with epsilon 0 and represents a fundamental unsolved problem in metric number theory another consequence is that w is of full hausdorff dimension if the above sum with epsilon 0 diverges i e the dimension analogue of the duffin schaeffer conjecture is true

global solutions to one dimensional compressible navier stokes poisson equations with density dependent viscosity in this paper we prove the global existence of weak solutions to one dimensional compressible isentropic navier stokes poisson equations with density dependent viscosity and free boundaries the initial density 0 w1 2n is bounded below by a positive constant and the initial velocity u0 l2n in contrast to jiang et al global weak solutions to 1d compressible isentropic navier stokes equations with density dependent viscosity methods appl anal 12 239 2005 the sobolev exponent n is less in this paper and the viscosity coefficient is a general function of including the cases c0 0 1 and c0 where c0 is a positive constant

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

extension of hilbert s inequality in this paper we make some further extensions of discrete hilbert s inequality by using euler maclaurin summation formula we give the improvements of some previously obtained results and also compare our results with some previously known from the literature

global existence of large solutions to initial boundary value problems for a viscous heat conducting one dimensional real gas abstract the existence of global classical solutions to initial boundary value problems in the dynamics of a one dimensional viscous heat conducting gas is established the nonlinear dissipative effects turn out to be sufficiently strong to prevent the development of singularities

on a more accurate class of discrete hilbert type inequalities motivated by some known results regarding a particular set of non homogeneous kernels in this article we study a more general class of discrete hilbert type inequalities we derive a more accurate form for this class of inequalities based on the application of the hermite hadamard inequality

global existence and large time behavior of solutions to the cauchy problem of one dimensional viscous radiative and reactive gas abstract in this paper we consider the global existence and large time behavior of solutions of one dimensional viscous radiative and reactive gas with large initial data in precedent studies the case for the initial boundary value problem in bounded domain has been investigated and our main purpose focuses on the corresponding problem in unbounded domain if the initial data is a large perturbation of a non vacuum constant equilibrium state and is assumed to be without vacuum mass concentrations or vanishing temperatures then we can show that its cauchy problem admits a unique global smooth non vacuum solution which tends time asymptotically to such an equilibrium state the key point in the analysis is to deduce the uniform positive lower and upper bounds on the specific volume and the absolute temperature

mathieu series and associated sums involving the zeta functions almost twelve decades ago mathieu investigated an interesting series s r in the study of elasticity of solid bodies since then many authors have studied various problems arising from the mathieu series s r in various diverse ways in this paper we present a relationship between the mathieu series s r and certain series involving the zeta functions by means of this relationship we then express the mathieu series s r in terms of the trigamma function j z or equivalently the hurwitz or generalized zeta function z s a accordingly various interesting properties of s r can be obtained from those of j z and z s a among other results certain integral representations of s r are deduced here by using the aforementioned relationships among s r j z and z s a

global strong solution to compressible navier stokes equations with density dependent viscosity and temperature dependent heat conductivity abstract we obtain existence and uniqueness of global strong solution to one dimensional compressible navier stokes equations for ideal polytropic gas flow with density dependent viscosity and temperature dependent heat conductivity under stress free and thermally insulated boundary conditions here we assume viscosity coefficient 1 and heat conductivity coefficient for all 0 and 0

global solutions of nonlinear magnetohydrodynamics with large initial data a free boundary problem for nonlinear magnetohydrodynamics with general large initial data is investigated the existence uniqueness and regularity of global solutions are established with large initial data in h1 it is shown that neither shock waves nor vacuum and concentration in the solutions are developed in a finite time although there is a complex interaction between the hydrodynamic and magnetodynamic effects an existence theorem of global solutions with large discontinuous initial data is also established

numerical investigation of cavitation in multidimensional compressible flows the compressible navier stokes equations for an ideal polytropic gas are considered in r n n 2 3 the question of possible vacuum formation an open theoretical problem is investigated numerically using highly accurate computational methods the flow is assumed to be symmetric about the origin with a purely radial velocity field the numerical results indicate that there are weak solutions to the navier stokes system in two and three space dimensions which display formation of vacuum when the initial data are discontinuous and sufficiently large the initial density is constant while the initial velocity field is symmetric points radially away from the origin and belongs to h s loc for all s n 2 in addition in the one dimensional case the numerical solutions are in agreement with known theoretical results

supplements to a class of logarithmically completely monotonic functions associated with the gamma function abstract in this article the authors present a necessary condition a sufficient condition and a necessary and sufficient condition for a class of functions associated with the gamma function to be logarithmically completely monotonic as a consequence of these results supplements to the recent investigation by chen and qi c p chen f qi logarithmically completely monotonic functions relating to the gamma function j math anal appl 321 2006 405 411 are provided and a new keckic vasic type inequality is concluded

