nature of complex singularities for the 2d euler equation abstract a detailed study of complex space singularities of the two dimensional incompressible euler equation is performed in the short time asymptotic regime when such singularities are very far from the real domain this allows an exact recursive determination of arbitrarily many spatial fourier coefficients using high precision arithmetic we find that the fourier coefficients of the stream function are given over more than two decades of wavenumbers by f k c k e k where k k cos sin the prefactor exponent typically between 5 2 and 8 3 is determined with an accuracy better than 0 01 it depends on the initial condition but not on the vorticity diverges as s where 7 2 and s is the distance to the complex singular manifold this new type of non universal singularity is permitted by the strong reduction of nonlinearity depletion which is associated to incompressibility spectral calculations show that the scaling reported above persists well beyond the time of validity of the short time asymptotics a simple model in which the vorticity is treated as a passive scalar is shown analytically to have universal singularities with exponent 5 2

singularity formation for prandtl s equations abstract we consider prandtl s equations for an impulsively started disk and follow the process of the formation of the singularity in the complex plane using the singularity tracking method we classify van dommelen and shen s singularity as a cubic root singularity we introduce a class of initial data uniformly bounded in h 1 which have a dipole singularity in the complex plane these data lead to a solution blow up whose time can be made arbitrarily short within the class this is numerical evidence of the ill posedness of the prandtl equations in h 1 the presence of a small viscosity in the streamwise direction changes the behavior of the singularities they stabilize at a distance from the real axis which depends on the amount of viscosity we show that the van dommelen and shen singularity and the singularity predicted by e and engquist in w e b engquist blowup of the solutions to the unsteady prandtl s equations comm pure appl math 50 1997 1287 1293 have different complex structures

sublayer of prandtl boundary layers the aim of this paper is to investigate the stability of prandtl boundary layers in the vanishing viscosity limit nu to 0 in grenier commun pure appl math 53 9 1067 1091 2000 one of the authors proved that there exists no asymptotic expansion involving one of prandtl s boundary layer with thickness of order sqrt nu which describes the inviscid limit of navier stokes equations the instability gives rise to a viscous boundary sublayer whose thickness is of order nu 3 4 in this paper we point out how the stability of the classical prandtl s layer is linked to the stability of this sublayer in particular we prove that the two layers cannot both be nonlinearly stable in l infty that is either the prandtl s layer or the boundary sublayer is nonlinearly unstable in the sup norm

curvature flows in the sphere we consider contracting and expanding curvature flows in mathbb s n 1 when the flow hypersurfaces are strictly convex we establish a relation between the contracting hypersurfaces and the expanding hypersurfaces which is given by the gaus map the contracting hypersurfaces shrink to a point x 0 while the expanding hypersurfaces converge to the equator of the hemisphere mathcal h x 0 after rescaling by the same scale factor the rescaled hypersurfaces converge to the unit spheres with centers x 0 resp x 0 exponentially fast in c infty mathbb s n

a value efficiency approach to incorporating preference information in data envelopment analysis we develop a procedure and the requisite theory for incorporating preference information in a novel way in the efficiency analysis of decision making units the efficiency of deci sion making units is defined in the spirit of data envelopment analysis dea complemented with decision maker s preference information concerning the desirable structure of inputs and outputs our procedure begins by aiding the decision maker in searching for the most preferred combination of inputs and outputs of decision making units for short most preferred solution which are efficient in dea then assuming that the decision maker s most preferred solution maximizes his her underlying unknown value function we approximate the indifference contour of the value function at this point with its possible tangent hyperplanes value efficiency scores are then calculated for each decision making unit comparing the inefficient units to units having the same value as the most preferred solution the resulting value efficiency scores are optimistic approximations of the true scores the procedure and the resulting efficiency scores are immediately applicable to solving practical problems

blowup of solutions of the unsteady prandtl s equation we prove that for certain class of compactly supported c initial data smooth solutions of the unsteady prandtl s equation blow up in nite time

