almost exponential decay of periodic viscous surface waves without surface tension we consider a viscous fluid of finite depth below the air occupying a three dimensional domain bounded below by a fixed solid boundary and above by a free moving boundary the fluid dynamics are governed by the gravity driven incompressible navier stokes equations and the effect of surface tension is neglected on the free surface the long time behavior of solutions near equilibrium has been an intriguing question since the work of beale commun pure appl math 34 3 359 392 1981 this paper is the third in a series of three guo in local well posedness of the viscous surface wave problem without surface tension anal pde 2012 to appear in decay of viscous surface waves without surface tension in horizontally infinite domains preprint 2011 that answers this question here we consider the case in which the free interface is horizontally periodic we prove that the problem is globally well posed and that solutions decay to equilibrium at an almost exponential rate in particular the free interface decays to a flat surface our framework contains several novel techniques which include 1 a priori estimates that utilize a geometric reformulation of the equations 2 a two tier energy method that couples the boundedness of high order energy to the decay of low order energy the latter of which is necessary to balance out the growth of the highest derivatives of the free interface 3 a localization procedure that is compatible with the energy method and allows for curved lower surface geometry our decay estimates lead to the construction of global in time solutions to the surface wave problem

growth rates for the linearized motion of fluid interfaces away from equilibrium we consider the motion of a two dimensional interface separating an inviscid incompressible irrotational fluid influenced by gravity from a region of zero density we show that under certain conditions the equations of motion linearized about a presumed time dependent solution are wellposed that is linear disturbances have a bounded rate of growth if surface tension is neglected the linear equations are well posed provided the underlying exact motion satisfies a condition on the acceleration of the interface relative to gravity similar to the criterion formulated by g i taylor if surface tension is included the linear equations are well posed without qualifications whether the fluid is above or below the interface an interesting qualitative structure is found for the linear equations a lagrangian approach is used like that of numerical work such as 3 except that the interface is assumed horizontal at infinity certain integral equations which occur involving double layer potentials are shown to be solvable in the present case 1993 john wiley sons inc

a remark on the two dimensional water wave problem with surface tension abstract we consider the motion of a two dimensional interface between air above and an irrotational incompressible inviscid infinitely deep water below with surface tension present we propose a new way to reduce the original problem into an equivalent quasilinear system which is related to the interface s tangent angle and a quantity related to the difference of tangential velocities of the interface in the lagrangian and the arc length coordinates the new way is relatively simple because it involves only taking differentiation and the real and the imaginary parts then if assuming that waves are periodic we establish a priori energy inequality

the clifford hilbert transform and riemann hilbert problems associated to the dirac operator in lebesgue spaces abstract in this paper we establish some properties of the hilbert transform in clifford analysis setting and mainly show the weak type 1 1 inequality for the clifford hilbert transform in euclidean space r m m 3 using the clifford algebra and analysis generalizations of the one dimensional proofs furthermore we prove kolmogorov s inequality for clifford hilbert transform with the help of the properties for the hilbert transform and clifford analytic techniques we establish the riemann hilbert problems in clifford value lebesgue p integrable spaces based on the newton embedding method we prove an existence and uniqueness for the nonlinear riemann hilbert problem and also give an error estimation for the approximate solutions in the newton embedding procedure in lebesgue spaces

zero surface tension limit of viscous surface waves we consider the free boundary problem for a layer of viscous incompressible fluid in a uniform gravitational field lying above a rigid bottom and below the atmosphere for the semi small initial data we prove the zero surface tension limit of the problem within a local time interval the unique local strong solution with surface tension is constructed as the limit of a sequence of approximate solutions to a special parabolic regularization for the small initial data we prove the global in time zero surface tension limit of the problem

capacitated facility location separation algorithms and computational experience we consider the polyhedral approach to solving the capacitated facility location problem the valid inequalities considered are the knapsack cover flow cover effective capacity single depot and combinatorial inequalities the flow cover effective capacity and single depot inequalities form subfamilies of the general family of submodular inequalities the separation problem based on the family of submodular inequalities is np hard in general for the well known subclass of flow cover inequalities however we show that if the client set is fixed and if all capacities are equal then the separation problem can be solved in polynomial time for the flow cover inequalities based on an arbitrary client set and general capacities and for the effective capacity and single depot inequalities we develop separation heuristics an important part of these heuristics is based on the result that two specific conditions are necessary for the effective cover inequalities to be facet defining the way these results are stated indicates precisely how structures that violate the two conditions can be modified to produce stronger inequalities the family of combinatorial inequalities was originally developed for the uncapacitated facility location problem but is also valid for the capacitated problem no computational experience using the combinatorial inequalities has been reported so far here we suggest how partial output from the heuristic identifying violated submodular inequalities can be used as input to a heuristic identifying violated combinatorial inequalities we report on computational results from solving 60 medium size problems

