on the vanishing viscosity limit for the 3d navier stokes equations with a slip boundary condition where and below and denote the div and curl operators respectively n is the outward normal and is the unit tangential vector of the investigation of vanishing viscosity limit of solutions of the navier stokes equations both in the two and three spacial dimensional cases is a classical issue there are two related questions arising from here one is how to describe the inviscid limiting behavior of the navier stokes equation and the other is that does the euler equation can be approximated by the navier stokes equations in the case that the solution to the this research is supported in part by zheng ge ru foundation and hong kong rgc earmarked research grants cuhk 4028 04p cuhk 4040 02p and cuhk 4279 00p

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

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

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

geometric regularity criteria for incompressible navier stokes equations with navier boundary conditions abstract we study the regularity criteria for weak solutions to the 3d incompressible navier stokes equations in terms of the direction of vorticity taking into account the boundary conditions a boundary regularity theorem is proved on regular curvilinear domains with a family of oblique derivative boundary conditions provided that the directions of vorticity are coherently aligned up to the boundary as an application we establish the boundary regularity for weak solutions to navier stokes equations in round balls half spaces and right circular cylindrical ducts subject to the classical navier and kinematic boundary conditions

uniform time of existence for the alpha euler equations we consider the euler equations on a bounded three dimensional domain with frictionless navier boundary conditions our main result is the existence of a strong solution on a positive time interval uniform in for sufficiently small combined with the convergence result in 6 this implies convergence of solutions of the euler equations to solutions of the incompressible euler equations when 0 in addition we obtain a new result on local existence of strong solutions for the incompressible euler equations on bounded three dimensional domains the proofs are based on new a priori estimates in conormal spaces

quasi neutral and zero viscosity limits of navier stokes poisson equations in the half space abstract the present paper is concerned with the quasi neutral and zero viscosity limits of navier stokes poisson equations in the half space we consider the navier slip boundary condition for velocity and dirichlet boundary condition for electric potential by means of asymptotic analysis with multiple scales we construct an approximate solution of the navier stokes poisson equations involving two different kinds of boundary layer and establish the linear stability of the boundary layer approximations by conormal energy estimate

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

uniform regularity estimates of solutions to three dimensional incompressible magnetic benard equations with navier slip type boundary conditions in half space abstract in this paper we are concerned with the uniform estimates of solutions to three dimensional 3d incompressible magnetic benard equations with navier slip type boundary conditions being imposed on both velocity and magnetic field in half space where the physical boundary is assumed to be insulation under the assumption that the viscosity and resistivity coefficients are same which are parameterized by a small parameter of e the uniform estimates of solutions to the viscous magnetic benard system of equations are established in the conormal sobolev space which is independent of e as a direct consequence the inviscid type limit between the solutions to viscous magnetic benard system and the ideal magnetic benard system is proved rigorously

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

maximal positive boundary value problems as limits of singular perturbation problems we study three types of singular perturbations of a symmetric positive system of partial differential equations on a domain 2 c r in all cases the limiting behavior is given by the solution of a maximal positive boundary value problem in the sense of friedrichs the perturbation is either a second order elliptic term or a term large on the complement of q the first corresponds to a sort of viscosity and the second to physical systems with vastly different properties in i2 and outside u the results show that in the limit of zero viscosity or infinitely large difference the behavior is described by a maximal positive boundary value problem in 2 the boundary condition is determined in a simple way from the system and the singular terms 1 introduction in this paper we study boundary value roblems for singularly perturbed systems of partial differential equations the unperturbed system is of first order and positive symmetric in the sense of friedrichs we study three sorts of singular perturbations 1 the addition of e times a positive second order elliptic system in a domain fi 2 3 the addition of xp or p d dt with a 1 and p strictly positive in the exterior of n and zero in s2 in the first case we study the solutions he to the dirichlet problem in s2 bardos brezis and brezis 1 have shown that in this case ue u weakly in l2 fi where u is uniquely determined as the solution of a maximal positive boundary value problem for the unperturbed opera tor here one has the phenomenon of loss of boundary conditions in the limit and the existence of boundary layers for uc near 3i2 we prove several more refined estimates on ue in particular a uniform bound in l 2 fi for any rj 0 these estimates yield a proof of the strong convergence of uc to u in all hs u for s this singular perturbation problem is well known and much studied in contrast to the problems of the second sort mentioned above in the latter problems we study the cauchy problems

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

detecting k theory by cyclic homology we discuss which part of the rationalized algebraic k theory of a group ring is detected via trace maps to hochschild homology cyclic homology periodic cyclic or negative cyclic homology

