large time behavior of energy in exponentially decreasing solutions of the navier stokes equations abstract we study large time behavior of exponentially decreasing global strong solutions of the navier stokes equations we show as the main result that these solutions are characterized by the asymptotic disappearance of low and high frequencies and concentration of energy in middle frequencies we also describe a relationship between the spectrum of the stokes operator and the class of exponentially decreasing global strong solutions

asymptotic stability of landau solutions to navier stokes system it is known that the three dimensional navier stokes system for an incompressible fluid in the whole space has a one parameter family of explicit stationary solutions that are axisymmetric and homogeneous of degree 1 we show that these solutions are asymptotically stable under any l2 perturbation

on energy inequality smoothness and large time behavior in l 2 for weak solutions of the navier stokes equations in exterior domains soit un domaine exterieur de r n n 3 4 a frontiere lisse sur le cylindre d espace temps 0 on considere le probleme de navier stokes non stationnaire u t u u u p f dans 0 u 0 dans 0 u 0 sur 0 u x 0 a x avec des donnees prescrites f et a

the variation with the poisson ratio of lamb modes in a free plate iii behaviour of individual modes abstract the way individual real valued lamb mode branches of a free elastic plate vary with the poisson ratio is studied the coverage extending over the whole range both positive and negative of the latter parameter after examining the zero order antisymmetric and symmetric branches an investigation is made of the non zero order branches of arbitrary order the overall behaviour is shown to be largely influenced by two cylicities the first of these applies over the full range of and occurs when the normal displacements within the plate are entirely dilatational while the second cyclicity applies only over the negative range of and occurs when the normal displacements within the plate are entirely shear illustrations covering the full range of are presented for the first few non zero order modes additionally guidance supplementing mindlin s rules of bounds is proffered for the easy sketching without need for any computations of sets of lamb mode spectra at any positive or negative value of onto grids of mindlin s bounds

on stability of the spatially inhomogeneous navier stokes boussinesq system with general nonlinearity this paper considers l 2 asymptotic stability of the spatially inhomogeneous navier stokes boussinesq system with general nonlinearity including both power nonlinear terms and convective terms we construct a local in time strong solution of the system by applying semigroup theory on hilbert spaces and fractional powers of the stokes laplace operator it is also shown that under some assumptions on an energy inequality the system has a unique global in time strong solution when the initial datum is sufficiently small furthermore we investigate the asymptotic stability of the global in time strong solution by using an energy inequality maximal l p in time regularity for hilbert space valued functions and fractional powers of linear operators in a solenoidal l 2 space we introduce new methods for showing the asymptotic stability by applying an energy inequality and maximal l p in time regularity for hilbert space valued functions our approach in this paper can be applied to show the asymptotic stability of energy solutions for various incompressible viscous fluid systems and the stability of small stationary solutions whose structure is not clear

conversion modes in a free elastic plate the spectra of conversion waves in a free infinite isotropic elastic plate and their relationship to the spectra of lamb waves are investigated over the whole range of the poisson ratio in addition to the known set of conversion modes the sum conversion modes in which for a double traverse of the plate a phase sum condition is satisfied there is a set denoted as the difference conversion modes in which a phase difference condition is fulfilled in the real solutions plane the antisymmetric and symmetric lamb branches of order h oscillate around the hth sum conversion mode crossing the latter at its intersections with the difference conversion modes when is less than 0 263 the hth pair of lamb branches also crosses the hth sum conversion mode at two additional points in the imaginary solutions plane the antisymmetric and symmetric lamb branches forming the lth plait oscillate about the lth difference conversion mode crossing it at its intersections with the sum conversion modes the degree of coupling of partial waves within the wavenumber plate thickness field shows a regular pattern related to the branches of the sum and difference conversion modes