fixed points for fuzzy mappings in this paper the problem of the existence and computation of fixed points for fuzzy mappings is approached a fuzzy mapping r over a set x is defined to be a function attaching to each x in x a fuzzy subset r of x an element x of x is called fixed point of r iff its membership degree to r is at least equal to the membership degree to r of any y x i e r r y y x two existence theorems for fixed points of a fuzzy mapping are proved and an algorithm for computing approximations of such a fixed point is described the convergence theorem of our algorithm is proved under the restrictive assumption that for any x in x the membership function of r has a complementary function examples of fuzzy mappings having this property are given but the problem of proving general criteria for a function to have a complementary remain open

characteristic boundary layers in real vanishing viscosity limits abstract in this paper we study the vanishing viscosity limit of initial boundary value problems for one dimensional mixed hyperbolic parabolic systems when the boundary is characteristic for both the viscous and the inviscid systems in particular we assume that an eigenvalue of the inviscid system vanishes uniformly we prove the stability of boundary layers expansions in small time i e before shocks for the inviscid system as long as the amplitude of the boundary layers remains sufficiently small in particular by using lagrangian coordinates we apply our result to physical systems like gasdynamics and magnetohydrodynamics with homogeneous dirichlet condition for the velocity at the boundary

well posedness in smooth function spaces for moving boundary 1 d compressible euler equations in physical vacuum we prove well posedness for the 3 d compressible euler equations with moving physical vacuum boundary with an equation of state given by the so called gamma gas law for gamma 1 the physical vacuum singularity requires the sound speed c to go to zero as the square root of the distance to the moving boundary and thus creates a degenerate and characteristic hyperbolic free boundary system wherein the density vanishes on the free boundary the uniform kreiss lopatinskii condition is violated and manifest derivative loss ensues nevertheless we are able to establish the existence of unique solutions to this system on a short time interval which are smooth in sobolev spaces all the way to the moving boundary and our estimates have no derivative loss with respect to initial data our proof is founded on an approximation of the euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term chosen to preserve as much of the geometric structure of the euler equations as possible we first construct solutions to this degenerate parabolic regularization using a new higher order hardy type inequality we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter solutions to the compressible euler equations are found in the limit as the artificial viscosity tends to zero our regular solutions can be viewed as degenerate viscosity solutions out methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws

boundary layers associated with incompressible navier stokes equations the noncharacteristic boundary case abstract the goal of this article is to study the boundary layer of wall bounded flows in a channel at small viscosity when the boundaries are uniformly noncharacteristic i e there is injection and or suction everywhere at the boundary following earlier work on the boundary layer for linearized navier stokes equations in the case where the boundaries are characteristic no slip at the boundary and non permeable we consider here the case where the boundary is permeable and thus noncharacteristic the form of the boundary layer and convergence results are derived in two cases linearized equation and full nonlinear equations we prove that there exists a boundary layer at the outlet downwind of the form e uz e where u is the speed of injection suction at the boundary z is the distance to the outlet of the channel and e is the kinematic viscosity we improve an earlier result of s n alekseenko 1994 siberian math j 35 no 2 209 230 where the convergence in l 2 of the solutions of the navier stokes equations to that of the euler equations at vanishing viscosity was established in the two dimensional case we are able to derive the physically relevant uniform in space l norm estimates of the boundary layer the uniform in space estimate is derived by properly developing our previous idea of better control on the tangential derivative and the use of an anisotropic sobolev imbedding to the best of our knowledge this is the first rigorously proved result concerning boundary layers for the full nonlinear navier stokes equations for incompressible fluids