global existence for thermomechanical processes with nonconvex free energies of ginzburg landau form abstract global existence of solutions is proved for the system of partial differential equations which governs the nonlinear thermomechanical processes in a nonviscous solid under the assumption that the corresponding free energy density is of ginzburg landau type to allow for possible solid solid phase transitions in the material the stress strain interrelation is not assumed to be monotone

regularity of 1d compressible isentropic navier stokes equations with density dependent viscosity abstract in this paper we consider one dimensional compressible isentropic navier stokes equations with the viscosity depending on density and with free boundary the viscosity coefficient is proportional to with 0 1 where is the density the existence and uniqueness of global weak solutions in h 1 0 1 have been established in s jiang z xin p zhang global weak solutions to 1d compressible isentropic navier stokes equations with density dependent viscosity methods appl anal 12 2005 239 252 we will establish the regularity of global solution under certain assumptions imposed on the initial data by deriving some new a priori estimates

global classical solutions to the one dimensional compressible fluid models of korteweg type with large initial data abstract this paper is concerned with the global existence of classical solutions with large initial data away from vacuum to the cauchy problem of the one dimensional isothermal compressible fluid models of korteweg type with density dependent viscosity coefficient and capillarity coefficient the case when the viscosity coefficient and the capillarity coefficient for some parameters r is considered under some conditions on we first show the global existence of large solutions around constant states if the far fields of the initial data are the same while if the far fields of the initial data are different we prove the global stability of rarefaction waves with large strength here global stability means the initial perturbation can be arbitrarily large our analysis is based on the elementary energy method and the technique developed by y kanel 29

global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in an exterior domain we consider the equations of a viscous polytropic ideal gas in the domain exterior to a ball in n n 2 or 3 and prove the global existence of spherically symmetric smooth solutions for large initial data with spherical symmetry the large time behavior of the solutions is also discussed to prove the existence we first study an approximate problem in a bounded annular domain and then obtain a priori estimates independent of the boundedness of the annular domain letting the diameter of the annular domain tend to infinity we get a global spherically symmetric solution as the limit

some families of mathieu a series and alternating mathieu a series the main purpose of this paper is to present a number of potentially useful integral representations for the familiar mathieu a series as well as for its alternating version these results are derived here from many different considerations and are shown to yield sharp bounding inequalities involving the mathieu and alternating mathieu a series relationships of the mathieu a series with the riemann zeta function and the dirichlet eta function are also considered such special functions as the classical bessel function j n z and the confluent hypergeometric functions 0f 1 and 1f 2 are characterized by means of certain fredholm type integral equations of the first kind which are associated with some of these mathieu type series several integrals containing mathieu type series are also evaluated finally some closely related new questions and open problems are indicated with a view to motivating further investigations on the subject of this paper

integral representation of a series which includes the mathieu a series intgeral expression is deduced for the series mathfrak s r mu nu mathbf a sum n 1 infty frac 2f 1 frac nu mu 1 2 frac nu mu 2 1 nu 1 frac r 2 a n 2 a n nu mu 1 a n 2 r 2 mu 1 2 where r 0 mu 1 2 nu 1 mu and mathbf a 0 a 1 a 2 cdots a n uparrow infty and 2f 1 is the gauss hypergemetric function the result precizes the intergal expression for the generalized qi type mathieu mathbf a series s r p mathbf a sum n 1 infty a n a n 2 r 2 p 1 given in j inequal pure appl math 4 2003 4 5 generalizing some other results by cerone and lenard tomovski and qi as well bounding inequalities are given for mathfrak s r mu nu mathbf a using the derived intergal expression

global solutions of the navier stokes equations for viscous compressible flows the compressible navier stokes equations for viscous flows with general large continuous initial data as well as with large discontinuous initial data are studied both a homogeneous free boundary problem with zero outer pressure and a fixed boundary problem are considered for the large initial data in h1 the existence uniqueness and regularity of global solutions in h1 for real viscous flows are established and it is showed that neither shock waves nor vacuum and concentration in the solutions are developed in a finite time for the large discontinuous data the global existence of large weak solutions for the perfect gases is also established using a different argument and it is indicated that the solutions do not develop vacuum or concentration although the solutions have large discontinuity for the free boundary problem the interface separating the flows from the zero outer pressure expands at a finite speed

