soliton solutions for generalized quasilinear schrodinger equations abstract by introducing a new variable replacement we study the existence of nontrivial solutions for generalized quasilinear schrodinger equations which appear from plasma physics as well as high power ultrashort laser in matter

on the local well posedness of quasilinear schrodinger equations in arbitrary space dimension abstract a class of quasilinear schrodinger equations is studied which contain strongly singular nonlinearities including derivatives of second order such equations have been derived as models of several physical phenomena the results of this paper apply to the superfluid film equation in fluid mechanics the local well posedness for smooth solutions of the cauchy problem is proved in arbitrary space dimension without any smallness assumption on the initial value this improves results in the literature which cover the one dimensional case only almost global well posedness results are obtained as well the proof combines nash moser techniques on implicit function theorems in frechet spaces with hyperbolic semigroup theory including evolution systems using the intersection of all sobolev spaces as basic function space the nondissipativity is overcome by a transformation procedure yielding the evolution operator and a priori estimates of higher order sobolev norms for the linearized problem here a certain trace has to vanish in view of apparent integrability conditions the transformations and the regularity theory are substantially different from the easier one dimensional case in particular continuity estimates are proved for the operator norms of the linear evolution operator in negative sobolev spaces

three dimensional steady state flow to a well in a randomly heterogeneous bounded aquifer 1 we consider flow in a confined aquifer of uniform thickness due to a well of zero radius that fully penetrates the aquifer and discharges at a constant rate if the lateral extent of the aquifer is infinite a steady state flow regime never develops it is however well known that if the aquifer is additionally uniform a quasi steady state region extends from the well out to a cylindrical surface whose radius expands as the square root of time on the expanding surface head is uniform and time invariant inside this surface head at any time is described by a steady state solution a rigorous analysis of the analogous situation in a randomly heterogeneous aquifer would require the solution of a three dimensional transient stochastic flow problem in an aquifer of infinite lateral extent here we take a different approach by developing a three dimensional steady solution for mean flow to a well in a randomly heterogeneous aquifer with a cylindrical prescribed head boundary in analogy to the uniform case we expect our solution to approximate a quasi steady state region whose radius is initially small in comparison to the horizontal correlation scale of log conductivity but grows with time to become eventually much larger we treat log conductivity as a statistically homogeneous random field characterized by a gaussian spatial covariance function that may have different horizontal and vertical correlation scales our solution consists of analytical expressions for the ensemble mean and variance of head in the aquifer to second order in the standard deviation of log conductivity it is based on recursive approximations of exact nonlocal moment equations that are free of distributional assumptions and so apply to both gaussian and non gaussian log conductivity fields the analytical solution is supported by numerical monte carlo simulations it clarifies the manner in which relationships between the horizontal and vertical scales of the quasi steady state region and those of statistical anisotropy impact the statistical moments of drawdown and the equivalent and apparent hydraulic conductivities of the aquifer both conductivities are shown to exhibit a scale effect by growing with distance from the well within a radius of one to two horizontal integral scales from it

multiples of pfister forms abstract the isotropy of multiples of pfister forms is studied some classical results are recalled and their consequences presented with certain of these statements being previously known but hitherto unpublished in particular a lower bound on the first witt index of pfister multiples is established and a number of its corollaries are outlined the relationship between a form and its pfister multiples is explored with particular attention being devoted to the case where the pfister form is generic isotropy statements are obtained in this context with related results demonstrating a strong correspondence between the properties of a form of those of its generic pfister multiples these correspondence results are applied to discriminate between properties inherited by pfister multiples with some novel examples being provided in this regard

on instability of excited states of the nonlinear schrodinger equation we introduce a new notion of linear stability for standing waves of the nonlinear schrodinger equation nls which requires not only that the spectrum of the linearization be real but also that the generalized kernel be not degenerate and that the signature of all the positive eigenvalues be positive we prove that excited states of the nls are not linearly stable in this more restrictive sense we then give a partial proof that this more restrictive notion of linear stability is a necessary condition to have orbital stability