the rigorous approximation of long wavelength capillary gravity waves in a previous paper we proved that long wavelength solutions of the water wave problem in the case of zero surface tension split up into two wave packets one moving to the right and one to the left where each of these wave packets evolves independently as a solution of a korteweg de vries kdv equation in this paper we examine the effect of surface tension on this scenario we find that we obtain three different physical regimes depending on the strength of the surface tension for weak surface tension the propagation of the wave packets is very similar to that in the zero surface tension case for strong surface tension the evolution is again governed by a pair of kdv equations but the coefficients in these equations have changed in such a way that the kdv soliton now represents a wave of depression on the fluid surface finally at a special intermediate value of the surface tension where the bond number equals the kdv description breaks down and it is necessary to introduce a new approximating equation the kawahara equation which is a fifth order nonlinear partial differential equation in each of these regimes we give rigorous estimates of the difference between the solution of the appropriate modulation equation and the solution of the true water wave problem

on the regularity of the flow map for the gravity capillary equations abstract we prove via explicitly constructed initial data that solutions to the gravity capillary wave system in r 3 representing a 2 d air water interface immediately fail to be c 3 with respect to the initial data if the initial data h 0 0 h s 1 2 h s for s 3 similar results hold in r 2 domains with a 1 d interface furthermore we discuss the related threshold for the pure gravity water wave system

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

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

well posedness for the motion of an incompressible liquid with free surface boundary we study the motion of an incompressible perfect liquid body in vacuum this can be thought of as a model for the motion of the ocean or a star the free surface moves with the velocity of the liquid and the pressure vanishes on the free surface this leads to a free boundary problem for euler s equations where the regularity of the boundary enters to highest order we prove local existence in sobolev spaces assuming a physical condition related to the fact that the pressure of a fluid has to be positive

global solutions for the gravity water waves equation in dimension 3 we show existence of global solutions for the gravity water waves equation in dimension 3 in the case of small data the proof combines energy estimates which yield control of l 2 related norms with dispersive estimates which give decay in l 8 to obtain these dispersive estimates we use an analysis in fourier space the study of space and time resonances is then the crucial point

sequence independent lifting for mixed integer programs with variable upper bounds we investigate the convex hull of the set defined by a single inequality with continuous and binary variables with variable upper bound constraints we extend the traditional flow cover inequality and show that it is valid for a restriction of the set in which some variables are fixed we also give conditions under which this inequality is facet defining and when it is not we show how it can be lifted to obtain valid inequalities for the entire set using sequence independent lifting in general computing the lifting function is np hard but under an additional restriction on the cover we obtain a closed form finally we show how these results imply and extend known results about the single node fixed charge flow polyhedron

zero surface tension limit of the free surface incompressible euler equations with damping abstract we consider the free boundary problem for a layer of an incompressible inviscid fluid in a uniform gravitational field lying above a rigid bottom and below the atmosphere in a horizontally periodic setting we establish the global in time zero surface tension limit of the problem with damping for the small initial data

oscillations and concentrations in weak solutions of the incompressible fluid equations the authors introduce a new concept of measure valued solution for the 3 d incompressible euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations one application of the concepts developed here is the following important result a sequence of leray hopf weak solutions of the navier stokes equations converges in the high reynolds number limit to a measure valued solution of 3 d euler defined for all positive times the authors present several explicit examples of solution sequences for 3 d incompressible euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations an extensions of the concept of young measure is developed to incorporate these complex phenomena in the measure valued solutions constructed here

on the cauchy problem for gravity water waves we are interested in the system of gravity water waves equations without surface tension our purpose is to study the optimal regularity thresholds for the initial conditions in terms of sobolev embeddings the initial surfaces we consider turn out to be only of c 3 2 epsilon class for some epsilon 0 and consequently have unbounded curvature while the initial velocities are only lipschitz we reduce the system using a paradifferential approach

well posedness of the free boundary problem in incompressible elastodynamics under the mixed type stability condition abstract we consider the free boundary problem for the flow of an incompressible inviscid elastic fluid the columns of the deformation gradient are tangent and the pressure vanishes along the free interface we prove the local existence of a unique smooth solution to the free boundary problem under the mixed type stability condition provided that the rayleigh taylor sign condition is satisfied at all the points of the initial interface where the non collinearity condition for the deformation gradient among three columns of the deformation gradient there are two non collinear vectors fails in particular we solve an open problem proposed by y trakhinin in the paper 29 under the incompressible setting