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

the initial boundary value problem for thin plate flow of incompressible non newtonian viscous fluids abstract in this paper the particular model that describes the thin plate flow of incompressible non newtonian viscous fluids is investigated this model which arises from polymer processing concerns the initial boundary problem of nonstationary thin plate flow of non newtonian viscous incompressible fluids by using monotone operator theory and schauder s fixed point theorem an existence and uniqueness theorem for a generalized solution is obtained and the regularity for this solution is tested

on the overlapping resonances concept of acoustic transmission through an elastic plate ii numerical examples and physical implications abstract in a companion paper 1 the fiorito madigosky and uberall approximate theory of transmittivity of an infinite isotropic elastic plate in terms of lamb mode resonances 2 is examined and extended by validating coherent addition of these overlapping resonances the present paper provides numerical and heuristic tests which yield convincing proof of the applicability of the overlapping resonances concept at various thicknesses of a water immersed steel plate including the half space and zero thickness limits the behaviour is shown to be compatible with known exact behaviour overlapping resonances explain the reduction towards zero of the transmittivity near the coincidence angles of the zero order lamb modes as these converge to the rayleigh angle contrary to earlier interpretations both zero order modes survive at the half space limit the a 0 mode is shown to contribute below fd critical becoming then highly damped and effectively non propagating the interpretation provided of the transition of the a 0 mode in this region bridges the gap between conventional lamb mode theory and thin plate theory

weighted estimates for the incompressible fluid in exterior domains for a strong solution u x t of the navier stokes equations in exterior domain in rn where n 2 3 we study the time decay of x u t lp for n 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 h o bae b j jin temporal and spatial decay rates of navier stokes solutions in exterior domains bull korean math soc 44 3 2007 547 567 h o bae b j jin asymptotic behavior for the navier stokes solutions in 2d exterior domains j funct anal 240 2006 508 529 and we will extend bae and jin s results by modifying their methods

asymptotic stability of large solutions with large perturbation to the navier stokes equations abstract consider weak solutions w of the navier stokes equations in serrin s class w l 0 l q for 2 3 q 1 with 3 q where is a general unbounded domain in r 3 we shall show that although the initial and external disturbances from w are large every perturbed flow v with the energy inequality converges asymptotically to w as v t w t l 2 0 v t w t l 2 o t 1 2 as t

gidas ni nirenberg results for finite difference equations estimates of approximate symmetry are positive solutions of finite difference boundary value problems hu f u in h u 0 on h as symmetric as the domain to answer this question we first show by examples that almost arbitrary non symmetric solutions can be constructed this is in striking difference to the continuous case where by the famous gidas ni nirenberg theorem b gidas wei ming ni l nirenberg symmetry and related problems via the maximum principle comm math phys 68 1979 209 243 positive solutions inherit the symmetry of the underlying domain then we prove approximate symmetry theorems for solutions on equidistantly meshed n dimensional cubes explicit estimates depending on the data are given which show that the solutions become more symmetric as the discretization gets finer the quality of the estimates depends on whether or not f 0 0 the one dimensional case stands out in two ways the proofs are elementary and the estimates for the defect of symmetry are o h compared to o 1 log h in the higher dimensional case

asymptotic stability for the navier stokes equations in the marginal class in this paper we consider the navier stokes equations in r n n 3 we prove the asymptotic stability for weak solutions in the marginal class u l 2 0 bmo where bmo denotes the bounded mean oscillation function with arbitrary initial and external perturbations

spatial stability of 3d exterior stationary navier stokes flows abstract in this paper we study the stability of stationary solutions w for the navier stokes flows in an exterior domain with zero velocity at infinity with suitable assumptions of w by the works of chen 1993 kozono ogawa 1994 and borchers miyakawa 1995 if u 0 w l r l 3 then one can obtain u t w p o t 3 2 1 r 1 p for 1 r p u t w p o t 3 2 1 r 1 p 1 2 for 1 r p 3 where u x t is a solution of the navier stokes equations with the initial condition u 0 in this paper we will prove that for any 0 3 if x u 0 w belongs to l r then one has x u t w l p o t 3 2 1 r 1 p 2 for p 3 r 3 r

a symmetry theorem and applications to nonlinear partial differential equations on donne des applications d un nouveau theoreme sur les proprietes de symetrie des solutions des equations differentielles non lineaires de la forme lu f x u dans u 0 ou u n 0 ou est une region bornee ou non bornee de r n

l2 decay for the navier stokes flow in halfspaces on etudie le comportement asymptotique dans l 2 des solutions faibles des equations de navier stokes dans des demi espaces