global smooth solutions and large time behavior of the one dimensional navier stokes equations the system of balance laws of mass momentum and energy for a viscous heat conductive one dimensional real gas navier stokes equations is considered the existence of globally defined smooth solution to an initial boundary value problem is established because of the boundary conditions effect vacuum will be developed as time tends to infinity

global existence of strong solutions of the navier stokes equations for isentropic compressible fluids with density dependent viscosity abstract this paper is concerned with global strong solutions of the isentropic compressible navier stokes equations with density dependent viscosity coefficient in one dimensional bounded intervals precisely the viscosity coefficient and the pressure p is proportional to with 1 the important point in this paper is that the initial density may vanish in an open subset we also show that the strong solution obtained above is unique provided that the initial data satisfies additional regularity and a compatible condition compared with former results obtained by hyunseok kim in h kim global existence of strong solutions of the navier stokes equations for one dimensional isentropic compressible fluids available at http com2mac postech ac kr papers 2001 01 38 pdf we deal with density dependent viscosity coefficient

a refined discrete hilbert inequality obtained via the hermite hadamard inequality in this paper we establish a general method for improving the discrete hilbert type inequalities via the hermite hadamard inequality the general result is then applied to homogeneous kernels moreover some particular homogeneous kernels are considered and a whole series of refinements of some recent results known from the literature are yielded in addition some particular non homogeneous cases are also considered

bohr sets and multiplicative diophantine approximation in two dimensions gallagher s theorem is a strengthening of the littlewood conjecture that holds for almost all pairs of real numbers we prove an inhomogeneous fiber version of gallagher s theorem sharpening and making unconditional a result recently obtained conditionally by beresnevich haynes and velani the idea is to find large generalized arithmetic progressions within inhomogeneous bohr sets extending a construction given by tao this precise structure enables us to verify the hypotheses of the duffin schaeffer theorem for the problem at hand via the geometry of numbers

global smooth solutions for 1d thermally radiative magnetohydrodynamics this paper is concerned with the global existence of smooth solutions to equations describing one dimensional motion of a self gravitating radiative of magnetohydrodynamics gas we have proved that for any arbitrary large smooth initial data the problem under consideration admits a unique globally smooth classical solution our results have improved the results given by zhang and xie j differ equations 52 1853 1882 2008

stabilization for equations of one dimensional viscous compressible heat conducting media with nonmonotone equation of state abstract we consider the navier stokes system describing motions of viscous compressible heat conducting and self gravitating media we use the state function of the form p u p 0 u p 1 u linear with respect to the temperature but we admit rather general nonmonotone functions p 0 and p 1 of u which allows us to treat various physical models of nuclear fluids for which p and u are the pressure and the specific volume or thermoviscoelastic solids for solutions to an associated initial boundary value problem with fixed free boundary conditions and arbitrarily large data we prove a collection of estimates independent of time interval including uniform two sided bounds for u and describe asymptotic behavior as t namely we establish the stabilization pointwisely and in l q for u in l 2 for and in l q for v the velocity for any q 2 for completeness the proof of the corresponding global existence theorem is also included

global solutions of real compressible reactive gas with density dependent viscosity and self gravitation for higher order kinetics abstract a mathematical model for viscous real compressible reactive fluid flows is considered the existence of global solutions for the free boundary problem with species diffusion in dynamic combustion is established when the viscosity depends on the density i e a 0 1 2 where a is a generic positive constant furthermore the equations of state depend nonlinearly on density and temperature unlike the case of perfect gases or radiative flows in addition the shock wave turbulence vacuum mass concentration or extremely hot spot will not be developed in any finite time if the initial data do not contain vacuum

global smooth solutions of the compressible navier stokes equations with density dependent viscosity abstract in this paper we study a free boundary problem for the viscous compressible heat conducting one dimensional real fluids more precisely the viscosity is assumed to be a power function of density i e where denotes the density of fluids and is a positive constant in addition the equations of state include and are more general than perfect flows which only depend linearly on temperature the global existence uniqueness of smooth solutions is established with 0 1 2 for general large initial data which improves the previous results moreover it is also shown that the solutions will not develop vacuum mass concentration or heat concentration in a finite time provided the initial data are bounded and smooth and do not contain vacuum