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

barrier lyapunov functions for the output tracking control of constrained nonlinear switched systems abstract in this paper we investigate the output tracking control problem of constrained nonlinear switched systems in lower triangular form first when all the states are subjected to constraints we employ a barrier lyapunov function blf which grows to infinity whenever its arguments approach some finite limits to prevent the states from violating the constraints based on the simultaneous domination assumption we design a continuous feedback controller for the switched system which guarantees that asymptotic output tracking is achieved without transgression of the constraints and all closed loop signals remain bounded provided that the initial states are feasible then we further consider the case of asymmetric time varying output constraints by constructing an appropriate blf finally the effectiveness of the proposed results is demonstrated with a numerical example

stability of large ekman boundary layers in rotating fluids the aim of this paper is to investigate the stability of ekman boundary layers for rotating fluids when the ekman number and the rossby number go to zero more precisely we prove that spectral stability implies linear and nonlinear stabilities of approximate solutions in particular we replace the smallness condition obtained with energy methods in 5 by a weaker spectral condition which is sharp

initial boundary layer associated with the nonlinear darcy brinkman system abstract we study the interaction of initial layer and boundary layer in the nonlinear darcy brinkman system in the vanishing darcy number limit in particular we show the existence of a function of corner layer type so called initial boundary layer in the solution of the nonlinear darcy brinkman system an approximate solution is constructed by the method of multiple scale expansion in space and in time we establish the optimal convergence rates in various sobolev norms via energy method

semi group theory for the stokes operator with navier type boundary conditions on l p spaces in this article we consider the stokes problem with navier type boundary conditions on a domain omega not necessarily simply connected since under these conditions the stokes problem has a non trivial kernel we also study the solutions lying in the orthogonal of that kernel we prove the analyticity of several semigroups generated by the stokes operator considered in different functional spaces we obtain strong weak and very weak solutions for the time dependent stokes problem with the navier type boundary condition under different hypotheses on the initial data u0 and external force f then we study the fractional and pure imaginary powers of several operators related with our stokes operators using the fractional powers we prove maximal regularity results for the homogeneous stokes problem on the other hand using the boundedness of the pure imaginary powers we deduce maximal l p l q regularity for the inhomogeneous stokes problem

well posedness for compressible euler equations with physical vacuum singularity an important problem in the theory of compressible gas flows is to understand the singular behavior of vacuum states the main difficulty lies in the fact that the system becomes degenerate at the vacuum boundary where the characteristic speeds u c coincide and have unbounded spatial derivative since c behaves like x 1 2 close to the boundary in this paper we overcome this difficulty by presenting a new formulation and new energy spaces we establish the local in time well posedness of one dimensional compressible euler equations for isentropic flows with the physical vacuum singularity in some spaces adapted to the singularity 2009 wiley periodicals inc

vanishing vertical viscosity limit of anisotropic navier stokes equation with no slip boundary condition abstract in this paper we consider the zero viscosity limit of the anisotropic incompressible navier stokes equations with no slip boundary condition see 1 1 in r 2 we prove that there exist t independently on e such that the strong solutions of 1 1 convergence to the solution of 1 2 away from the boundary and to solution of 1 3 near the boundary in l 0 t l 2 l r 2 when the vertical viscosity vanish provided that the initial velocity is regular enough and we obtain the optimal convergence rate

ekman layers of rotating fluids the case of general initial data in this paper we describe the weak solutions of the navier stokes equations with a large coriolis term as the rossby and the ekman numbers go to zero in a special domain with various boundary conditions this work extends the results in the case of well prepared initial data in 19 30 2000 john wiley sons inc

boundary layer analysis of the navier stokes equations with generalized navier boundary conditions we study the weak boundary layer phenomenon of the navier stokes equations with generalized navier friction boundary conditions u n 0 s u n tan au 0 in a bounded domain in r3 when the viscosity e 0 is small here s u is the symmetric gradient of the velocity u and a is a type 1 1 tensor on the boundary when a i we obtain navier boundary conditions and when a is the shape operator we obtain the conditions u n curlu n 0 by constructing an explicit corrector we prove the convergence as e tends to zero of the navier stokes solutions to the euler solution both in the natural energy norm and uniformly in time and space

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

viscous boundary layers for the navier stokes equations with the navier slip conditions we tackle the issue of the inviscid limit of the incompressible navier stokes equations when the navier slip with friction conditions are prescribed on impermeable boundaries we justify an asymptotic expansion which involves a weak amplitude boundary layer with the same thickness as in prandtl s theory and a linear behavior this analysis holds for general regular domains in both dimensions two and three

a new boundary condition for the three dimensional navier stokes equation and the vanishing viscosity limit in this paper we propose a new vorticity boundary condition for the three dimensional incompressible navier stokes equation for a general smooth domain in r3 this boundary condition is motivated by the generalized navier slip boundary condition involving the vorticity it is shown first that such an initial boundary value problem is well posed at least local in time furthermore more importantly we obtain some estimates on rate of convergence in c 0 t h1 and c 0 t h2 of the solutions to the corresponding solutions of the ideal euler equations with the standard slip boundary condition

