solutions to the 2d euler equations with velocity unbounded at infinity abstract we prove existence of solutions to the two dimensional euler equations with vorticity bounded and with velocity locally bounded but growing at infinity at a rate slower than a power of the logarithmic function we place no integrability conditions on the initial vorticity this result improves upon a result of serfati which gives existence of a solution to the two dimensional euler equations with bounded velocity and vorticity

hessian measures on the heisenberg group abstract in this paper we study the properties of k convex functions on the heisenberg group h n for 1 k 2 n and prove the weak continuity of k hessian measures with respect to local uniform convergence in the heisenberg setting our approach through monotonicity formulae makes use of previous research in the corresponding euclidean case of trudinger and wang the case k 2 n provides an analogue of the monge ampere measure of aleksandrov for h n we also answer a conjecture of garofalo and tournier on monotonicity for the cases n 2

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

effective hydraulic conductivity of bounded strongly heterogeneous porous media we develop analytical expressions for the effective hydraulic conductivity ke of a three dimensional heterogeneous porous medium in the presence of randomly prescribed head and flux boundaries the log hydraulic conductivity y forms a gaussian statistically homogeneous and anisotropic random field with an exponential autocovariance by effective hydraulic conductivity of a finite volume in such a field we mean the ensemble mean expected value of all random equivalent conductivities that one could associate with a similar volume under uniform mean flow we start by deriving a first order approximation of an exact expression developed in 1993 by neuman and orr we then generalize this to strongly heterogeneous media by invoking the landau lifshitz conjecture upon evaluating our expressions we find that ke decreases rapidly from the arithmetic mean ka toward an asymptotic value as distance between the prescribed head boundaries increases from zero to about eight integral scales of y the more heterogeneous is the medium the larger is ke relative to its asymptote at any given separation distance our theory compares well with published results of spatially power averaged expressions and with a first order expression developed intuitively by kitanidis in 1990

new properties of convex functions in the heisenberg group we prove some new properties of the weakly h convex functions recently introduced by danielli garofalo and nhieu as an interesting application of our results we prove a theorem of busemann feller alexandrov type in the heisenberg groups h n 1 2

upscaling of permeability of anisotropic heterogeneous formations 1 the general framework in numerical simulations of flow through heterogeneous formations the domain is partitioned into numerical elements the solution requires assigning physical properties to each numerical block the process of transferring information from the scale of actual heterogeneity to that of the numerical elements is known as upscaling we consider steady one phase flow and the only property of interest is the permeability which is regarded as a random space function and the same is true for the upscaled conductivity in this paper we establish the necessary and sufficient conditions to be satisfied by upscaling the necessary conditions are expressed with the aid of the global response of the formation e g the total flux caused by a constant pressure head drop applied on the boundaries the requirement is that the expected value and the variance of the total flux are the same in the actual and upscaled domains these are supplemented by local conditions which stipulate that space averages of flow variables over the numerical blocks in the upscaled domain tend to those in the actual formation when the elements size tends to 0 the result of this analysis is that the upscaled permeability must be such as to ensure the equality in a statistical sense between the space averaged dissipation in the two media by assuming an unbounded domain average uniform flow and stationarity simplified relationships between the moments of the dissipation in the upscaled and actual media are derived it is shown that the effective permeability of the two media is the same the approach is illustrated for stratified formations for which exact composition rules are possible even in this simple case the upscaled permeability is a tensor characterized at second order by two mean components and three covariance functions

valuations on convex sets non commutative determinants and pluripotential theory abstract a new method of constructing translation invariant continuous valuations on convex subsets of the quaternionic space h n is presented in particular new examples of sp n sp 1 invariant translation invariant continuous valuations are constructed this method is based on the theory of plurisubharmonic functions of quaternionic variables developed by the author in two previous papers bull sci math 127 1 2003 1 j geom anal 13 2 2003 183

the continuity and range of the quaternionic monge ampere operator on quaternionic space in this paper we use the quaternionic closed positive currents to establish some pluripotential results for quaternionic monge ampere operator by introducing a new quaternionic capacity we prove a sufficient condition which implies the weak convergence of quaternionic monge ampere measures triangle u j n rightarrow triangle u n we also obtain an equivalent condition of convergence in c n 1 capacity by using methods from xing proc am math soc 124 2 457 467 1996 as an application the range of the quaternionic monge ampere operator is discussed

lipschitz continuity aleksandrov theorem and characterizations for h convex functions in the geometries of stratified groups we show that h convex functions locally bounded from above are locally lipschitz continuous and that the class of v convex functions exactly corresponds to the class of upper semicontinuous h convex functions as a consequence v convex functions are locally lipschitz continuous in every stratified group in the class of step 2 groups we characterize locally lipschitz h convex functions as measures whose distributional horizontal hessian is positive semidefinite in euclidean space the same results were obtained by dudley and reshetnyak we prove that a continuous h convex function is a e twice differentiable whenever its second order horizontal derivatives are radon measures

