boundary layer for a class of nonlinear pipe flow we establish the mathematical validity of the prandtl boundary layer theory for a family of nonlinear parallel pipe flow the convergence is verified under various sobolev norms including the physically important space time uniform norm as well as thel 1 h 1 norm higher order asymptotics is also studied

on the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half plane we consider the navier stokes equations for viscous incompressible flows in the half plane under the no slip boundary condition by using the vorticity formulation we prove the local in time convergence of the navier stokes flows to the euler flows outside a boundary layer and to the prandtl flows in the boundary layer in the inviscid limit when the initial vorticity is located away from the boundary 2014 wiley periodicals inc

finite size effects for percolation on apollonian networks we study the percolation problem on the apollonian network model the apollonian networks display many interesting properties commonly observed in real network systems such as small world behavior scale free distribution and a hierarchical structure by taking advantage of the deterministic hierarchical construction of these networks we use the real space renormalization group technique to write exact iterative equations that relate percolation network properties at different scales more precisely our results indicate that the percolation probability and average mass of the percolating cluster approach the thermodynamic limit logarithmically we suggest that such ultraslow convergence might be a property of hierarchical networks since real complex systems are certainly finite and very commonly hierarchical we believe that taking into account finite size effects in real network systems is of fundamental importance

the inviscid limit of navier stokes equations for analytic data on the half space in their classical work sammartino and caflisch commun math phys 192 2 433 461 1998a commun math phys 192 2 463 491 1998b proved the inviscid limit of the incompressible navier stokes equations for well prepared data with analytic regularity in the half space their proof is based on the detailed construction of prandtl s boundary layer asymptotic expansions in this paper we give a direct proof of the inviscid limit for general analytic data without having to construct prandtl s boundary layer correctors our analysis makes use of the boundary vorticity formulation and the abstract cauchy kovalevskaya theorem on analytic boundary layer function spaces that capture unbounded vorticity

bounds of percolation thresholds in the enhanced binary tree it is argued that the lower critical percolation threshold of the enhanced binary tree is bounded as p c1 0 355 059 by studying its subgraphs while the upper threshold is bounded both from

zero viscosity limit of the navier stokes equations in the analytic setting in this paper we consider the zero viscosity limit of the navier stokes equations in a half space with non slip boundary condition based on the vorticity formulation and the use of conormal derivative we develop an energy method to justify the zero viscosity limit for the analytic data

the inviscid limit for the navier stokes equations with data analytic only near the boundary we address the inviscid limit for the navier stokes equations in a half space with initial datum that is analytic only close to the boundary of the domain and that has sobolev regularity in the complement we prove that for such data the solution of the navier stokes equations converges in the vanishing viscosity limit to the solution of the euler equation on a constant time interval

gevrey stability of prandtl expansions for 2 dimensional navier stokes flows we investigate the stability of boundary layer solutions of the 2 dimensional incompressible navier stokes equations we consider shear flow solutions of prandtl type u t x y ue t y ubl t y 0 0 1 we show that if ubl is monotonic and concave in y y then u is stable over some time interval 0 t t independent of under perturbations with gevrey regularity in x and sobolev regularity in y we improve in this way the classical stability results of sammartino and caflisch in analytic class both in x and y moreover in the case where ubl is steady and strictly concave our gevrey exponent for stability is optimal the proof relies on new and sharp resolvent estimates for the linearized orr sommerfeld operator

local in time existence and uniqueness of solutions to the prandtl equations by energy methods we prove local existence and uniqueness for the two dimensional prandtl system in weighted sobolev spaces under oleinik s monotonicity assumption in particular we do not use the crocco transform or any change of variables our proof is based on a new nonlinear energy estimate for the prandtl system this new energy estimate is based on a cancellation property that is valid under the monotonicity assumption to construct the solution we use a regularization of the system that preserves this nonlinear structure this new nonlinear structure may give some insight into the convergence properties from the navier stokes system to the euler system when the viscosity goes to 0 2015 wiley periodicals inc

generalized cell dual cell transformation and exact thresholds for percolation suggested by scullard s recent star triangle relation for correlated bond systems we propose a general cell dual cell transformation which allows in principle an infinite variety of lattices with exact percolation thresholds to be generated we directly verify scullard s site percolation thresholds and derive the bond thresholds for his martini lattice p c 1 sqrt 2 and the a lattice p c 0 625457 dots solution to p 5 ensuremath 4 p 4 3 p 3 2 p 2 ensuremath 1 0 we also present a precise monte carlo test of the site threshold for the a lattice