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

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

singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite using classical methods the authors develop a general theory of such problems this theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto fluid dynamics including new constructive local existence theorems for the time singular limit equations in particular the authors give an entirely self contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the mach number becomes small the theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures similar results are developed also for the compressible and incompressible navier stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero

vanishing diffusivity limit for the 3d heat conductive boussinesq equations with a slip boundary condition abstract in this paper we consider the 3d heat conductive boussinesq equations with a slip boundary condition for the velocity field and neumann boundary condition for the temperature we prove that there exists an interval of time which is uniform for the heat conductivity coefficient and for which the 3d heat conductive boussinesq equations have a strong solution the solution is uniformly bounded in some spaces with respect to based on these uniform estimates we establish the vanishing diffusivity limit for the boussinesq equations and also obtain the convergence rates for the velocity field and the temperature

relevance of the slip condition for fluid flows near an irregular boundary we consider the navier stokes equation in a domain with a rough boundary the roughness is modeled by a small amplitude and small wavelength oscillation with typical scale varepsilon ll 1 for periodic oscillation it is well known that the best homogenized that is regular in varepsilon boundary condition is of navier type such result still holds for random stationary irregularities as shown recently by the first author 5 15 we study here arbitrary irregularity patterns

from the boltzmann equation to the stokes fourier system in a bounded domain we prove that the renormalized solutions of the boltzmann equation considered in a bounded domain with different types of kinetic boundary conditions converge to the stokes fourier system with different types of fluid boundary conditions when the main free path goes to zero this extends the work of f golse and d levermore 9 to the case of a bounded domain 2003 wiley periodicals inc

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

zero viscosity limit of the incompressible navier stokes equations with sharp vorticity gradient abstract it is well known that the 3d incompressible euler equations admit some local in time solutions for which the vorticity is piecewise smooth and discontinuous across a smooth time dependent hypersurface which evolves with the flow in this paper we prove that such a solution can be obtained as zero viscosity limit of strong solutions to the navier stokes equations whose vorticity has sharp variations near the hypersurface associated with the inviscid limit indeed we exhibit some sequences of exact solutions to the navier stokes equations with vanishing viscosity which are given by multi scale asymptotic expansions involving some characteristic boundary layers given by some linear pdes the convergence and the validity of the expansion are guaranteed on the time interval associated with the solution to the euler equations

the incompressible limit of the non isentropic euler equations we study the euler equations for slightly compressible fluids that is after rescaling the limits of the euler equations of fluid dynamics as the mach number tends to zero in this paper we consider the general non isentropic equations and general data we first prove the existence of classical solutions for a time independent of the small parameter then on the whole space d we prove that the solution converges to the solution of the incompressible euler equations

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

uniform regularity and relaxation limit for the outer pressure problem of gas dynamics with several thermal nonequilibrium modes abstract we study in this paper the uniform regularity and relaxation limit for the outer pressure problem of one dimensional gas dynamics with several thermal nonequilibrium modes the uniform regularity is established for the initial data satisfying the compatible conditions on the boundary the relaxation limit to equilibrium flow or frozen flow is also investigated by a compactness argument

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

some properties on the surfaces of vector fields and its application to the stokes and navier stokes problems with mixed boundary conditions abstract in this paper we are concerned with the stationary and non stationary stokes and navier stokes problems with mixed boundary conditions involving velocity pressure rotation stress and normal derivative of velocity together relying on the relations among strain rotation normal derivative of velocity and shape of boundary surface we get variational formulations of the stokes and navier stokes problems with the mixed boundary conditions also we study existence and uniqueness of solutions to the problems

wall laws for fluid flows at a boundary with random roughness the general concern of this paper is the effect of rough boundaries on fluids we consider a stationary flow governed by incompressible navier stokes equations in an infinite domain bounded by two horizontal rough plates the roughness is modeled by a spatially homogeneous random field with characteristic size e a mathematical analysis of the flow for small e is performed the navier s wall law is rigorously deduced from this analysis this substantially extends former results obtained in the case of periodic roughness notably in 16 17 2007 wiley periodicals inc

uniform regularity and vanishing dissipation limit for the full compressible navier stokes system in three dimensional bounded domain in the present paper we study the uniform regularity and vanishing dissipation limit for the full compressible navier stokes system whose viscosity and heat conductivity are allowed to vanish at different orders the problem is studied in a three dimensional bounded domain with navier slip type boundary conditions it is shown that there exists a unique strong solution to the full compressible navier stokes system with the boundary conditions in a finite time interval which is independent of the viscosity and heat conductivity the solution is uniformly bounded in w 1 infty and is a conormal sobolev space based on such uniform estimates we prove the convergence of the solutions of the full compressible navier stokes to the corresponding solutions of the full compressible euler system in l infty 0 t l 2 l infty 0 t h 1 and l infty 0 t times omega with a rate of convergence