quasicontinuity and maximality of quaternionic plurisubharmonic functions abstract in this paper we establish the quaternionic versions of several results in the complex pluripotential theory by using the quaternionic closed positive currents which were introduced in 24 we show that quasicontinuity one of the most important properties of complex plurisubharmonic functions holds also for quaternionic plurisubharmonic functions in h n moreover we prove an equivalent characterization of the maximality which is central to the pluripotential theory a locally bounded plurisubharmonic function u is maximal if and only if it satisfies the homogeneous quaternionic monge ampere equation u n 0

sharp comparison and maximum principles via horizontal normal mapping in the heisenberg group in this paper we solve a problem raised by gutierrez and montanari about comparison principles for h convex functions on subdomains of heisenberg groups our approach is based on the notion of the sub riemannian horizontal normal mapping and uses degree theory for set valued maps the statement of the comparison principle combined with a harnack inequality is applied to prove the aleksandrov type maximum principle describing the correct boundary behavior of continuous h convex functions vanishing at the boundary of horizontally bounded subdomains of heisenberg groups this result answers a question by garofalo and tournier the sharpness of our results are illustrated by examples

stochastic analysis of three dimensional flow in a bounded domain a commonly accepted first order approximation of the equation for steady state flow in a fully saturated spatially random medium has the form of poisson s equation this form allows for the advantageous use of green s functions to solve for the random output hydraulic heads in terms of a convolution over the random input the logarithm of hydraulic conductivity a solution for steady state three dimensional flow in an aquifer bounded above and below is presented consideration of these boundaries is made possible by use of green s functions to solve poisson s equation within the bounded domain the medium hydraulic conductivity is assumed to be a second order stationary random process as represented by a simple three dimensional covariance function upper and lower boundaries are taken to be no flow boundaries the mean flow vector lies entirely in the horizontal dimensions the resulting hydraulic head covariance function exhibits nonstationary effects resulting from the imposition of boundary conditions comparisons are made with existing infinite domain solutions

a probabilistic approach to the two dimensional navier stokes equations we turn the navier stokes equations for a 2 dimensional viscous incompressible fluid into a system of functional integrals in the trajectory space of a suitable diffusion process using probabilistic techniques as girsanov s transformation and bismut elworthy formula we prove the existence of a unique global solution of this system in a constructive way

a higher order approximation to effective conductivity in media of anisotropic random structure properties of the effective conductivity tensor keff are studied by deriving the second order terms in its expansion in the variance 2 of normally distributed log conductivity it is shown that for media of anisotropic structure the components of the effective conductivity tensor are expressed by a functional of the log conductivity covariance that is it depends on the shape of the correlation function and not only on anisotropy ratios variance 2 and space dimensions however the trace of keff is independent of the log conductivity autocovariance and for a given mean conductivity depends only on 2 the dependence of the effective conductivity on the correlation structure is illustrated for gaussian and exponential autocovariances of log conductivity and for two and three dimensional flows

holder continuous solutions to quaternionic monge ampere equations abstract we prove the holder continuity of the unique solution to quaternionic monge ampere equation with densities in l p p 2 on a bounded strictly pseudoconvex domains

on the euler flow in 2 we give a global existence and uniqueness theorem for the euler flow in 2 for suitable initial velocity fields possibly diverging at infinity

subharmonic functions on carnot groups many results of classical potential theory are extended to sub laplacians on carnot groups some characterizations of subharmonicity representation formulas of poisson jensen s kind and nevanlinna type theorems are proved we also characterize the riesz measure related to bounded above subharmonic functions in n

the quaternionic monge ampere operator and plurisubharmonic functions on the heisenberg group many fundamental results of pluripotential theory on the quaternionic space mathbb h n are extended to the heisenberg group we introduce notions of a plurisubharmonic function the quaternionic monge ampere operator differential operators d 0 and d 1 and a closed positive current on the heisenberg group the quaternionic monge ampere operator is the coefficient of d 0d 1u n we establish the chern levine nirenberg type estimate the existence of quaternionic monge ampere measure for a continuous quaternionic plurisubharmonic function and the minimum principle for the quaternionic monge ampere operator unlike the tangential cauchy riemann operator overline partial b on the heisenberg group which behaves badly as partial b overline partial b ne overline partial b partial b the quaternionic counterpart d 0 and d 1 satisfy d 0d 1 d 1d 0 this is the main reason that we have a good theory for the quaternionic monge ampere operator than partial b overline partial b n

on the linear algebra in the quaternionic pluripotential theory abstract we clarify the linear algebra used in the quaternionic pluripotential theory so that proofs of several results there can be greatly simplified in particular we characterize and normalize real 2 forms with respect to the quaternionic structure and show that the moore determinant of a quaternionic hyperhermitian matrix is the coefficient of the exterior product of the associated real 2 form as a corollary the quaternionic monge ampere operator is the coefficient of the exterior product of the baston operator

viscosity solutions to quaternionic monge ampere equations abstract quaternionic monge ampere equations have recently been studied intensively using methods from pluripotential theory we present an alternative approach by using the viscosity methods we study the viscosity solutions to the dirichlet problem for quaternionic monge ampere equations d e t f f q f with boundary value f g on here is a bounded domain on the quaternionic space h n g c and f q t is a continuous function on r r which is non decreasing in the second variable we prove a viscosity comparison principle and a solvability theorem moreover the equivalence between viscosity and pluripotential solutions is shown