the turning bands method for simulation of random fields using line generation by a spectral method the turning bands method tbm for the simulation of multidimensional random fields is presented these fields commonly occur in the monte carlo simulation of hydrologic processes particularly groundwater flow and mass transport the general tbm equations for two and three dimensional fields are derived with particular emphasis on the more complicated two dimensional case for stationary two dimensional fields the unidimensional line process is generated by a simple spectral method a technique which can be generally applied to any two dimensional covariance function and which is easily extended to anisotropic and areal averaged processes theoretically and by example the tbm is shown to be ergodic even for a finite number of lines and it is demonstrated that it rapidly converges to the true statistics of the field guidelines are presented for the selection of model parameters which will be helpful in the design of simulation experiments the tbm is compared to other methods in terms of cost and accuracy demonstrating that the tbm is as accurate as and much less expensive than multidimensional spectral techniques and more accurate than the most expensive approaches which use matrix inversion such as the nearest neighbor approach the unidimensional spectral technique presented here permits for the first time the inexpensive and accurate tbm simulation of any proper two dimensional covariance function and should be of some help in the stochastic analysis of hydrologic processes

isotropy over function fields of pfister forms abstract the question of which quadratic forms become isotropic when extended to the function field of a given form is studied a formula for the minimum dimension of the minimal isotropic forms associated to such extensions is given and some consequences thereof are outlined especial attention is devoted to function fields of pfister forms here the relationship between excellence concepts and the isotropy question is explored moreover in the case where the ground field is formally real and has finite hasse number the isotropy question is answered for forms of sufficiently large dimension

radially convergent flow in heterogeneous porous media we present an analytical solution for apparent effective transmissivity under radially convergent steady state flow conditions produced by constant pumping from a single well of finite radius rw apparent effective transmissivity te is defined as the value that relates the expected values of flow and head gradient at a certain location the domain is two dimensional of annular shape and the size of the pumping well is explicitly taken into account the solution for the steady state heads is obtained by solving the perturbed flow equation and substituting it into darcy s law to obtain a consistent second order expansion for te we show that apparent effective transmissivity is a scalar for any choice of isotropic covariance model with an expression given in integral form our main result is that te in a heterogeneous statistically isotropic random field under radial steady state flow conditions is a monotonie increasing function of r distance from the well that rises from the harmonic mean of the point transmissivity values close to the well and tends asymptotically towards the geometric mean far from the well the asymptotic value is reached at a distance of a few integral scales 1 5 2 for the gaussian model and 3 5 for the exponential one the apparent effective transmissivity versus normalized r curves are in excellent agreement with previously published numerical work carried out using monte carlo method

quasilinear elliptic equations with critical growth via perturbation method abstract we consider a class of quasilinear schrodinger equations which include the modified nonlinear schrodinger equations a new perturbation approach is used to treat the critical exponent case giving new existence results

on quadratic forms of height two and a theorem of wadsworth let x7 and b be anisotropic quadratic forms over a field f of characteristic not 2 their function fields f ep and f b are said to be equivalent over f if 7 0 f b and b 0 f ep are isotropic we consider the case where dim p 2n and p is divisible by an n 2 fold pfister form we determine those forms b for which p becomes isotropic over f b if n 4 these results imply that if f ep and f b are equivalent and dim p dim 0 then p is similar to b over f this together with already known results yields that if p is of height 2 and degree 1 or 2 and if dimxp dim b then f ep and f b are equivalent iff f ep and f b are isomorphic over f

infinitely many nonradial solutions of a euclidean scalar field equation abstract we study the equation u b x u x u x r n u h 1 r n the existence of a nonradial solution has been an open problem for some time even in the autonomous case under suitable hypotheses mainly on we find an unbounded sequence of nonradial solutions if n 4 or n 6 we also obtain infinitely many solutions if b and are not rotationally symmetric with respect to x but satisfy only a weaker symmetry

existence of ground state sign changing solutions for a class of generalized quasilinear schrodinger maxwell system in r3 abstract in this paper we study the existence of ground state sign changing solutions for the following generalized quasilinear schrodinger maxwell system d i v g 2 u u g u g u u 2 v x u g u g u k x f u x r 3 g 2 u x r 3 where g c 1 r r v x and k x are positive continuous functions and is a positive parameter by making a change of variable as u g 1 v and g u 0 u g t d t we obtain one ground state sign changing solution v g 1 u by using some new analytical skills and non nehari manifold method furthermore the energy of v g 1 u is strictly larger than twice that of the ground state solutions of nehari type we also establish the convergence property of v g 1 u as the parameter 0 our results improve and generalize some results obtained by chen and tang 2016 zhu et al 2016