the free boundary euler equations with large surface tension abstract we study the free boundary euler equations with surface tension in three spatial dimensions showing that the equations are well posed if the coefficient of surface tension is positive then we prove that under natural assumptions the solutions of the free boundary motion converge to solutions of the euler equations in a domain with fixed boundary when the coefficient of surface tension tends to infinity

large time existence for 1d green naghdi equations we consider here the 1d green naghdi equations that are commonly used in coastal oceanography to describe the propagation of large amplitude surface waves we show that the solution of the green naghdi equations can be constructed by a standard picard iterative scheme so that there is no loss of regularity of the solution with respect to the initial condition

well posedness in smooth function spaces for the moving boundary three dimensional compressible euler equations in physical vacuum we prove well posedness for the three dimensional compressible euler equations with moving physical vacuum boundary with an equation of state given by p c for 1 the physical vacuum singularity requires the sound speed c to go to zero as the square root of the distance to the moving boundary and thus creates a degenerate and characteristic hyperbolic free boundary system wherein the density vanishes on the free boundary the uniform kreiss lopatinskii condition is violated and manifest derivative loss ensues nevertheless we are able to establish the existence of unique solutions to this system on a short time interval which are smooth in sobolev spaces all the way to the moving boundary and our estimates have no derivative loss with respect to initial data our proof is founded on an approximation of the euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term chosen to preserve as much of the geometric structure of the euler equations as possible we first construct solutions to this degenerate parabolic regularization using a higher order version of hardy s inequality we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter solutions to the compressible euler equations are found in the limit as the artificial viscosity tends to zero our regular solutions can be viewed as degenerate viscosity solutions our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws

darcy s flow with prescribed contact angle well posedness and lubrication approximation we consider the spreading of a thin two dimensional droplet on a solid substrate we use a model for viscous fluids where the evolution is governed by darcy s law at the contact point where air and liquid meet the solid substrate a constant non zero contact angle partial wetting is assumed we show local and global well posedness of this free boundary problem in the presence of the moving contact point our estimates are uniform in the contact angle assumed by the liquid at the contact point in the so called lubrication approximation long wave limit we show that the solutions converge to the solution of a one dimensional degenerate parabolic fourth order equation which belongs to a family of thin film equations the main technical difficulty is to describe the evolution of the non smooth domain and to identify suitable spaces that capture the transition to the asymptotic model uniformly in the small parameter varepsilon

on capacitated network design cut set polyhedra this paper provides an analysis of capacitated network design cut set polyhedra we give a complete linear description of the cut set polyhedron of the single commodity single facility capacitated network design problem then we extend the analysis to single commodity multifacility and multicommodity multifacility capacitated network design problems valid inequalities described here are applicable to directed network design problems with any number of facility types and any level of capacities we report results from a computational study done for testing the effectiveness of the new inequalities

global regularity of 2d density patches for inhomogeneous navier stokes this paper is about lions open problem on density patches lions in mathematical topics in fluid mechanics vol 1 volume 3 of oxford lecture series in mathematics and its applications clarendon press oxford university press new york 1996 whether or not inhomogeneous incompressible navier stokes equations preserve the initial regularity of the free boundary given by density patches using classical sobolev spaces for the velocity we first establish the propagation of c 1 gamma regularity with 0 gamma 1 in the case of positive density furthermore we go beyond this to show the persistence of a geometrical quantity such as the curvature in addition we obtain a proof for c 2 gamma regularity

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

existence and stability of noncharacteristic boundary layers for the compressible navier stokes and viscous mhd equations for a general class of hyperbolic parabolic systems including the compressible navier stokes and compressible mhd equations we prove existence and stability of noncharacteristic viscous boundary layers for a variety of boundary conditions including classical navier stokes boundary conditions our first main result using the abstract framework established by the authors in the companion work gues et al in j differ equ 244 309 387 2008 is to show that existence and stability of arbitrary amplitude exact boundary layer solutions follow from a uniform spectral stability condition on layer profiles that is expressible in terms of an evans function uniform evans stability whenever this condition holds we give a rigorous description of the small viscosity limit as the solution of a hyperbolic problem with residual boundary conditions our second is to show that uniform evans stability for small amplitude layers is equivalent to evans stability of the limiting constant layer which in turn can be checked by a linear algebraic computation finally for a class of symmetric dissipative systems including the physical examples mentioned above we carry out energy estimates showing that constant and thus small amplitude layers always satisfy uniform evans stability this yields existence of small amplitude multi dimensional boundary layers for the compressible navier stokes and mhd equations for both equations these appear to be the first such results in the compressible case

boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems abstract in this paper we study viscous perturbations of quasilinear hyperbolic systems in several dimensions as the viscosity goes to zero the boundary is noncharacteristic for the hyperbolic system we in particular describe the boundary layer which arises near the boundary and give a sufficient condition for the convergence of the solution to the solution of some mixed hyperbolic problem with some nonlinear maximal dissipative boundary conditions a counterexample is given when this condition is not satisfied and the solution blows up as the viscosity goes to 0

decay rates for the viscous incompressible mhd equations with and without surface tension abstract in this paper we consider a layer of a viscous incompressible electrically conducting fluid interacting with the magnetic field in a horizontally periodic setting the upper boundary is bounded by a free boundary and below bounded by a flat rigid interface we prove the global well posedness of the problem for both the case with and without surface tension moreover we show that the global solution decays to the equilibrium exponentially in the case with surface tension however the global solution decays to the equilibrium at an almost exponential rate in the case without surface tension

valid linear inequalities for fixed charge problems many problems in the operations research management science literature can be formulated with both zero one and continuous variables however the exact optimization of such mixed zero one models remains a computational challenge in this paper we propose to study mixed problems from a mathematical point of view that is similar in spirit to recent research on purely combinatorial problems that has investigated systems of defining linear inequalities or facets of the underlying polytope at least two numerical studies have validated this line of research computationally and the advances in the problem solving capabilities are considerable we expect that similar gains are possible for the mixed zero one problem more precisely we consider the mixed integer programs whose feasible region x is composed of i a simple additive constraint in the continuous variables xj for j 1 2 n and ii constraints 0 xj mjyj defined by binary variables yj for j 1 2 n this type of feasible region arises in a variety of mixed integer problems and particularly in network problems with fixed charges on the arcs we derive two classes of facet defining linear inequalities of the convex hull of x and show that the second of these classes gives a complete description of the convex hull when mj m for all j we also develop methods to detect violated inequalities from these classes so that these facets can be used as cutting planes to strengthen the formulations of certain mixed integer problems

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

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

global well posedness of the free surface incompressible euler equations with damping abstract we consider a layer of an incompressible inviscid fluid bounded below by a fixed solid boundary and above by a free moving boundary in a horizontally periodic setting the fluid dynamics is governed by the gravity driven incompressible euler equations with damping and the effect of surface tension is neglected on the free surface we prove that the problem is globally well posed for the small initial data and that solutions decay to the equilibrium at an almost exponential rate

fuzzy expected utility abstract the aim of this paper is to state the foundations of fuzzy decision theory under uncertainty attention is focused on the analysis of a fuzzy expected utility given a fuzzy decision problem which does not rest on a borel field but on a weak borel field and according to the information available we shall use successively the probability and the possibility of a fuzzy subset when the imprecise preference structure of the decision maker verifies four axioms we can determine a function of fuzzy expected utility this latter is a numerical translation of the imprecise preferences of the decision maker in an uncertain world

nonlinear stability of the current vortex sheet to the incompressible mhd equations in this paper we solve a long standing open problem nonlinear stability of the current vortex sheet in the ideal incompressible magnetohydrodynamics under the syrovatskij stability condition this result gives the first rigorous confirmation of the stabilizing effect of the magnetic field on kelvin helmholtz instability 2017 wiley periodicals inc

the zero surface tension limit of two dimensional water waves we consider two dimensional water waves of infinite depth periodic in the horizontal direction it has been proven by wu in the slightly different nonperiodic setting that solutions to this initial value problem exist in the absence of surface tension recently ambrose has proven that solutions exist when surface tension is taken into account in this paper we provide a shorter more elementary proof of existence of solutions to the water wave initial value problem both with and without surface tension our proof requires estimating the growth of geometric quantities using a renormalized arc length parametrization of the free surface and using physical quantities related to the tangential velocity of the free surface using this formulation we find that as surface tension goes to 0 the water wave without surface tension is the limit of the water wave with surface tension far from being a simple adaptation of previous works our method requires a very original choice of variables these variables turn out to be physical and well adapted to both cases c