the finite product method in the theory of waves and stability this paper presents a method of analysing the dispersion relation and field shape of any type of wave field for which the dispersion relation is transcendental the method involves replacing each transcendental term in the dispersion relation by a finite product polynomial the finite products chosen must be consistent with the low frequency low wavenumber limit but the method is nevertheless accurate up to high frequencies and high wavenumbers full details of the method are presented for a non trivial example that of anti symmetric elastic waves in a layer the method gives a sequence of polynomial approximations to the dispersion relation of extraordinary accuracy over an enormous range of frequencies and wavenumbers it is proved that the method is accurate because certain gamma function expressions which occur as ratios of transcendental terms to finite products largely cancel out nullifying runge s phenomenon the polynomial approximations which are unrelated to taylor series introduce no spurious branches into the dispersion relation and are ideal for numerical computation the method is potentially useful for a very wide range of problems in wave theory and stability theory

algebraic decay of weak solutions to 3d navier stokes equations in general unbounded domains abstract in this paper we study the decay properties of a weak solution to the nonstationary navier stokes equations by means of the spectral decomposition method of fractional powers of the stokes operator in an arbitrary unbounded domain the explicit l 2 decay rates are given as t which improves greatly the decay results in 8 9

search extension method for multiple solutions of a nonlinear problem abstract based on the eigensystem j j of multiple solutions of nonlinear problem u u 3 0 min u 0 on are approximated a new method called the search extension method sem composed of three steps is proposed numerical examples in a square and the unit circle are presented to show the efficiency of our method

dynamic response of an elastic plate containing periodic masses abstract this paper develops an analytical model that incorporates an infinite number of periodically spaced discrete masses into the equations of elasticity of a two dimensional solid that is excited by a harmonic force in both time and space two specific problems are addressed the first is that of a plate with the masses on the bottom edge and the second is that of a plate with the masses embedded in the medium the equations of elasticity are written as stress field expressions with the appropriate boundary conditions in the spatial frequency domain an infinite number of indexed equations are generated using an orthogonalization procedure once this is accomplished all the indexed equations of the system are written together in a single matrix equation the problem is then solved using a truncated set of terms and the displacement fields are transferred into the wavenumber frequency domain for analysis these results are compared to previously available low frequency model results for solutions involving the flexural wave in the plate a numerical example is then solved at high frequency that includes higher order wave motion and this example is discussed

interpolation between hp spaces the complex method the corollary follows immediately from theorem 1 and fefferman s theorem see 111 that bmo is the dual space of h before proceeding to further results we make some comments on the history of theorem 1 unfortunately the theorem h lm e lp l p 1 0 has been folklore for many years and is stated e g in 2 221 this is due to a small but inopportune typographical error in 111 it is stated on page 157 of that paper that h lp lq l q 1 8 e p 1 p co

revisiting the edge resonance for lamb waves in a semi infinite plate the resonance for the elastic plate with a free edge is studied from the point of view of complex resonance the variations of the real part and of the imaginary part of the complex resonance frequency as a function of the poisson ratio are determined numerically the results confirm the real resonance frequency theoretically predicted in i roitberg et al q j mech appl math 51 1 13 1998 for a zero poisson ratio 1 0 and a real resonance frequency that corresponds to a lame mode is discovered for a poisson ratio 2 0 2248 it is shown that both real resonance frequencies may exist at these two particular values of because of the decoupling between the propagating lamb mode and the set of evanescent lamb modes

the existence of periodic solution and behavior of the generalized solution when t of boundary problem of non newtonian fluids1this research was supported by national science foundation of china 1 in this paper we discuss the existence of periodic solution and the behavior of the generalized solution when t of the first boundary problem of incompressible non newtonian fluids this problem arises from polymer processing which concerns with the first initial boundary value problem of non stationary thin plate flow of the non newtonian viscous incompressible fluids by using the monotone operator theory and the schauder s fixed point theorem we have obtained the existence theorem of periodic solution and the characteristics of generalized solution when t

stability results for fluids of oldroyd b type on exterior domains consider the set of equations describing fluids of oldroyd b type on an exterior domain it is shown that the solution of the linearized equation is governed by a bounded analytic semigroup on lp w1 p 1 p which is strongly stable moreover it is shown that the trivial solution of the full system is asymptotically stable in the sense that any solution starting in small ball around 0 0 converges towards 0 as t

constrained mountain pass algorithm for the numerical solution of semilinear elliptic problems a new numerical algorithm for solving semilinear elliptic problems is presented a variational formulation is used and critical points of a c1 functional subject to a constraint given by a level set of another c1 functional or an intersection of such level sets of finitely many functionals are sought first constrained local minima are looked for then constrained mountain pass points the approach is based on the deformation lemma and the mountain pass theorem in a constrained setting several examples are given showing new numerical solutions in various applications