global classical large solutions to 1d compressible navier stokes equations with density dependent viscosity and vacuum abstract in this paper we investigate an initial boundary value problem for 1d compressible isentropic navier stokes equations with large initial data density dependent viscosity external force and vacuum making full use of the local estimates of the solutions in cho and kim 2006 3 and the one dimensional properties of the equations and the sobolev inequalities we get a unique global classical solution u where c 1 0 t h 1 0 1 and u h 1 0 t h 2 0 1 for any t 0 as it is pointed out in xin 1998 31 that the smooth solution u c 1 0 t h 3 r 1 t is large enough of the cauchy problem must blow up in finite time when the initial density is of nontrivial compact support it seems that the regularities of the solutions we obtained can be improved which motivates us to obtain some new estimates with the help of a new test function 2 u t t such as lemmas 3 2 3 6 this leads to further regularities of u where c 1 0 t h 3 0 1 u h 1 0 t h 3 0 1 it is still open whether the regularity of u could be improved to c 1 0 t h 3 0 1 with the appearance of vacuum since it is not obvious that the solutions in c 1 0 t h 3 0 1 to the initial boundary value problem must blow up in finite time

some mathieu type series associated with the fox wright function closed form integral expressions are derived here for a family of convergent mathieu type series and its alternating variant when their terms contain the fox wright p j q function some two sided exponential bounding inequalities are then obtained for a class of fox wright p j q functions thereby generalizing certain results of luke finally by means of the integral expressions obtained here a number of two sided exponential bounding inequalities are given for the aforementioned series

nonlinearly exponential stability of compressible navier stokes system with degenerate heat conductivity abstract we study the large time behavior of strong solutions to the one dimensional compressible navier stokes system for a viscous and heat conducting ideal polytropic gas when the viscosity is constant and the heat conductivity is proportional to a positive power of the temperature both the specific volume and the temperature are proved to be bounded from below and above independently of time moreover it is shown that the global solution is nonlinearly exponentially stable as time tends to infinity note that the conditions imposed on the initial data are the same as those of the constant heat conductivity case kazhikhov shelukhin 1977 18 kazhikhov 1981 17 and can be arbitrarily large therefore our result can be regarded as a natural generalization of the kazhikhov s ones for the constant heat conductivity case to the degenerate and nonlinear one

two sided inequalities for the extended hurwitz lerch zeta function abstract recently srivastava et al 2011 2 unified and extended several interesting generalizations of the familiar hurwitz lerch zeta function z s a by introducing a fox wright type generalized hypergeometric function in the kernel for this newly introduced special function two integral representations of different kinds are investigated here by means of a known result involving a fox wright generalized hypergeometric function kernel which was given earlier by srivastava et al 2011 2 and by applying some mathieu a series techniques finally by appealing to each of these two integral representations two sets of two sided bounding inequalities are proved thereby extending and generalizing the recent work by jankov et al 2011 15

symmetric flows for compressible heat conducting fluids with temperature dependent viscosity coefficients abstract we consider the navier stokes equations for compressible heat conducting ideal polytropic gases in a bounded annular domain when the viscosity and thermal conductivity coefficients are general smooth functions of temperature a global in time spherically or cylindrically symmetric classical solution to the initial boundary value problem is shown to exist uniquely and converge exponentially to the constant state as the time tends to infinity under certain assumptions on the initial data and the adiabatic exponent the initial data can be large if is sufficiently close to 1 these results are of nishida smoller type and extend the work liu et al 2014 16 restricted to the one dimensional flows

exponential stability for a nonlinear one dimensional heat conductive viscous real gas abstract in this paper we prove the existence and the exponential stability in h i i 1 2 an incomplete metric subspace of hi hi hi i 1 2 of a nonlinear c0 semigroup s t for a nonlinear one dimensional heat conductive viscous real gas in bounded domain 0 1 the results in this paper improve those previously related results

blow up criterions of strong solutions to 3d compressible navier stokes equations with vacuum abstract in the paper we establish a blow up criterion in terms of the integrability of the density for strong solutions to the cauchy problem of compressible isentropic navier stokes equations in r 3 with vacuum under the assumptions on the coefficients of viscosity 29 3 this extends the corresponding results in huang et al 2011 sun et al 2011 20 36 where a blow up criterion in terms of the upper bound of the density was obtained under the condition 7 as a byproduct the restriction 7 in fan et al 2010 sun et al 2011 12 37 is relaxed to 29 3 for the full compressible navier stokes equations by giving a new proof of lemma 3 1 besides we get a blow up criterion in terms of the upper bound of the density and the temperature for strong solutions to the cauchy problem of the full compressible navier stokes equations in r 3 the appearance of vacuum could be allowed this extends the corresponding results in sun et al 2011 37 where a blow up criterion in terms of the upper bound of 1 was obtained without vacuum the effective viscous flux plays a very important role in the proofs

on mathieu s inequality the inequalities formula hold for all real numbersx 0 the constants 1 2 3 and 1 6 are best possible