nash fuzzy equilibrium theory and application to a spatial duopoly abstract consider a non cooperative n persons game each gambler has a set of mixed strategies at his disposal the payoffs are some physical or immaterial objects the game is a fuzzy game because 1 gamblers have more or less precise preferences for the payoffs and 2 the outcoming of payoffs is uncertain the uncertainty can be expressed either by a distribution of possibility or by a distribution of probability the product set of a gambler s mixed strategies is convex and compact and the payoff functions are continuous then a closed and convex fuzzy point to set mapping is defined on the product set of strategies and by using a butnariu theorem the existence of a fixed point for this fuzzy point to set mapping is proved the issue allows us to generalize a famous nash result a n persons non cooperative fuzzy game with mixed strategies has at least one equilibrium point in the second part of the paper an economic application is devoted to the statement of the equilibrium existence conditions in a spatial duopoly the model is not only more general than the classical ones but also more relevant because new results are obtained

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

water wave propagation in unbounded domains part ii numerical methods for fractional pdes this paper concerns numerical solutions for a fractional partial differential equation arising as a nonreflecting boundary condition in water wave propagation the fractional derivative operator is written as divergence of a singular integrable convolution which allows the equation to be viewed as a conservation law with a linear nonlocal flux a semi discrete finite volume scheme is presented using conservative piecewise polynomial reconstruction of the solution the convolution with the singular kernel is then integrated exactly time integration uses runge kutta schemes of matching order stability is discussed convergence is established and numerical examples are presented

uniform regularity for the navier stokes equation with navier boundary condition we prove that there exists an interval of time which is uniform in the vanishing viscosity limit and for which the navier stokes equation with the navier boundary condition has a strong solution this solution is uniformly bounded in a conormal sobolev space and has only one normal derivative bounded in l this allows us to obtain the vanishing viscosity limit to the incompressible euler system from a strong compactness argument

global wellposedness of the 3 d full water wave problem we consider the problem of global in time existence and uniqueness of solutions of the 3 d infinite depth full water wave problem in the setting that the interface tends to the horizontal plane the velocity and acceleration on the interface tend to zero at spatial infinity we show that the nature of the nonlinearity of the water wave equation is essentially of cubic and higher orders for any initial interface that is sufficiently small in its steepness and velocity we show that there exists a unique smooth solution of the full water wave problem for all time and the solution decays at the rate 1 t

uniform regularity and vanishing viscosity limit for the free surface navier stokes equations we study the inviscid limit of the free boundary navier stokes equations we prove the existence of solutions on a uniform time interval by using a suitable functional framework based on sobolev conormal spaces this allows us to use a strong compactness argument to justify the inviscid limit our approach does not rely on the justification of asymptotic expansions in particular we get a new existence result for the euler equations with free surface from the one for navier stokes

non cooperative fuzzy games in normal form a survey in this paper we present a review of most of the approaches for solving non cooperative fuzzy games in normal form applications of these games are reviewed as well we also present a new class of fuzzy games in normal form the class of non cooperative games involving fuzzy parameters finally some potential directions of research are provided throughout the paper and in the conclusion

remarks about the inviscid limit of the navier stokes system in this paper we prove two results about the inviscid limit of the navier stokes system the first one concerns the convergence in hs of a sequence of solutions to the navier stokes system when the viscosity goes to zero and the initial data is in hs the second result deals with the best rate of convergence for vortex patch initial data in 2 and 3 dimensions we present here a simple proof which also works in the 3d case the 3d case is new

almost global wellposedness of the 2 d full water wave problem we consider the problem of global in time existence and uniqueness of solutions of the 2 d infinite depth full water wave equation it is known that this equation has a solution for a time period 0 t e for initial data of the form e where t depends only on in this paper we show that for such data there exists a unique solution for a time period 0 e t e this is achieved by better understandings of the nature of the nonlinearity of the full water wave equation

stability of one dimensional boundary layers by using green s functions the aim of this paper is to investigate the stability of one dimensional boundary layers of parabolic systems as the viscosity goes to 0 in the noncharacteristic case and more precisely to prove that spectral stability implies linear and nonlinear stability of approximate solutions in particular we replace the smallness condition obtained by the energy method 10 13 by a weaker spectral condition 2001 john wiley sons inc