on the overlapping resonances concept of acoustic transmission through an elastic plate i an examination of properties abstract the fiorito madigosky and uberall approximate representation 1 of the acoustic transmission coefficient of an infinite isotropic elastic plate by resonances associated with lamb modes is examined and clarification of its properties is obtained some theoretical extension is provided which validates coherent addition of these overlapping resonances minor limitations shown are that the theory is inapplicable near grazing incidence and that it gives only a partial solution at incidence beyond the second critical angle when the frequency thickness product is the variable existing overlapping resonances theory is shown not to yield a unique relationship between the incident wave and a set of physical motions within the plate and an approach to overcome this non uniqueness is suggested an equivalence is demonstrated between the 3 db resonance half width of each lamb mode and the imaginary part of the modal wavenumber per unit wavenumber of the fluid and this facilitates the numerical checking of the theory which is carried out in a companion paper 2 and yields convincing proof of the applicability of the approach

energy decay for a weak solution of the navier stokes equation with slowly varying external forces we consider the navier stokes system with slowly decaying external forces formula we show that the energy norm of a weak solution has non uniform decay u t 2 0 t under suitable conditions on the data f and a which make the energy of solution bounded in time also we show the exact rate of the decay uniform decay of the energy u t 2 c 1 t e for external forces with a given explicit decay rate

an efficient and stable method for computing multiple saddle points with symmetries in this paper an efficient and stable numerical algorithm for computing multiple saddle points with symmetries is developed by modifying the local minimax method established in y li and j zhou siam j sci comput 23 2001 pp 840 865 y li and j zhou siam j sci comput 24 2002 pp 840 865 first an invariant space is defined in a more general sense and a principle of invariant criticality is proved for the generalization then the orthogonal projection to the invariant space is used to preserve the invariance and to reduce computational error across iterations simple averaging formulas are used for the orthogonal projections numerical computations of examples with various symmetries of which some can and others cannot be characterized by a compact group of linear isomorphisms are carried out to confirm the theory and to illustrate applications the mathematical features of various problems demonstrated in these examples fall into two categories nodal solutions of saddle point type with large morse indices and nonradial positive solutions via symmetry breaking in radially symmetric elliptic problems the new numerical algorithm generates these rather unstable solutions in an efficient and stable way the existence of many unstable solutions and their behavior found in this paper remain to be investigated

stability and convergence of a finite element method for a semilinear elliptical problem with small viscosity abstract in this paper we propose and analyze a new finite element method for a semilinear elliptical problem with small viscosity firstly we prove the existence and uniqueness of solution for its variational problem then we obtain the stability and convergence of the corresponding finite element galerkin approximation furthermore we derive the optimal error estimates in the l 2 and h 1 norms for the finite element approximations respectively finally several numerical experiments are provided to confirm the above theoretical results

the vibration with the poisson ratio of lamb modes in a free plate i general spectra abstract this study of lamb modes in a free elastic plate builds largerly upon the work of mindlin to which it makes various extensions and a few corrections in addition to dealing with the more familiar region in which the poisson ratio is positive the increasingly important region in which it is negative also is investigated and a set of lamb mode spectra is presented covering the full range of the poisson ratio this set enables observations to be made of trends both in the real and in the imaginary wavenumber domains the work described in this paper has led to the identification of a number of problems which are investigated in two companion papers

high order numerical method for the nonlinear helmholtz equation with material discontinuities in one space dimension the nonlinear helmholtz equation nlh models the propagation of electromagnetic waves in kerr media and describes a range of important phenomena in nonlinear optics and in other areas in our previous work we developed a fourth order method for its numerical solution that involved an iterative solver based on freezing the nonlinearity the method enabled a direct simulation of nonlinear self focusing in the nonparaxial regime and a quantitative prediction of backscattering however our simulations showed that there is a threshold value for the magnitude of the nonlinearity above which the iterations diverge in this study we numerically solve the one dimensional nlh using a newton type nonlinear solver because the kerr nonlinearity contains absolute values of the field the nlh has to be recast as a system of two real equations in order to apply newton s method our numerical simulations show that newton s method converges rapidly and in contradistinction with the iterations based on freezing the nonlinearity enables computations for very high levels of nonlinearity in addition we introduce a novel compact finite volume fourth order discretization for the nlh with material discontinuities our computations corroborate the design fourth order convergence of the method the one dimensional results of the current paper create a foundation for the analysis of multidimensional problems in the future