asymptotic expansion of mathieu power series and trigonometric mathieu series abstract we consider a generalized mathieu series where the summands of the classical mathieu series are multiplied by powers of a complex number the mellin transform of this series can be expressed by the polylogarithm or the hurwitz zeta function from this we derive a full asymptotic expansion generalizing known expansions for alternating mathieu series another asymptotic regime for trigonometric mathieu series is also considered to first order by applying known results on the asymptotic behavior of trigonometric series

certain classes of series associated with the zeta and related functions a fascinatingly large number of seemingly independent solutions of the so called basler problem of evaluating the riemann zeta function z s when s 2 which was of vital importance to leonhard euler 1707 1783 and the bernoulli brothers jakob bernoulli 1654 1705 and johann bernoulli 1667 1748 have appeared in the mathematical literature ever since euler first solved this problem in the year 1736 the main object of this two part series of lectures is to present some recent developments on the evaluations and representations of z s when email protected email protected 1 n being the set of natural numbers we emphasize upon several interesting classes of rapidly convergent series representations for z 2n 1 email protected n which have been developed in recent years in two of many computationally useful special cases considered here it is observed that z 3 can be represented by means of series which converge much more rapidly than that in euler s celebrated formula as well as the series used recently by roger apery 1916 1994 in his proof of the irrationality of z 3 symbolic and numerical computations using mathematica version 4 0 for linux show among other things that only 50 terms of one of these series are capable of producing an accuracy of seven decimal places

universal attractor in h4 for the nonlinear one dimensional compressible navier stokes equations abstract this paper is concerned with the regularity exponential stability of solutions and existence of universal attractor in h 4 for a nonlinear one dimensional compressible navier stokes equations describing a motion of heat conductive viscous real gas in a bounded domain 0 1 some new ideas and more delicated estimates are introduced to prove these results

lyapunov functional method for 1d radiative and reactive viscous gas dynamics we consider the compressible navier stokes system for 1d flows of a viscous heat conducting gas with the pressure law open image in new window and a one order kinetics to include radiative effects and reactive processes the mass force and the ignition phenomenon are also taken into account for large data and under general assumptions on the heat conductivity we establish global in time bounds and exponential stabilization for solutions in lq and h1 norms to this end we construct new global lyapunov functionals and show that they describe the dynamics of solutions for any t 0 a short proof of the corresponding global existence is also included for completeness

the duffin schaeffer conjecture with extra divergence ii in 2012 the authors set out a programme to prove the duffin schaeffer conjecture for measures arbitrarily close to lebesgue measure in this paper we take a new step in this direction given a non negative function psi mathbb n rightarrow mathbb r let w psi denote the set of real numbers x such that nx a 0 our main result is that w psi is of full lebesgue measure if there exists a c 0 such that begin aligned sum n ge 16 frac varphi n psi n n exp c log log n log log log n infty end aligned

global spherically symmetric flows for a viscous radiative and reactive gas in an exterior domain abstract we consider the equations for a viscous compressible radiative and reactive gas pressure p r a 4 3 internal energy e c v a 4 over an unbounded exterior domain in r n where n 2 is the space dimension the existence uniqueness and large time behavior of global spherically symmetric solutions are established for large initial data the key point in the analysis is to deduce certain uniform a priori estimates on the solutions especially on lower and upper bounds of the specific volume and temperature

on the solvability of a boundary value problem for model equations of linearly viscous materials this paper deals with model equations for linearly viscous materials viscosity is allowed to vary with temperature the global solvability of an initial boundary value problem for nonlinear equations is proved by continuation of a local solution with the help of a priori estimates the main attention is paid to the global estimates

on a simplified compressible navier stokes equations with temperature dependent viscosity abstract we consider a simplified compressible navier stokes equations with cylindrical symmetry when viscosity coefficient and heat conductivity coefficient depend on temperature we obtain global existence of strong solution and vanishing shear viscosity limit to the initial boundary value problem in eulerian coordinates the analysis for the global existence is based on the assumption that const 0 1 c m c 1 m q for m 0 1 q m for the part of vanishing shear viscosity limit we require in addition that 1 c 1 m c 1 m in the paper the acceleration effect in one direction is neglected however we do not need any smallness assumption for the initial data

global solutions to the one dimensional compressible navier stokes equation with radiation abstract we investigate the global existence of a smooth solution to the one dimensional compressible navier stokes equation with radiation the transport coefficients and are both degenerate the pressure p t x varies in different temperature regions our discussion considers the case when pressure p t x satisfies the benedict webb rubin equation at high temperature the result obtained for this problem is presented for the first time