nonlocal properties of nonuniform averaged flows in heterogeneous media the properties of nonuniform average potential flows in media of stationary random conductivity are studied the mathematical model of average flow is derived as a system of governing equations to be satisfied by mean velocity and mean head the averaged darcy s law determines the effective conductivity as an integral operator of the convolution type relating the mean velocity to the mean head gradient in a nonlocal way in fourier domain the mean velocity is proportional to the mean head gradient the coefficient of proportionality is referred to as the effective conductivity tensor and is derived by perturbation methods in terms of functionals of the conductivity spatial moments it is shown that the effective conductivity cannot be defined uniquely for potential flows however this nonuniqueness does not affect the spatial distribution of the mean head and velocity analytical expressions of the effective conductivity tensor are derived for two and three dimensional flows and for exponential and gaussian correlations of isotropic conductivity the fundamental solution of the governing equations in effective media mean green function is calculated for the same cases two new asymptotic models of the averaged darcy s law are developed to be applicable to large and small scales of heterogeneity several asymptotic expansions of tue two dimensional and three dimensional mean green functions are derived for exponential and gaussian correlations

effective groundwater model parameter values influence of spatial variability of hydraulic conductivity leakance and recharge a stochastic simulation approach was used to inspect the influence of spatial variability in aquifer and recharge properties on effective averaged groundwater model parameters a two dimensional unconfined aquifer model was set up for an area similar to that near livermore california monte carlo simulations were generated for different sets of spatial correlation structures assuming stationarity and an exponential semivariogram for each monte carlo realization groundwater flow was simulated and estimates of the means and standard deviations of the hydraulic head were obtained nine different cases were studied involving different correlation lengths of hydraulic conductivity conditional simulations parameter zonation and spatial recharge distributions results indicate that there is no single uniform value of the hydraulic conductivity that reproduces the expected head values spatial averaging to obtain effective values was based on the p norm power norm which ranges from 1 for the arithmetic mean to 1 for the harmonic mean with 0 representing the geometric mean for the unconditional case the 0 4 p norm seems to best reproduce the expected values the 0 2 p norm gave the best result for the conditional simulation case if the geometric mean were used instead the heads would deviate from the expected heads by about 2 m near the wells some value between the arithmetic and the geometric means of riverbed leakance will give results close to the expected head values the arithmetic mean of aerially distributed recharge produced results with small deviations from the expected head values these particular results may only apply to this modeled system as the pumping pattern and magnitudes exhibited strong influences on zones of maximum head deviation the implication of this work is that if an effective hydraulic conductivity is used in a simulation model it must be selected for a particular set of wells and pumping rates in an aquifer with significant pumping centers the best effective value for a two dimensional unconfined flow model is most likely between the geometric and harmonic mean if the wells are turned off the effective hydraulic conductivity for that flow model reverts to the geometric mean

pumping tests in heterogeneous aquifers an analytical study of what can be obtained from their interpretation using jacob s method interpretation of pumping tests to estimate hydraulic parameter values is typically based on the assumption of aquifer homogeneity the applicability of the traditional methods of interpretation in real aquifers can be questioned since the evaluation of the drawdown curves observed at different locations in a single test may not result in one consistent set of hydraulic parameters thus most hydrogeologists tend to look at estimated transmissivities t as some average property of the medium while estimated storativities s are disregarded in some cases particularly when they are obtained from data measured at the pumping well an analytical study of drawdown under radially convergent flow toward a single point in heterogeneous aquifers shows that large time drawdown values form a straight line on a drawdown versus log time plot jacob s method consists of obtaining estimates for t and s from the slope and intercept of this line we find that even in a heterogeneous field these estimates provide valuable information about the aquifer estimated t values for different observation points tend to converge to a single value which corresponds to the effective t derived under parallel flow conditions estimated storativities however display higher variability but the geometric mean of the sest values can be used as an unbiased estimator of the actual s thus it appears that although jacob s method was originally derived for homogeneous media it can provide valuable information in real aquifers

an evaluation of jacob s method for the interpretation of pumping tests in heterogeneous formations most pumping tests are interpreted using the classical theis assumption of large scale homogeneity with various corrections to account for early time behavior of drawdown curves when drawdowns are plotted versus log time late time data often delineate a straight line which is consistent with jacob s approximation of theis solution but may seem surprising in view of the heterogeneity of natural media the aim of our work is to show that jacob s method leads to a good approximation of the effective transmissivity of heterogeneous media when constrained to late time data a review of several multiwell pumping tests demonstrates that when drawdown curves from each observation well are interpreted separately they produce very similar transmissivity t estimates however the corresponding estimates for storativity span a broad range this behavior is verified numerically for several models of formation heterogeneity a very significant finding of the numerical investigation is that t values estimated using simulated drawdown from individual observation wells are all very close to the effective t value for parallel flow this was observed even in nonmultigaussian t fields where high t zones are well connected and where the effective t is larger than the geometric average of point values this implies that jacob s method can be used for estimating effective t values in many if not most formations