mode structure for fluid solid media as derived by low frequency asymptotics abstract the modes in a laterally homogeneous fluid solid medium can be classified according to their low frequency behaviour for each mode the horizontal wavenumber k will tend to a complex limit q possibly infinite but that appears to be exceptional as the angular frequency tends to zero the modes with vanishing wavenumber limits q can be listed explicitly they are finite in number for each particular medium the non zero finite limits q appear as the zeros of certain entire analytic functions one particular function for each fluid or solid region in the medium and also for each riemann sheet in the presence of homogeneous half spaces the resulting low frequency mode structure can be uncovered for each fluid solid medium by computing the wavenumber limits q as provided by the different regions efficient and reliable numerical techniques for this purpose are proposed for homogeneously layered media based on compound matrix factorization the mode structure can be carried to higher frequencies by tracking dispersion curves the behaviour at double roots and branch points needs special consideration two examples are studied in detail one from plate acoustics and one from underwater acoustics

newton s method s basin of attraction for sign changing solutions of concave and convex nonlinearities abstract in this paper we will illustrate and visualize the chaotic behavior for solving the sign changing solutions of concave convex nonlinearities we propose contractor iteration algorithm and bifurcation method to improve the computation speed and avoid local convergence of newton raphson method by this method sign changing numerical solutions will be computed and visualized in various domains

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

lamb mode spectra versus the poisson ratio in a free isotropic elastic plate the variation with material parameters of lamb modes is investigated vibration spectra of traction free elastic plates are generally presented for a given isotropic material as a set of dispersion curves corresponding to the various lamb mode branches here the spectrum variations with the poisson ratio are plotted in a dimensionless co ordinate system in the form of a bundle of curves for each lamb mode except for the fundamental anti symmetric mode a0 this representation highlights the same behavior for all lamb modes vt denoting the shear wave velocity the k plane can be divided into two angular sectors separated by the line of slope vt2 in the upper one corresponding to a phase velocity v k larger than vt2 dispersion curves are very sensitive to the plate material parameters in the lower sector v vt2 all the branches whatever the value of the poisson ratio 0 0 5 are gathered into a thin pencil moreover curves of a given bundle cross the boundary line at coincidence po

l 2 asymptotic stability of the mild solution to the 3d mhd equation in this paper we investigate the l 2 asymptotic stability of a global in time mild solution of the small initial value problem to the 3d magnetohydrodynamic mhd equations we construct a weak solution to the mhd equation and then we show this weak solution converges to the mild solution in the sense of l2 energy norm

bifurcation method for computing the multiple positive solutions to p henon equation in this paper we proposed the algorithms to solving the d 4 symmetric positive solutions to the boundary value problem of p henon equation taking r in p henon equation as a bifurcation parameter the symmetry breaking bifurcation point on the branch of the d 4 symmetric positive solutions is found via the extended systems based on the liapunov schmidt reduction other symmetric positive solutions are computed by the branch switching method

asymptotic behavior of weak solutions to the damped navier stokes equations abstract whether or not weak solutions of classical navier stokes equations decay to zero in l 2 as time tends to infinity was posed by leray in his pioneering paper 12 13 and has been well solved see 15 for instance this paper examines the three dimensional navier stokes equations with damping term u 1 u and proves that the weak solutions decay to zero in l 2 as time tends to infinity for any 1 by developing local in space estimate for weak solutions

the variation with the poisson ratio of lamb modes in a free plate ii at transitions and coincidence values abstract a set of spectra of real and imaginary valued lamb modes has been presented in a companion paper each spectrum being for a fixed value of the poisson ratio and the set covering the full range of the analysis is extended in this paper by examining in more detail situations in which transitions need to be interpreted or in which coincidences of modes of like symmetry lead to discontinuities requiring explanation this includes studies of the upper end of the range of where the complicated lamb mode spectra of an elastic plate must change to the much simpler spectrum of a fluid plate and of the transition from positive to negative the special behaviour of the real lamb mode branches at 0 being explained in terms of mindlin s rules of bounds for negative discontinuous changes of slope of individual real modes are shown to occur under certain conditions and for both positive and negative a different type of slope discontinuity is demonstrated in association with occurrences of phase and group velocities in opposite directions