asymptotic behavior for the navier stokes equations with nonzero external forces abstract we estimate the asymptotic behavior for the stokes solutions with external forces first we found that if there are external forces then the energy decays slowly even if the forces decay quickly then we also obtain the asymptotic behavior in the temporal spatial direction for weak solutions of the navier stokes equations we also provide a simple example of external forces which shows that the stokes solution does not decay quickly

a variational iterative technique applied to quantum mechanical problems a technique for solving nonlinear operator equations iteratively is applied to quantum mechanical calculations it is shown that alternative methods of applying the technique lead to another way of approaching variational calculations and generalisations of perturbation theory respectively

decay rate for the incompressible flows in half spaces we show that the time decay rate of l 2 norm of weak solution for the stokes equations and for the navier stokes equations on the half spaces are t frac n 2 frac 1 r frac 1 2 frac 1 2 if the initial data u 0 in l 2 cap l r and int mathbb r n y n u 0 y r dy infty for 1 r 2 we also show that the decay rate is determined by the linear part of the weak solution we use the heat kernel and ukai s solution formula for the stokes equations it has been known up to now that the decay rate on the half space was t frac n 2 frac 1 r frac 1 2 which was obtained by borchers and miyakawa 1 and ukai 9

numerical methods for the eigenvalue determination of second order ordinary differential equations an accurate method for the numerical solution of the eigenvalue problem of second order ordinary differential equation using the shooting method is presented the method has three steps firstly initial values for the eigenvalue and eigenfunction at both ends are obtained by using the discretized matrix eigenvalue method secondly the initial value problem is solved using new highly accurate formulas of the linear multistep method thirdly the eigenvalue is properly corrected at the matching point the efficiency of the proposed methods is demonstrated by their applications to bound states for the one dimensional harmonic oscillator anharmonic oscillators the morse potential and the modified poschl teller potential in quantum mechanics

regularity and large time behaviour of solutions of a conservation law without convexity a l aide de la methode des caracteristiques generalisees on etudie la regularite et le comportement a grand des solutions faibles admissibles d une seule loi de conservation a 1 variable d espace avec un point d inflexion

eventual c regularity and concavity for flows in one dimensional porous media we study the regularity and the asymptotic behavior of the solutions of the initial value problem for the porous medium equation u t eft u m ight xx inq r imes eft 0 nfty ight ueft x 0 ight u 0 eft x ight forx n r

quantum theory of anharmonic oscillators a variational and systematic general approximation method this is an investigation of the energy levels and wavefunctions of an anharmonic oscillator characterised by the potential 1 2 omega 2q2 lambda q4 as a lowest order approximation an extremely simple formula for energy levels ei 0 i 1 2 1 4 3 alphai alphai is derived i being the quantum number of the energy level which covers any lambda i alphai is the real positive root of a cubic equation yi alphai3 alphai2 1 0 with yi 6lambda 2i2 2i 1 2i 1 this formula reproduces the exact energy levels within an error of about 1 for any lambda i the worst case is 2 for i 0 lambda to infinity systematically higher orders of this perturbation theory are developed which contains the usual perturbation theory for the limiting case of small lambda but this perturbation theory is valid for any i lambda the second order perturbation theory reduces the errors of our lowest order results by a factor of about 1 5 in general various ranges large intermediate small of i lambda are investigated and compared with the exact values obtained by the montroll group 1975 1978 for i 0 1 even the fourth order perturbation calculation can be elaborated explicitly which reduces the error to about 0 01 for any lambda for small lambda it gives correct numerical coefficients up to lambda4 terms as it should

the summation of the ordinary and renormalized perturbation series for the ground state energy of the quartic sextic and octic anharmonic oscillators using nonlinear sequence transformations the strongly divergent rayleigh schrodinger perturbation expansion for the ground state energy of the quartic sextic and octic anharmonic oscillator and a corresponding renormalized perturbation expansion f vinette and j ci ek j math phys 32 3392 1991 are summed by pade approximants by levin s sequence transformation d levin int j comput math b 3 371 1973 and by a closely related sequence transformation which was suggested recently e j weniger comput phys rep 10 189 1989 it is shown that the renormalized perturbation expansion can be summed much more easily than the original perturbation expansion from which it was derived and that levin s sequence transformation diverges and is not able to sum the perturbation expansions the pade summation of the renormalized perturbation expansions gives relatively good results in the quartic and sextic case in the case of the octic anharmonic oscillator even the renormalized perturbation expansion is not pade summable the best results are clearly obtained by the new sequence transformation which for instance is able to sum the renormalized perturbation expansions for the infinite coupling limits of the quartic sextic and octic anharmonic oscillator and which produces at least in the quartic and sextic case extremely accurate results