global solutions for the gravity water waves system in 2d we consider the gravity water waves system in the case of a one dimensional interface for sufficiently smooth and localized initial data and prove global existence of small solutions this improves the almost global existence result of wu invent math 177 1 45 135 2009 we also prove that the asymptotic behavior of solutions as time goes to infinity is different from linear unlike the three dimensional case germain et al ann math 175 2 691 754 2012 wu invent math 184 1 125 220 2011 in particular we identify a suitable nonlinear logarithmic correction and show modified scattering the solutions we construct in this paper appear to be the first global smooth nontrivial solutions of the gravity water waves system in 2d

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

well posedness in sobolev spaces of the full water wave problem in 2 d we consider the motion of the interface of 2 d irrotational incompressible inviscid water wave with air above water and surface tension zero we show that the interface is always not accelerating into the water region normal to itself as rapidly as the normal acceleration of gravity as long as the interface is nonself intersect we therefore obtain the existence and uniqueness of solutions of the full water wave problem locally in time for any initial interface which is nonself intersect including the case that the interface is of multiple heights

well posedness of the free surface incompressible euler equations with or without surface tension we provide a new method for treating free boundary problems in perfect fluids and prove local in time well posedness in sobolev spaces for the free surface incompressible 3d euler equations with or without surface tension for arbitrary initial data and without any irrotationality assumption on the fluid this is a free boundary problem for the motion of an incompressible perfect liquid in vacuum wherein the motion of the fluid interacts with the motion of the free surface at highest order

well posedness of the water waves equations 1 1 presentation of the problem the water waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect incompressible irrotational fluid under the influence of gravity in this paper we restrict our attention to the case when the surface is a graph parameterized by a function x where t denotes the time variable and x xi xd rd the horizontal spatial variables the method developed here works equally well for any integer d 1 but the only physically relevant cases are of course d 1 and d 2 the layer of fluid is also delimited from below by a not necessarily flat bottom parameterized by a time independent function b x we denote by q t the fluid domain at time t the incompressibility of the fluids is expressed by 1 1 div v 0 in qu t 0

symmetric positive systems with boundary characteristic of constant multiplicity the theory of maximal positive boundary value problems for symmetric positive systems is developed assuming that the boundary is characteristic of constant multiplicity no such hypothesis is needed on a neighborhood of the boundary both regularity theorems and mixed initial boundary value problems are discussed many classical ideas are sharpened in the process

lifted flow cover inequalities for mixed 0 1 integer programs we investigate strong inequalities for mixed 0 1 integer programs derived from flow cover inequalities flow cover inequalities are usually not facet defining and need to be lifted to obtain stronger inequalities however because of the sequential nature of the standard lifting techniques and the complexity of the optimization problems that have to be solved to obtain lifting coefficients lifting of flow cover inequalities is computationally very demanding we present a computationally efficient way to lift flow cover inequalities based on sequence independent lifting techniques and give computational results that show the effectiveness of our lifting procedures

small time existence for the three dimensional navier stokes equations for an incompressible fluid with a free surface in 1980 beale 2 studied the problem of describing the motion of a layer of heavy viscous incompressible fluid lying above an infinite rigid bottom and having a non compact free surface with no surface tension he proved its local solvability in an anisotropic sobolev space for any initial data its global solvability was discussed by sylv sxer 20 but a crucial point of her proof does not appear clear to me the regularity of the free surface with surface tension taken into account beale 31 and beale nishida 4 respectively studied the large time existence and regularity and the large time behavior of the solution to this problem with initial data near equilibrium teramoto 24 25 considered these problems for fluids lying above an inclined plane for any initial data the same problem was solved locally in time by allain 11 in the two dimensional case the aim of this paper is to establish the analogous result in the three dimensional case a similar problem describing the motion of a finite isolated mass of incompressible viscous fluid was studied thoroughly by solonnikov he proved local solvability in a h61der space in 12 and global solvability in the space w 2 1 p n where n is the dimension of the domain in 16 without surface tension and in 13 15 17 19 21 with surface tension let us formulate our problem given an initial domain f r 3 with x3 being the vertical component and an initial velocity vector field vo in f we want to know the domain f t t 0 occupied by the fluid which is bounded by the fixed bottom sb and the free surface se t the velocity vector field v v x t vl v2 v3 and the pressure p p x t so that