zero viscosity limit for analytic solutions of the navier stokes equation on a half space i existence for euler and prandtl equations this is the first of two papers on the zero viscosity limit for the incompressible navier stokes equations in a half space in this paper we prove short time existence theorems for the euler and prandtl equations with analytic initial data in either two or three spatial dimensions the main technical tool in this analysis is the abstract cauchy kowalewski theorem for the euler equations the projection method is used in the primitive variables to which the cauchy kowalewski theorem is directly applicable for the prandtl equations cauchy kowalewski is applicable once the diffusion operator in the vertical direction is inverted

vorticity measures and the inviscid limit we consider a sequence of leray hopf weak solutions of the 2d navier stokes equations on a bounded domain in the vanishing viscosity limit we provide sufficient conditions on the associated vorticity measures away from the boundary which ensure that as the viscosity vanishes the sequence converges to a weak solution of the euler equations the main assumptions are local interior uniform bounds on the l 1 norm of vorticity and the local uniform convergence to zero of the total variation of vorticity measure on balls in the limit of vanishing ball radii

exact site percolation thresholds using a site to bond transformation and the star triangle transformation i construct a two dimensional lattice on which the inhomogeneous site percolation threshold is exactly calculable and use this result to find two more lattices on which the site thresholds can be determined the primary lattice studied here the martini lattice is a hexagonal lattice with every second site transformed into a triangle the site threshold of this lattice is found to be 0 764826 i e the solution to p4 3p3 1 0 while the others have square root 5 1 2 the inverse of the golden ratio and 1 square root 2 this last solution suggests a possible approach to establishing the bound for the hexagonal site threshold pc 1 square root 2 to derive these results i solve a correlated bond problem on the hexagonal lattice by use of the star triangle transformation and then by a particular choice of correlations derived from a site to bond transformation solve the site problem on the martini lattice

vanishing viscosity limit for incompressible flow inside a rotating circle abstract in this article we consider circularly symmetric incompressible viscous flow in a disk the boundary condition is no slip with respect to a prescribed time dependent rotation of the boundary about the center of the disk we prove that if the prescribed angular velocity of the boundary has finite total variation then the navier stokes solutions converge strongly in l 2 to the corresponding stationary solution of the euler equations when viscosity vanishes our approach is based on a semigroup treatment of the symmetry reduced scalar equation

well posedness of the prandtl equation in sobolev spaces we develop a new approach to study the well posedness theory of the prandtl equation in sobolev spaces by using a direct energy method under a monotonicity condition on the tangential velocity field instead of using the crocco transformation precisely we firstly investigate the linearized prandtl equation in some weighted sobolev spaces when the tangential velocity of the background state is monotonic in the normal variable then to cope with the loss of regularity of the perturbation with respect to the background state due to the degeneracy of the equation we apply the nash moser hormander iteration to obtain a well posedness theory of classical solutions to the nonlinear prandtl equation when the initial data is a small perturbation of a monotonic shear flow

a note on prandtl boundary layers this note concerns nonlinear ill posedness of the prandtl equation and an invalidity of asymptotic boundary layer expansions of incompressible fluid flows near a solid boundary our analysis is built upon recent remarkable linear illposedness results established by gerard varet and dormy and an analysis by guo and tice we show that the asymptotic boundary layer expansion is not valid for nonmonotonic shear layer flows in sobolev spaces we also introduce a notion of weak well posedness and prove that the nonlinear prandtl equation is not well posed in this sense near nonstationary and nonmonotonic shear flows on the other hand we are able to verify that oleinik s monotonic solutions are well posed 2011 wiley periodicals inc

vorticity boundary conditions and boundary vorticity generation for two dimensional viscous incompressible flows abstract in this paper we present boundary conditions appropriate for the vorticity formulation of the two dimensional incompressible viscous navier stokes equations these boundary conditions are incorporated into a finite difference scheme and the resulting method is of the vorticity creation type i e vorticity is generated at the boundary to ensure that the tangential velocity boundary condition is satisfied the results of computations with this finite difference method are presented for flow past a circular cylinder a difference scheme and computational results for a model problem the prandtl boundary layer equations describing flow over a semi infinite flat plate are also presented

spline smoothing in a partly linear model on donne des estimations qui minimisent la somme des sommes residuelles des carres et une penalite de difficulte

on the vanishing viscosity limit in a disk we say that a solution of the navier stokes equations converges in the vanishing viscosity limit to a solution of the euler equations if their velocities converge in the energy l2 norm uniformly in time as the viscosity vanishes we show that a necessary and sufficient condition for the vanishing viscosity limit to hold in a disk is that the space time energy density of the solution to the navier stokes equations in a boundary layer of width proportional to vanish with and that one need only consider spatial variations whose frequencies in the radial or tangential direction lie in a band centered around 1