function fields of pfister neighbors a quadratic form q is called a special pfister neighbor if q is similar to a form of the shape p0 ap1 where p0 is pfister a k and p1 is a nonzero subform of p0 the pfister form p0 ap0 which is uniquely determined by q is called the associated pfister form of q if p is an anisotropic pfister form of dimension 8 then every subform q of p of codimension 4 is a special pfister neighbor and there exists an example with dim p 16 and codim q 5 which is not special special pfister neighbors of the same dimension and with the same associated pfister form define the same function field but there exists an example in dimension 5 which shows that such forms need not be similar

existence and nonexistence of positive solutions for a class of generalized quasilinear schrodinger equations involving a kirchhoff type perturbation with critical sobolev exponent in this paper by variational methods we study the existence and nonexistence of positive solutions for a class of generalized quasilinear schrodinger equations involving a kirchhoff type perturbation with critical sobolev exponent in particular the first result is based on the ambrosetti rabinowitz condition moreover the second results are established via the poho aev identity finally the last results are not based on the ambrosetti rabinowitz condition and we use generalized conditions to prove the results

soliton solutions for quasilinear schrodinger equations ii abstract for a class of quasilinear schrodinger equations we establish the existence of ground states of soliton type solutions by a variational method

steady flow toward wells in heterogeneous formations mean head and equivalent conductivity we consider steady flow of water in a confined aquifer toward a fully penetrating well of radius rw figure 1 the hydraulic conductivity k is modeled as a three dimensional stationary random space function the two point covariance of y in k kg is of axisymmetric anisotropy with i and i the horizontal and vertical integral scales respectively and kg the geometric mean of k unlike previous studies which assumed constant flux the well boundary condition is of given constant head figure 1 the aim of the study is to derive the mean head h and the mean specific discharge q as functions of the radial coordinate r and of the parameters y2 e i i and rw i an approximate solution is obtained at first order in y2 by replacing the well by a line source of strength proportional to k and by assuming ergodicity i e equivalence between space averages over the vertical and h q ensemble means an equivalent conductivity keq is defined as the fictitious one of a homogeneous aquifer which conveys the same discharge q as the actual one for the given head hw in the well and a given head in a piezometer at distance r from the well this definition corresponds to the transmissivity determined in a pumping test by an observer that measures hw andq the main result of the study is the relationship 19 keq ka 1 kefu where ka is the conductivity arithmetic mean and kefu is the effective conductivity for mean uniform flow in the horizontal direction in the same aquifer the weight coefficient 10 has the simple approximate expression ln r i in r rw near the well 0 and keq ka which is easily understood since for rw i 1 the formation behaves locally like a stratified one in contrast far from the well 1 and keq kefu the flow being slowly varying there since ka kefu our result indicates that the transmissivity is overestimated in a pumping test in a steady state and it decreases with the distance from the well however the difference between ka and kefu is small for highly anisotropic formations for which e 1 a nonlocal effective conductivity which depends only on the heterogeneous structure is derived in appendix a along the lines of indelman and abramovich 1994

type curve estimation of statistical heterogeneity 1 the analysis of pumping tests has traditionally relied on analytical solutions of groundwater flow equations in relatively simple domains consisting of one or at most a few units having uniform hydraulic properties recently attention has been shifting toward methods and solutions that would allow one to characterize subsurface heterogeneities in greater detail on one hand geostatistical inverse methods are being used to assess the spatial variability of parameters such as permeability and porosity on the basis of multiple cross hole pressure interference tests on the other hand analytical solutions are being developed to describe the mean and variance first and second statistical moments of flow to a well in a randomly heterogeneous medium we explore numerically the feasibility of using a simple graphical approach without numerical inversion to estimate the geometric mean integral scale and variance of local log transmissivity on the basis of quasi steady state head data when a randomly heterogeneous confined aquifer is pumped at a constant rate by local log transmissivity we mean a function varying randomly over horizontal distances that are small in comparison with a characteristic spacing between pumping and observation wells during a test experimental evidence and hydrogeologic scaling theory suggest that such a function would tend to exhibit an integral scale well below the maximum well spacing this is in contrast to equivalent transmissivities derived from pumping tests by treating the aquifer as being locally uniform on the scale of each test which tend to exhibit regional scale spatial correlations we show that whereas the mean and integral scale of local log transmissivity can be estimated reasonably well based on theoretical ensemble mean variations of head and drawdown with radial distance from a pumping well estimating the log transmissivity variance is more difficult we obtain reasonable estimates of the latter based on theoretical variation of the standard deviation of circumferentially averaged drawdown about its mean