a method for computing phase shifts for scattering abstract the method developed by papageorgiou and raptis for computing the eigenvalues is now applied to scattering problems in particular a new approach is developed for the computation of the phase shifts of the one dimensional schrodinger equation the algorithm is very simple to code and very efficient for computational purposes the method is also general and can be applied to any kind of potential functions coulomb type etc

lower bound of l2 decay of the navier stokes flow in the half space r n and its asymptotic behavior in the frequency space abstract we consider the asymptotic behavior of weak solutions of the navier stokes equations in the half space r n we obtain the lower bound of the energy decay of the navier stokes flow by means of the profile of the initial data indeed we construct a class of the initial data which causes the slow decay of the navier stokes flow with an explicit rate furthermore we investigate the asymptotic behavior of concentration in the frequency space

how parabolic free boundaries approximate hyperbolic fronts a rather complete study of the existence and qualitative behaviour of the boundaries of the support of solutions of the cauchy problem for nonlinear first order and second order scalar conservation laws is presented among other properties it is shown that under appropriate assumptions parabolic interfaces converge to hyperbolic ones in the vanishing viscosity limit

on a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium the main result of the paper is the uniqueness of nonnegative solutions of the cauchy problem and of the first and mixed boundary value problems for a class of degenerate parabolic equations which includes the model equation ut um xx un x where m 1 and n 0 in particular n is allowed to be smaller than one the proof is based on a refined test function argument the condition that u be nonnegative is crucial but the restriction to one space variable is not

the perturbed three dimensional oscillator a renormalized version of inner product and hypervirial theories is used to calculate the energy eigenvalues for four states of a perturbed three dimensional oscillator and to obtain the rayleigh schrodinger energy perturbation series

algebraic l 2 decay for weak solutions of a viscous boussinesq system in exterior domains abstract the algebraic decay rates for the total kinetic energy of weak solutions of the n dimensional viscous boussinesq system in exterior domains are established by means of the spectral decomposition method of fractional powers of the laplacian and stokes operators the results reveal that l 2 decay of the velocity field with zero divergence becomes slower than that of the temperature

decay results of solutions to the incompressible navier stokes flows in a half space abstract in a half space we consider the asymptotic behavior of the strong solution for the non stationary navier stokes equations in particular the decay rates of the second order derivatives of the navier stokes flows in l r r n n 2 with 1 r are derived by using l q l r estimates and a clever analysis on the fractional powers of the stokes operator in addition we prove that the strong solution and its first and second derivatives decay in time more rapidly than observed in general if the initial datum lies in a suitable weighted space

notes on the space time decay rate of the stokes flows in the half space in this paper a stokes equations in the half space r n n 2 has been considered we derive a rapid decay rate of the stokes flow in space and time when the initial data decreases fast enough and satisfies some additional condition initial data decreasing too slowly to be x h l1 r n are also considered

asymptotic behavior for the navier stokes equations in 2d exterior domains we show that the lp spatial temporal decay rates of solutions of incompressible flow in an 2d exterior domain when a domain has a boundary pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary to overcome the difficulty we adopt the ideas in he xin c he z xin weighted estimates for nonstationary navier stokes equations in exterior domain methods appl anal 7 3 2000 443 458 and our previous results h o bae b j jin asymptotic behavior of stokes solutions in 2d exterior domains j math fluid mech in press h o bae b j jin temporal and spatial decay rates of navier stokes solutions in exterior domains submitted for publication for the spatial decay rate estimate we first extend temporal decay rate result of the navier stokes solutions for general lp space when the initial velocity is in l 2 lr 1 r q 1 r q

regularity of solutions and interfaces of the porous medium equation via local estimates on demontre la regularite c pour des interfaces mobiles de solutions locales de l equation de milieu poreux

asymptotic behavior for the stokes flow and navier stokes equations in half spaces abstract using the solution formula in ukai 1987 27 for the stokes equations we find asymptotic profiles of solutions in l 1 r n n 2 for the stokes flow and non stationary navier stokes equations since the projection operator p l 1 r n l 1 r n is unbounded we use a decomposition for p u u to overcome the difficulty and prove that the decay rate for the first derivatives of the strong solution u of the navier stokes system in l 1 r n is controlled by t 1 2 1 t n 2 2 for any t 0