bayesian confidence intervals for the cross validated smoothing spline summary we consider the model y t g ti ei i 17 2 n where g t t 0 1 is a smooth function and the ei are independent n 0 a2 errors with g2 unknown the cross validated smoothing spline can be used to estimate g non parametrically from

patchy percolation on a hierarchical network with small world bonds the bond percolation properties of the recently introduced hanoi networks are analyzed with the renormalization group unlike scale free networks these networks are meant to provide an analytically tractable interpolation between finite dimensional lattice based models and their mean field limits in percolation the hierarchical small world bonds in the hanoi networks impose order by uniting otherwise disconnected local clusters this patchy order results in merely a finite probability to obtain a spanning cluster for certain ranges of the bond probability unlike the usual 0 1 transition found on ordinary lattices the various networks studied here exhibit a range of phase behaviors depending on the prevalence of those long range bonds fixed points in general exhibit nonuniversal behavior

almost global existence for the prandtl boundary layer equations we consider the prandtl boundary layer equations on the half plane with initial datum that lies in a weighted h1 space with respect to the normal variable and is real analytic with respect to the tangential variable the boundary trace of the horizontal euler flow is taken to be a constant we prove that if the prandtl datum lies within varepsilon of a stable profile then the unique solution of the cauchy problem can be extended at least up to time t varepsilon geqq rm exp varepsilon 1 rm log varepsilon 1

the euler limit of the navier stokes equations and rotating fluids with boundary in this paper we study the convergence of weak solutions of the navier stokes equations in some particular domains with different horizontal and vertical viscosities when they go to zero with different speeds the difficulty here comes from the dirichlet boundary conditions precisely we show that if the ratio of the vertical viscosity to the horizontal viscosity also goes to zero then the solutions converge to the solution of the euler system we study the same limit for rotating fluids with rossby number also going to zero

observations on the vanishing viscosity limit whether in the presence of a boundary solutions of the navier stokes equations converge to a solution to the euler equations in the vanishing viscosity limit is unknown in a seminal 1983 paper tosio kato showed that the vanishing viscosity limit is equivalent to having sufficient control of the gradient of the navier stokes velocity in a boundary layer of width proportional to the viscosity in a 2008 paper the present author showed that the vanishing viscosity limit is equivalent to the formation of a vortex sheet on the boundary we present here several observations that follow on from these two papers compiled on sunday 20 march 2016 1 definitions and past results 3 2 a 3d version of vorticity accumulation on the boundary 6 3 l norms of the vorticity blow up for p 1 7 4 improved convergence when vorticity bounded in l 8 5 some kind of convergence always happens 9 6 width of the boundary layer 2d 10 7 optimal convergence rate 2d 12 8 a condition on the boundary equivalent to v v 2d 14 9 examples where condition on the boundary holds 2d 17 10 on a result of bardos and titi 2d 22 appendix a a trace lemma 23 acknowledgements 25 references 26 the navier stokes equations for a viscous incompressible fluid in a domain rd d 2 with no slip boundary conditions can be written ns tu u u p u f in div u 0 in u 0 on date compiled on sunday 20 march 2016 2010 mathematics subject classification primary 76d05 76b99 76d10

zero viscosity limit for analytic solutions of the navier stokes equation on a half space ii construction of the navier stokes solution this is the second of two papers on the zero viscosity limit for the incompressible navier stokes equations in a half space in either 2d or 3d under the assumption of analytic initial data we construct solutions of navier stokes for a short time which is independent of the viscosity the navier stokes solution is constructed through a composite asymptotic expansion involving the solutions of the euler and prandtl equations which were constructed in the first paper plus an error term this shows that the navier stokes solution goes to an euler solution outside a boundary layer and to a solution of the prandtl equations within the boundary layer the error term is written as a sum of first order euler and prandtl corrections plus a further error term the equation for the error term is weakly nonlinear its linear part is the time dependent stokes equation this error equation is solved by inversion of the stokes equation through expressing the solution as a regular euler like part plus a boundary layer prandtl like part the main technical tool in this analysis is the abstract cauchy kowalewski theorem

zero viscosity limit for analytic solutions of the primitive equations the aim of this paper is to prove that the solutions of the primitive equations converge in the zero viscosity limit to the solutions of the hydrostatic euler equations we construct the solution of the primitive equations through a matched asymptotic expansion involving the solution of the hydrostatic euler equation and boundary layer correctors as the first order term and an error that we show to be o sqrt nu the main assumption is spatial analyticity of the initial datum

a note on smoothness priors and nonlinear regression abstract a generalization of a nonparametric estimator proposed by shiller 1984 is considered the estimator is shown to be directly related to penalized least squares estimation and spline smoothing some simplifications concerning the computation of shiller s estimator and corresponding posterior covariances are indicated