on the free boundary problem of three dimensional incompressible euler equations in this paper we consider in three dimensions the motion of a general inviscid incompressible fluid with a free interface that separates the fluid region from the vacuum we assume that the fluid region is below the vacuum and that there is no surface tension on the free surface then we prove the local well posedness of the free boundary problem in sobolev space provided that there is no self intersection point on the initial surface and under the stability assumption that with being restricted to the initial surface 2007 wiley periodicals inc

well posedness in sobolev spaces of the full water wave problem in 3 d we consider the motion of the interface separating an inviscid incompressible irrotational fluid from a region of zero density in three dimensional space we assume that the fluid region is below the vacuum the fluid is under the influence of gravity and the surface tension is zero assume that the density of mass of the fluid is one the gravitational field is 0 0 1 the free interface is t at time t 0 and the fluid occupies the region t the motion of the fluid is described by vt v v 0 0 1 p on t t 0 euler s equation 1 1 div v 0 on t t 0 incompressible 1 2 curl v 0 on t t 0 irrotational 1 3 where v v1 v2 v3 is the fluid velocity p is the fluid pressure since we neglect the surface tension the pressure is zero on the interface so on the interface p 0 on t 1 4 1 v is tangent to the free surface t t 1 5 we want to find solutions of system 1 1 1 5 taking prescribed initial data such that for every fixed t 0 t approaches the xy plane at infinity and v x y z t 0 vt x y z t 0 as x y z since the fluid is assumed irrotational incompressible we can reduce the study of the entire motion to the motion of the free surface the above model is a 3 d water wave model it is generally known that when surface tension is neglected the motion of the interface between an inviscid fluid and vacuum under the influence of gravity can be subject to taylor instability 8 22 in a previous work 24 we studied the 2 dimensional water wave model we showed that for a 2 d water wave the sign condition relating to taylor instability always holds for nonself intersecting interface that is the motion of the interface is not subject to taylor instability we showed further that the 2 d full nonlinear water wave problem is uniquely solvable in sobolev spaces locally in time for any initially nonself intersecting interface earlier works on the well posedness in sobolev spaces of the 2 d water wave problem include nalimov 18 yosihara 25 and walter craig 6 where the main results concern the well posedness of the

a polyhedral study of integer variable upper bounds we study the polyhedron of the single node capacitated network design model with integer variable upper bounds we first give a characterization of valid inequalities that is useful in proving the validity of several classes of inequalities next we derive several classes of valid inequalities and we give conditions for them to be facet defining sequence independent lifting is used to obtain additional facets we conclude by reporting computational results with a branch and cut algorithm

geometry and a priori estimates for free boundary problems of the euler s equation in this paper we derive estimates to the free boundary problem for the euler equation with surface tension and without surface tension provided the rayleigh taylor sign condition holds we prove that as the surface tension tends to 0 when the rayleigh taylor condition is satisfied solutions converge to the euler flow with zero surface tension 2007 wiley periodicals inc

global existence for capillary water waves consider the capillary water waves equations set in the whole space with infinite depth and consider small data i e sufficiently close to zero velocity and constant height of the water we prove global existence and scattering the proof combines in a novel way the energy method with a cascade of energy estimates the space time resonance method and commuting vector fields 2015 wiley periodicals inc

the cauchy integral calderon commutators and conjugations of singular integrals in ⁿ we consider the cauchy integral and hilbert transform for lipschitz domains in the clifford algebra based on r 1 the hilbert transform is shown to be the generating function for the calderon commutators in r 1 we make use of an intrinsic characterization of these commutators to obtain l2 estimates these estimates are used to show the analyticity of the hilbert transform and of the conjugation of singular integral operators by bi lipschitz changes of variable in r 1

valid inequalities for problems with additive variable upper bounds we study the facial structure of a polyhedron associated with the single node relaxation of network flow problems with additive variable upper bounds this type of structure arises for example in production planning problems with setup times and in network certain expansion problems we derive several classes of valid inequalities for this polyhedron and give conditions under which they are facet defining our computational experience with large network expansion problems indicates that these inequalities are very effective in improving the quality of the linear programming relaxations