decay properties of solutions to the non stationary magneto hydrodynamic equations in half spaces abstract l 1 decay properties of the strong solution including the first and second order spacial derivatives to the non stationary magneto hydrodynamic mhd equations are established in a half space and the weighted cases are also considered additionally

a simple iterative solution of the schrodinger equation in matrix representation form a simple procedure for solving the schrodinger equation is presented it is based upon an iterative solution of the secular equation a large enough convergence rate is obtained by using a basis set of properly scaled functions the effect of the scaling parameter on the convergence rate is studied in order to improve the calculation the method is applied to simple though non trivial quantum mechanical models such as the quartic sextic and octic anharmonic oscillators a double well potential and the linear confining potential highly accurate eigenvalues for all values of the coupling parameter are obtained

time decay rates of non newtonian flows in rn abstract this paper is concerned with time decay rates of the weak solutions of an incompressible non newtonian fluid motion model in half spaces r n for n 3 with the use of the spectral decomposition of the stokes operator and l p l q estimates it is shown that the weak solutions decay in l 2 norm like t n 2 1 r 1 2 when the initial velocity u 0 l 2 l r for 1 r 2 the higher decay rates t n 2 1 r 1 2 1 2 are obtained if u 0 satisfies the additional moment condition r n x n u 0 x r d x 1 r 2 moreover the error estimates between the non newtonian flow and the navier stokes flow are discussed

localization and boundedness of the solutions of the neumann problem for a filtration equation on etudie l existence et l unicite des solutions faibles du probleme de neumann u t u xx dans d t u x 0 u 0 x sur 0 u x 0 t h t sur 0 t ici c 1 r c 2 r 0 avec s 0 si s 0 pour t 0 d t 0 0 t on suppose que h 0 t r est bornee sur des sous ensembles compacts de 0 t et u 0 l 1 0 est telle que u 0 x est bornee sur 0

decay results of higher order norms for the navier stokes flows in 3d exterior domains using the nonstationary stokes solution formula and the regularity theory of the steady stokes equations we establish the decay results including the weighted cases of higher order spatial derivatives for the navier stokes flows in 3d exterior domains which is a long standing problem the higher order norm weighted decay results in this article are inspired by the work in bae and jin proc r soc edinburgh sect a 135 461 477 2005 schonbek and wiegner proc r soc edinburgh sect a 126 677 685 1996 respectively which are considered in the whole space

temporal decays in l 1 and l for the stokes flow we estimate the time decay rates in l1 in the hardy space and in l of solutions for the stokes equations on the half spaces for the estimates in the hardy space we adopt the ideas in math z 231 1999 383 temporal and spatial decays for the stokes flows submitted for publication and also use the heat kernel and the solution formula for the stokes equations

finite difference calculations of eigenvalues for various potentials the energy levels of the schrodinger equation for the potentials v x sigma 2n 2 lambda 2nx2n v x z2x2 x2n and v x x2 lambda x2n 1 gx2 with 2n 4 6 8 18 20 have been calculated by using a finite difference method for various values of lambda and the quantum number n the obtained results are compared with previously available results

quantum theory of anharmonic oscillator energy levels of a three dimensional oscillator with quartic anisotropic perturbation the energy levels of the schrodinger equation for a three dimensional oscillator with quartic perturbation v x y z 1 2 x2 y2 z2 axxx4 ayyy4 azz z4 2axyx2y2 2axzx2z2 2ayz y2z2 are calculated using matrix diagonalization for various values of the perturbation parameters and state numbers nx ny nz

pointwise decay estimate of navier stokes flows in the half space with slowly decreasing initial value abstract in this paper we study the spatial and temporal decay estimate of the navier stokes flow in the half space corresponding to auniformly but slowly decreasing initial velocity we show the local in time solvability of the navier stokes equations with u x t c 0 1 x t and u x t c 0 t 1 2 1 x t when 1 x t e t a h l r n 0 and t 1 2 1 x t e t a h l r n 0 0 n asymptotically it holds that u x t e t a h x o t x 0 n as x the solution exists globally in time for 1 when the data is small enough and it holds that u x t e t a h x o x t 1 n as x t

decay rate of solutions of navier stokes equations in 3 dimensional half space abstract in this article we derive the l 2 spatial decay rate of a weak solution of the incompressible flow in the half space when the half space or some other fluid region with boundary is concerned pressure estimate is main obstacle since we do not have enough information of the pressure on the boundary in this paper we give an idea which does not require any pressure information on the boundary and we hope that our idea could be applied to other problems such as exterior domain problem