reply to the comment on monte carlo simulation study of the two stage percolation transition in enhanced binary trees we discuss the nature of the two stage percolation transition on the enhanced binary tree in order to explain the disagreement in the estimation of the second transition probability between the one in our recent paper 2009 j phys a math theor 42 145001 and the other in the comment to it from baek et al 2009 j phys a math theor 42 478001 we point out some reasons that the finite size scaling analysis used by them is not proper for the enhanced tree due to its nonamenable nature which is verified by some numerical results

percolation on hyperbolic lattices the percolation transitions on hyperbolic lattices are investigated numerically using finite size scaling methods the existence of two distinct percolation thresholds is verified at the lower threshold an unbounded cluster appears and reaches from the middle to the boundary this transition is of the same type and has the same finite size scaling properties as the corresponding transition for the cayley tree at the upper threshold on the other hand a single unbounded cluster forms which overwhelms all the others and occupies a finite fraction of the volume as well as of the boundary connections the finite size scaling properties for this upper threshold are different from those of the cayley tree and two of the critical exponents are obtained the results suggest that the percolation transition for the hyperbolic lattices forms a universality class of its own

percolation in the hyperbolic plane the purpose of this paper is to study percolation in the hyperbolic plane and in transitive planar graphs that are quasi isometric to the hyperbolic plane

on the nonlinear instability of euler and prandtl equations in this paper we give examples of nonlinearly unstable solutions of euler equations in the whole space 2 the half space the periodic strip the strip 1 1 and the periodic torus 2 with an application to vortex sheets using the same methods we prove an instability result for prandtl type boundary layers that appear in and 2000 john wiley sons inc

on the ill posedness of the prandtl equation the concern of this paper is the cauchy problem for the prandtl equation this problem is known to be well posed for analytic data or for data with monotonicity properties we prove here that it is linearly ill posed in sobolev type spaces the key of the analysis is the construction at high tangential frequencies of unstable quasimodes for the linearization around solutions with non degenerate critical points interestingly the strong instability is due to vicosity which is coherent with well posedness results obtained for the inviscid version of the equation a numerical study of this instability is also provided

kernel smoothing in partial linear models on considere deux methodes d estimation l une reliee aux splines de lissage partiels l autre motivee par une analyse de residus partielle

smoothness priors and nonlinear regression abstract smoothness priors represent prior information that an unknown function does not change slope quickly and hence that the function describes a simple curve e g wahba 1978 in this article such priors for the multiple nonlinear regression model are developed in such a way that estimates and standard errors can be obtained as a natural and conceptually straightforward extension of linear multiple regression estimation with the addition of dummy variables and dummy observations relations to spline and polynomial interpolation are described an illustrative example of cost function estimation is provided

spectral instability of general symmetric shear flows in a two dimensional channel abstract in this paper we prove the spectral instability of general symmetric shear flows of the incompressible navier stokes equations at a high reynolds number in a two dimensional channel this includes shear flows that are spectrally stable to the corresponding euler equations and thus for the first time provides a complete mathematical proof of the viscous destabilization phenomenon pointed out by heisenberg 1924 5 c c lin 1944 9 and tollmien 1947 17 among others precisely we construct exact unstable eigenvalues and eigenfunctions of the linearized navier stokes equations around symmetric shear flows showing that the solution could grow slowly at the rate of e t r where r is the sufficiently large reynolds number and is the small spatial frequency that remains between lower and upper marginal stability curves low r r 1 7 and up r r 1 11 we introduce a new operator based approach which avoids to deal with matching inner and outer asymptotic expansions but instead involves a careful study of singularity in the critical layers by deriving pointwise bounds on the green function of the corresponding rayleigh and airy operators

percolation in hierarchical scale free nets we study the percolation phase transition in hierarchical scale free nets depending on the method of construction the nets can be fractal or small world the diameter grows either algebraically or logarithmically with the net size assortative or disassortative a measure of the tendency of like degree nodes to be connected to one another or possess various degrees of clustering the percolation phase transition can be analyzed exactly in all these cases due to the self similar structure of the hierarchical nets we find different types of criticality illustrating the crucial effect of other structural properties aside from the scale free degree distribution of the nets

universal condition for critical percolation thresholds of kagome like lattices lattices that can be represented in a kagom e like form are shown to satisfy a universal percolation criticality condition expressed as a relation between p 3 the probability that all three vertices in the triangle connect and p 0 the probability that none connect a linear approximation for p 3 p 0 is derived and appears to provide a rigorous upper bound for critical thresholds a numerically determined relation for p 3 p 0 gives thresholds for the kagom e site bond honeycomb 3 text ensuremath 12 2 lattice and stack of triangle lattices that compare favorably with numerical results