on the stability of boundary layers of incompressible euler equations abstract in this paper we investigate the stability and instability of boundary layers of incompressible euler equations copyright 2000 academic press dans cet article on etudie la stabilite et l instabilite de couches limites des equations d euler

singularities of euler flow not out of the blue does three dimensional incompressible euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time this blowup problem is still open after briefly reviewing what is known and pointing out some of the difficulties we propose to tackle this issue for the class of flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations we present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup

flow of nonconvex hypersurfaces into spheres the flow of surfaces by functions of their principal curvatures has been intensively studied it started with the work of brakke 1 who used the formalism of geometric measure theory a more classical approach had been chosen by huisken who looked at the so called inward flow in 5 the outward flow of surfaces this term will be explained in the sequel is scaling invariant and therefore more natural than the inward flow producing more general results so far huisken 6 and urbas 8 proved that the outward flow of convex surfaces by suitable functions of their principal curvatures converges to spheres the convexity assumption is essential in their work in this paper we would like to present a different method for proving the convergence of star shaped surfaces into spheres via the outward flow let be a symmetric positive function homogeneous of degree one being defined on an open cone of rn with vertex in the origin which contains the positive diagonal i e all tuples of the form

inverse curvature flows in riemannian warped products abstract the long time existence and umbilicity estimates for compact graphical solutions to expanding curvature flows are deduced in riemannian warped products of a real interval with a compact fibre notably we do not assume the ambient manifold to be rotationally symmetric nor the radial curvature to converge nor a lower bound on the ambient sectional curvature the inverse speeds are given by powers 0 p 1 of a curvature function satisfying few common properties

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

the size of the singular set in mean curvature flow of mean convex sets in this paper we study the singularities that form when a hypersurface of positive mean curvature moves with a velocity that is equal at each point to the mean curvature of the surface at that point it is most convenient to describe the results in terms of the level set flow also called biggest flow i2 of chen giga goto cgg and evans spruck es under the level set flow any closed set k in r generates a one parameter family of closed sets ft k t 0 with f0 k k if the boundary of k is a smooth compact hypersurface then so is the boundary of ft k for t in some interval 0 and for such t s the evolution coincides with motion by mean curvature as defined classically by partial differential equations and differential geometry as in h1 however if k is compact then the boundary of ft k will necessarily become singular at some finite time our goal is to show that the singular sets are necessarily quite small this we do provided the initial set k is compact and mean convex in the sense that ft k interior k for all t 0 in case m k is a smooth hypersurface we have the following equivalent characterization of mean convexity k is mean convex if and only if the mean curvature of m with respect to the inward unit normal is everywhere non negative given a compact set k let k be the region in spacetime swept out by the ft k k x t r r t 0 x ft k a point x x t in the boundary of k is called a regular point if 1 x has a neighborhood in which k is a smooth manifold with boundary and 2 the tangent plane to k at x is not horizontal i e is not r 0 note that if x x t is a regular point then in a neighborhood of x ft k is a smooth manifold with boundary in r a point x x t in k with t 0 that is not a regular point is called a singular point

a well posedness theory for the prandtl equations in three space variables abstract the well posedness of the three space dimensional prandtl equations is studied under some constraint on its flow structure together with the instability result given in 28 it gives an almost necessary and sufficient structural condition for the stability of the three dimensional prandtl equations it reveals that the classical burgers equation plays an important role in determining this type of flow with special structure that avoids the appearance of the secondary flow an unstabilizing factor in the three dimensional prandtl boundary layers and the sufficiency of the monotonicity condition on the tangential velocity field for the existence of solutions to the prandtl boundary layer equations is illustrated in the three dimensional setting moreover it is shown that this structured flow is linearly stable for any smooth three dimensional perturbation