equivalent transmissivity of heterogeneous leaky aquifers for steady state radial flow 1 the purpose of this study is to estimate the equivalent transmissivity for two dimensional steady state radial flow toward a fully penetrating well in a heterogeneous leaky aquifer the log transmissivity of the pumped aquifer is modeled as a multivariate random spatial function with stationary mean and exponential semivariogram while the aquitard conductance is assumed to be spatially uniform an approximate expression for the upscaled or block transmissivity defined as the transmissivity of an equivalent homogeneous leaky aquifer system with the same pumping is proposed the proposed relation expresses the upscaled transmissivity as a weighted average of the log transforms of the point transmissivity values monte carlo simulations are used to numerically validate the proposed upscaling expression for wide ranges of aquifer parameters the relation of the weighing function to the statistical parameters of the transmissivity field is discussed the first and second moments of the upscaled transmissivity are also estimated as a function of the problem parameters the expected value of the equivalent transmissivity is shown to increase from the geometric mean to the arithmetic mean as the aquitard conductance increases from zero to infinity

nodal soliton solutions for generalized quasilinear schrodinger equations this paper is concerned with constructing nodal radial solutions for generalized quasilinear schrodinger equations in rn which arise from plasma physics fluid mechanics as well as high power ultashort laser in matter for any given integer k 0 by using a change of variables and minimization argument we obtain a sign changing minimizer with k nodes of a minimization problem

nodal soliton solutions for quasilinear schrodinger equations with critical exponent this paper is concerned with constructing nodal radial solutions for quasilinear schrodinger equations in rn with critical growth which have appeared as several models in mathematical physics for any given integer k 0 by using a change of variables and minimization argument we obtain a sign changing minimizer with k nodes of a minimization problem since the critical exponent appears and the lower order term may change sign we should use more delicate arguments

global existence of small solutions to a relativistic nonlinear schrodinger equation on etudie le probleme de cauchy associea une equation de schrodinger non lineaire modelisant une impulsion laser ultra intense et ultra courte dans un plasma les termes non lineaires nouveaux sont dus aux effets relativistes et a la force ponderomotrice nous prouvons l existence locale et l unicite de solutions petites lorsque la dimension de l espace transverse est egale a 2 ou 3 ainsi que l existence locale pour des donnees initiales de taille arbitraire en dimension un d espace transverse

prediction of steady state flow in nonuniform geologic media by conditional moments exact nonlocal formalism effective conductivities and weak approximation we consider the effect of measuring randomly varying local hydraulic conductivities k x on one s ability to predict steady state flow within a bounded domain driven by random source and boundary functions more precisely we consider the prediction of local hydraulic head h x and darcy flux q x by means of their unbiased ensemble moments h x and q x conditioned on measurements of k x these predictors satisfy a deterministic flow equation in which q x x h x r x where x is a relatively smooth unbiased estimate of k x and r x is a residual flux we derive a compact integral expression for r x which is rigorously valid for a broad class of k x fields including fractals it demonstrates that q x is nonlocal and non darcian so that an effective hydraulic conductivity does not generally exist we show analytically that under uniform mean flow the effective conductivity may be a scalar a symmetric or a nonsymmetric tensor or a set of directional scalars which do not form a tensor we demonstrate numerically that in two dimensional mean radial flow it may increase from the harmonic mean of k x near interior and boundary sources to the geometric mean far from such sources for cases where r x can neither be expressed nor approximated by a local expression we propose a weak integral approximation closure which appears to work well in media with pronounced heterogeneity and improves as the quantity and quality of k x measurements increase the nonlocal deterministic flow equation can be solved numerically by standard methods our theory shows clearly how the scale of grid discretization should relate to the scale quantity and quality of available data after providing explicit approximations for the second moments of head and flux prediction errors we conclude by discussing practical methods to compute x from noisy measurements of k x and to calculate required second moments of the associated estimation errors when k x is lognormal