some finite difference methods for computing eigenvalues and eigenvectors of special two point boundary value problems abstract direct numerical integration methods will be discussed for calculating eigenvalues and eigenvectors of two point boundary value problems involving the differential equation y a p x y 0 with p x p x the derived results are compared with previously derived numerical data and with available exact values

the characterization of reaction convection diffusion processes by travelling waves it has long been known that the heat equation displays infinite speed of propagation this is to say that if the initial data are nonnegative and have nonempty compact support the solution of an initial value problem is positive everywhere after any infinitesimal time however since the nineteen fifties it has also been known that certain nonlinear diffusion equations of degenerate parabolic type do not display this phenomenon for these equations the generalized solution of an initial value problem with compactly supported initial data will have bounded support with respect to the spatial variable at all times in this paper the necessary and sufficient criterion for finite speed of propagation for the general nonlinear reaction convection diffusion equation formula is determined the assumptions on the coefficientsa bandcare such that the classification unifies and generalizes previously known results the technique employed is comparision of an arbitrary solution of the equation with suitably constructed travelling wave solutions and subsolutions basically the central conclusion is that the equation exhibits finite speed of propagation if and only if it admits a travelling wave solution with bounded support concurrently the search for a travelling wave solution with bounded support can be reduced to the study of a singular nonlinear integral equation whose solution must satisfy a certain constraint

inner product perturbation theory for a perturbed two dimensional oscillator with mixed parity potential ground and excited state energy levels of the schrodinger equation for various model potentials in two dimensional space are calculated using inner product and renormalized series techniques over wide ranges of relevant perturbation parameters mixed parity potentials are treated whereas earlier works have treated only even parity perturbations

the eigenvalues of the schrodinger equation for spherically symmetric states for various types of potentials in two three and n dimensions by using perturbative and non perturbative methods the energy levels of the schrodinger equation involving various model potentials in two three and n dimensional space are calculated by using perturbative and non perturbative methods the methods have been used effectively to calculate the energy eigenvalues even for high state numbers and large values of the perturbation parameters

long time behavior for the nonstationary navier stokes flows in l1 r n the large time decay for the navier stokes flows in l1 r n is a long standing unsolved question the main difficulties are that usual lq lr estimates for the stokes flow fail in this case and the projection operator p l1 r n l 1 r n becomes unbounded using the stokes solution formula we find a crucial and new estimate for the stokes flow in l1 r n which plays a fundamental role in studying the time l1 behavior for the navier stokes equations in addition we decompose the operator p into two parts and reduce its unboundedness to establish an l1 estimate for an elliptic problem with neumann boundary condition which is overcome by using the weighted estimates of the gaussian kernel s convolution the main results in this article are motivated by bae s works 2006 2008 1 2 and brandolese s work 2001 11

sufficient and necessary initial mass conditions for the existence of a waiting time in nonlinear convection processes diaz departamento de matematica aplicada universidad complutense de madrid 28040 spain submitted by avner friedman received april 17 1989 we study the initial behavior of the fronts for interfaces generated by the solu tions of the equation u u b u where m b i 0 are real numbers we prove a mass comparison principle that allows us to give necessary and sufficient conditions in order to have waiting time at the fronts different regions in the i m parameter space must be introduced leading to answer of very different nature

upper and lower bounds of temporal and spatial decays for the navier stokes equations we obtain the lower bounds of the temporal spatial decays for weak solutions of the navier stokes equations c0 1 t 5 2i4 1 2 i 2u t l2 c1 1 t 5 2i4 for i 1 2 the upper bound parts are estimated in several papers for example bae and jin submitted and he and xin proc roy soc edinburg section a 131 3 2001 597 by the interpolation arguments we also have c0 1 t 54 2 1 x 2 2u t l2 c1 1 t 54 2 for 0 2

c regularity for the porous medium equation on etablit un resultat de regularite optimale pour l equation d un milieu poreux u t u m xx 0 x r t 0 u 0 u 0 pour m 1 et la donnee initiale u 0 positive continue a support compact connexe

long time behavior for navier stokes flows in a two dimensional exterior domain abstract there are several long standing problems on the incompressible navier stokes flows in 2d exterior domains which claim how to characterize l 1 summability of the 2d n s flows whether the total net force exerted on the boundary is finite and how to establish decay results of higher order spatial derivatives including the weighted cases in order to solve these questions we firstly find some types of new technical inequalities which are used to overcome the difficulties caused by the domain boundary using l q l r properties for nonstationary stokes flows together with elliptic estimates for the steady stokes system we can avoid the strong singularity and answer these mentioned problems completely it should be pointed out that main results in this article are motivated by the works in 5 37 respectively