structural comparison of data envelopment analysis and multiple objective linear programming the concept of efficiency as it applies to decision making units dmus solutions alternatives plays an important role both in data envelopment analysis dea and multiple objective linear programming molp despite this and other apparent similarities dea and molp research has developed separately we show that structurally the dea formulation to identify efficient units is quite similar to the molp model based on the reference point or the reference direction approach to generate efficient solutions dea and molp should not be seen as substitutes but rather as complements we show that they cross fertilize each other molp provides interesting extensions to dea and dea provides new areas of application to molp

singularity formation for complex solutions of the 3d incompressible euler equations abstract moore s approximation method first formulated for vortex sheets is generalized and applied to axi symmetric flow with swirl and with smooth initial data the approximation preserves the forward cascade of energy but neglects any backflow of energy it splits the euler equations into two sets of equations one for u u r z t containing all non negative wavenumbers in z and the second for u u the equations for u are exactly the euler equations but with complex initial data traveling waves solutions u u r z i t with imaginary wave speed are found numerically for this problem the asymptotic properties of the resulting fourier coefficients show a singularity forming in finite time at which the velocity blows up

tracing complex singularities with spectral methods abstract a numerical method for investigating the possibility of blow up after a finite time is introduced for a large class of nonlinear evolution problems with initial data analytic in the space variable s the solutions have for any t 0 complex space singularities at the edge of an analyticity strip of width t loss of regularity corresponds to the vanishing of t numerical integration by high resolution spectral methods reveals the large wavenumber behavior of the fourier transform of the solutions from which t is readily obtained its time evolution can be traced down to about one mesh length by extrapolation of t such numerical experiments provide evidence suggesting finite time blow up or all time regularity the method is tested on the inviscid and viscous burgers equations and is applied to the one dimensional nonlinear schrodinger equation with quartic potential and to the two dimensional incompressible euler equation all with periodic boundary conditions in the latter case evidence is found suggesting that existing all time regularity results can be substantially sharpened

a multi objective distance friction minimization model for performance assessment through data envelopment analysis abstract the distance friction minimization dfm model proposed by suzuki nijkamp rietveld pels 2010 is an important data envelopment analysis dea model for performance improvement this model has been widely applied to improve the performance of government finance and transportation sectors and energy environment management the dfm model is constructed to set input and output improvements based on the optimal weight vector of the traditional ccr model however non unique optimal weight vectors of the ccr model may be obtained when different linear programming solvers are used thus different values for the two distance friction df objectives may be obtained furthermore the df values obtained using one optimal weight vector may be dominated by those obtained using another optimal vector to address this issue we propose an improved dfm model that considers all the possible optimal weight vectors of the ccr model to determine a set of non dominated df values definitively our dfm model is a multi objective quadratic and nonlinear programming model that is solvable by an augmented constraint method that ensures all solutions are pareto efficient we demonstrate the applicability of our new model by using it to analyse china s transportation sectors

inscribed radius estimates for inverse curvature flow in sphere and hyperbolic space abstract in this paper we study a fully nonlinear inverse curvature flow in sphere and hyperbolic space and prove a non collapsing property for this flow using maximum principle precisely when ambient space is sphere we show that upon some conditions on speed function the radius of the largest touching interior ball is bounded below by a multiple of the reciprocal of the speed where ambient space is hyperbolic we obtain an upper bound of the curvature of the largest touching interior ball for star shaped inverse mean curvature flow

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

embedded minimal tori in s3 and the lawson conjecture we show that any embedded minimal torus in s 3 is congruent to the clifford torus this answers a question posed by h b lawson jr in 1970

complex space singularities of 2d euler flow in lagrangian coordinates we show that for 2d space periodic incompressible flow the solution can be evaluated numerically in lagrangian coordinates with the same accuracy that is achieved in standard eulerian spectral methods this allows the determination of complex space lagrangian singularities lagrangian singularities are found to be closer to the real domain than eulerian singularities and seem to correspond to fluid particles which escape to complex infinity by the current time various mathematical conjectures regarding eulerian lagrangian singularities are presented