spatial averaging of transmissivity in heterogeneous fields with flow toward a well steady state nonuniform flow in a heterogeneous medium is investigated using a combined numerical empirical approach the objective of the study is to determine the effective transmissivity of a square field with a constant head external boundary and a constant rate well at its center transmissivity at the point scale is modeled as an isotropic and multivariate lognormal spatial random function block transmissivities are defined using an empirical scaling up of point support values within the field the scaling process is a weighted spatial geometric averaging where log transmissivities are weighted by the inverse square of their distance from the well block transmissivities thus obtained and true effective values calculated using a numerical flow model are compared for realizations a finely discretized transmissivity field the principal finding of this study is that effective transmissivities obtained from the numerical flow model and block values obtained by spatial averaging are in excellent agreement for low to moderate variances of log transmissivity and moderately eccentric field geometries the geostatistical model for point scale transmissivity and the deterministic spatial averaging law are combined in order to create a geostatistical model for transmissivity at the block scale exact expressions are derived for the ensemble moments of block averaged transmissivities and well drawdowns in both the nonconditional case and the conditional case when transmissivity at the well bore is known the ensemble mean of block averaged transmissivity is found to decrease from the ensemble arithmetic mean toward the ensemble geometric mean as the field size becomes large compared to the integral range of spatial correlation this result contradicts earlier published work numerical results presented in this study also disagree with the analytical results of a recent stochastic analysis of specific discharge in radial systems

solutions for a quasilinear schrodinger equation a dual approach abstract we consider quasilinear stationary schrodinger equations of the form 1 u u 2 u g x u x r n introducing a change of unknown we transform the search of solutions u x of 1 into the search of solutions v x of the semilinear equation 2 v 1 1 2f 2 v g x f v x r n where f is suitably chosen if v is a classical solution of 2 then u f v is a classical solution of 1 variational methods are then used to obtain various existence results

bayesian identification of the local transmissivity using time drawdown data from pumping tests 1 conventional pumping test analysis techniques focus mostly on estimating the effective transmissivity of the perturbed subsurface volume surrounding the extraction and observation wells ignoring the existing local scale heterogeneity the purpose of this paper is to explore whether transient drawdown data derived from pumping tests can also be used to identify the statistical spatial structure of the local transmissivity field the log transmissivity is modeled as a multivariate random spatial function with stationary mean and exponential semivariogram monte carlo simulations are used to compute the transient drawdown for different values of the log transmissivity variance and integral scale on the basis of these simulations a bayesian procedure is developed to estimate the probability density functions of these two statistical parameters the main benefit of using a bayesian approach is that it readily accounts for the uncertainty of the estimated parameters application of the proposed procedure to simulated drawdown data shows that it is possible to characterize the local transmissivity from pumping test data and that the reliability of the estimates increases with an increase in the number of available tests

radial solutions of superlinear equations on ℝn part i global variational approach we present a global variational approach to the search for multiple nodal solutions u h 1 n to a class of elliptic equations of type u x f x u x x n where n 2 f is superlinear and subcritical and f x 0

analysis of flow through heterogeneous random aquifers 2 unsteady flow in confined formations the direct problem of determining the head field in a formation of given stationary isotropic random structure is solved approximately for unsteady flow conditions under restrictive assumptions of sufficiently small conductivity variance and average head gradient slowly varying in time and space closed form expressions are derived for the effective conductivity or storativity and for the head variogram two types of unsteady flow are considered transients and periodic flows in the case of transients it is assumed that initially the head is constant no flow and that ultimately the flow becomes uniform it is found that the effective conductivity and transmissivity are time dependent and drop from the arithmetic mean initially to the steady state value during a relaxation time the latter is much larger for one dimensional flows than for two or three dimensional ones the head variogram also relaxes from the one corresponding to the initial lack of correlation to the steady state variogram during a relaxation time which grows with the lag while in three dimensional structure the relaxation time scales are relatively short and steady state analysis can be used accurately even under transient conditions the tendency to steady state may be very slow in the case of transmissivity because of its large integral scale hence steady state analysis for transients might not be warranted for regional transient flows in the case of periodic flows of seasonal nature the head gradient is assumed to be made up from a steady component and a periodic one if the amplitude of the latter is sufficiently small compared to the steady one steady state value of effective transmissivity and storativity can be adopted the head variogram has also its steady state form for large lags and quasisteady for small lags hence the variogram lies between the steady one steady component of head gradient and the quasisteady one instantaneous head gradient