some inner product techniques for computing eigenvalues for three dimensional anharmonic oscillators with quartic and sextic perturbations abstract an algorithm previously used by the author for calculating the energy eigenvalues for a three dimensional oscillator with limited range of perturbation parameters eigenstates and accuracy is revisited and implemented in a more effective way to give excellent accuracy even for higher values of and nx ny nz x y z fifty coefficients of renormalized perturbation series are calculated in contrast to the previous works which were limited to 20 coefficients of the perturbation series due to computational facilities some of the results calculated by the inner product technique are compared with results produced by other techniques

decay rates for the incompressible navier stokes flows in 3d exterior domains abstract the exterior nonstationary problem is studied for the 3d navier stokes equations for which the associated total net force to the boundary may not vanish the time decay properties of the strong solution including the first and second derivatives are shown in l q and weighted spaces in particular the relation of weighted l 1 summability for smooth solutions is discussed in details between the time decay and the total net force exerted by the fluid to the body the conclusions in this article improve and extend results in h bae b jin asymptotic behavior for the navier stokes equations in 2d exterior domains j funct anal 240 2006 508 529 h bae b jin temporal and spatial decay rates of navier stokes solutions in exterior domains bull korean math soc 44 2007 547 567 and in c he t miyakawa on l 1 summability and asymptotic profiles for smooth solutions to navier stokes equations in a 3d exterior domain math z 245 2003 387 417 c he t miyakawa on weighted norm estimates for nonstationary incompressible navier stokes flows in a 3d exterior domain j differential equations 246 2009 2355 2386 respectively

energy levels of double well potentials in a three dimensional system the renormalized inner product perturbation technique is used to compute the energy levels for a double well potential v x y z z2 z2 x2 y2 z2 axxx4 ayyy4 azzz4 2axyx2y2 2axzx2z2 2ayzy2z2 in the three dimensional system results are produced for a wide range of parameters z2 and of state numbers nx ny nz comparison is made with results obtained by other means

large time behavior for the nonstationary navier stokes flows in the half space abstract asymptotic behavior of higher order spatial derivatives in l r is established for incompressible navier stokes flows in the half space which is a long time unsolved problem the main tools employed for solving this problem are the nonstationary stokes solution formula and the a priori estimates of the steady stokes system in the half space another main result is devoted to studying time l 1 decay and a partial answer to this open problem is given by means of a crucial and new estimate for the stokes flow two theorem 1 1 theorem 1 2 in this article can be regarded as a great improvement and powerful extension of the work in 6 38

lower bounds of rates of decay for solutions to the navier stokes equations where a p 3 2 p 1 and the constant c depends on the l2 and lnorms of the initial data u0 this paper deals with the more subtle problem of deriving lower bounds on the energy decay rates we show that for a certain class of initial data the solutions u x t to the 2d and 3d navier stokes equations admit an algebraic lower bound on the energy decay specifically there are two cases to consider in the first case the average of the initial data f u0 dx is nonzero this case was treated in the earlier paper 5 where it was established that

weighted decay properties for the incompressible stokes flow and navier stokes equations in a half space abstract using the stokes solution formula and l q l r estimates of the stokes operator semigroup we establish the weighted decay properties for the stokes flow and navier stokes equations including their spatial derivatives in half spaces in addition the unboundedness of the projection operator p l r n l r n is overcome by employing a decomposition for the nonlinear term and l asymptotic behavior for the second derivatives of navier stokes flows in half spaces is given

asymptotic behavior of solutions to the nonstationary magneto hydrodynamic equations abstract asymptotic behavior of the higher order spacial derivatives of the strong solution to the incompressible non stationary magneto hydrodynamic mhd equations is given in a half space which is a long time difficult question the main tools employed in this article are the non stationary stokes solution formula and some a priori estimates of the steady stokes system in the half space

numerical methods for solving radial schro dinger equations abstract an algorithm previously introduced by brown et al 1963 for solving radial schrodinger equations is revisited and implemented in a more accurate way the method is firstly applied to equations where potentials are present which are finite at the origin and which have an asymptotic behaviour v r 0 as r typical examples of potentials belonging to that class are the woods saxon and the morse potential furthermore the method is also used for coulomb like potentials such as the hulthen and the hellmann potential a comparison between the approximated numerical values and other available numerical and exact bound state energies is made