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

the van dommelen and shen singularity in the prandtl equations abstract in 1980 van dommelen and shen provided a numerical simulation that predicted the spontaneous generation of a singularity in the prandtl boundary layer equations from a smooth initial datum for a nontrivial euler background in this paper we prove this numerical conjecture by rigorously establishing the finite time blowup of the boundary layer thickness

a blow up criterion for classical solutions to the prandtl equations in this paper we obtain a blow up criterion for classical solutions to the two dimensional prandtl equations under a monotonicity assumption in weighted sobolev spaces our proof is based on nonlinear energy estimates inspired by masmoudi and wong commun pure appl math 68 1683 1741 2015 and maximum principles for parabolic equations

on the global existence of solutions to the prandtl s system abstract in this paper we establish a global existence of weak solutions to the two dimensional prandtl s system for unsteady boundary layers in the class considered by oleinik j appl math mech 30 1966 951 provided that the pressure is favourable this generalizes the local well posedness results due to oleinik 1966 mathematical models in boundary layer theory chapman hall london 1999 for the proof we introduce a viscous splitting method so that the asymptotic behaviour of the solution near the boundary can be estimated more accurately by methods applicable to the degenerate parabolic equations

local in time well posedness for compressible mhd boundary layer abstract in this paper we are concerned with the motion of electrically conducting fluid governed by the two dimensional non isentropic viscous compressible mhd system on the half plane with no slip condition on the velocity field perfectly conducting wall condition on the magnetic field and dirichlet boundary condition on the temperature on the boundary when the viscosity heat conductivity and magnetic diffusivity coefficients tend to zero in the same rate there is a boundary layer which is described by a prandtl type system under the non degeneracy condition on the tangential magnetic field instead of monotonicity of velocity by applying a coordinate transformation in terms of the stream function of magnetic field as motivated by the recent work 27 we obtain the local in time well posedness of the boundary layer system in weighted sobolev spaces

a new approach to the bi dimensional representation of the dea efficient frontier with multiple inputs and outputs abstract this paper presents a new approach to the graphical presentation of dea results whatever the number of inputs and outputs are an adequate normalization of their weights is enough to generate a simple bi dimensional graph similar to that of the ccr frontier with one input and one output an advantage over other approaches to the same representation problem is that no complementary techniques are required to plot the frontier it is also proved that the distance of a dmu to the frontier is related to its efficiency the proposed approach is also valid for the bcc model

a borel transform method for locating singularities of taylor and fourier series given a taylor series with a finite radius of convergence its borel transform defines an entire function a theorem of polya relates the large distance behavior of the borel transform in different directions to singularities of the original function with the help of the new asymptotic interpolation method of van der hoeven we show that from the knowledge of a large number of taylor coefficients we can identify precisely the location of such singularities as well as their type when they are isolated there is no risk of getting artefacts with this method which also gives us access to some of the singularities beyond the convergence disk the method can also be applied to fourier series of analytic periodic functions and is here tested on various instances constructed from solutions to the burgers equation large precision on scaling exponents up to twenty accurate digits can be achieved

a distance friction minimization approach in data envelopment analysis a comparative study on airport efficiency abstract this paper aims to present a newly developed distance friction minimization dfm method in the context of data envelopment analysis dea in order to generate an appropriate non radial efficiency improving projection model for both input reduction and output increase in this approach a generalized distance function based on a euclidean distance metric in weighted spaces is proposed to assist a decision making unit dmu to improve its performance by an appropriate movement towards the efficiency frontier surface a suitable form of multidimensional projection function for efficiency improvement is given by a multiple objective quadratic programming moqp model the paper describes the various steps involved in a systematic manner the above mentioned dfm model is illustrated empirically by using a data set on 30 european airports where the aim is to present a comparative analysis of the efficiency of operational management in these airports in addition the comparative analysis of these airports is able to assess both input slacks and output slacks or a combination of input reduction and output rise