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

kahlerian twistor spaces the work of r penrose has shown the deep relationship between the conformal structure of minkowski space and the complex geometry of lines in projective three space this transformation applies to a more general class of conformal structures and in particular there is a riemannian version described in 5 which is itself of interest to mathematical physicists basically to each oriented riemannian 4manifold x one associates canonically a 6 manifold z the twistor space of x together with an almost complex structure if this structure is integrable then z becomes a complex 3 manifold with a distinguished family of projective lines on it whose geometry describes the conformal geometry of the manifold x it is natural to ask what sort of complex manifold one obtains by this process in this paper we ask the more specific question which compact 4c manifolds have ka hlerian twistor spaces 1 the answer rather surprisingly is that there are just two examples the 4 sphere s and the complex projective plane p2 c each with their standard homogeneous conformal structures the twistor space of s 4 is the 3 dimensional projective space p3 c and that of p 2 q is the manifold f2 c of flags in c 3 the essential point of the proof of this result is the remark that if z is a twistor space with a kahler metric then the canonical bundle k is negative using vanishing theorems and the riemann roch theorem we then narrow down the possibilities for z to four cases projective space the flag manifold the intersection of two quadrics in p5 and the double covering of p3 branched over a non singular quartic surface the last two candidates cannot be realized however because the value of their euler characteristic is incompatible with their being twistor spaces throughout the proof we make essential use of a real structure on z which reduces considerably the occurrence of singularities at various stages aside from this our methods are the standard ones of algebraic geometry the four examples to which the proof leads us are all fano threefolds if we drop the requirement of compactness on x then it is quite possible for an open set of a fano threefold to be a twistor space in particular we show that the family of conies lying on the intersection of two

eigenvalue estimates for the dirac operator on quaternionic kahler manifolds we consider the dirac operator on compact quaternionic kahler manifolds and prove a lower bound for the spectrum this estimate is sharp since it is the first eigenvalue of the dirac operator on the quaternionic projective space

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

sur le groupe d automorphismes du groupe libre i transvections abstract in this study we define a new family of free automorphisms called transvections which generalise the whitehead automorphisms this leads to a new presentation of the group of free automorphisms and to a partial generalisation of mccool s well known result on the presentation of certain of its subgroups we study the linear characteristics of transvections with the use of fox free differential calculus the groups generated by these automorphisms form the family of transvection groups that one could name the pre classical groups we give some simple examples of these groups which we shall study further throughout the following papers

compact 8 manifolds with holonomy spin 7 in berger s classification 4 of the possible holonomy groups of a nonsymmetric irreducible riemannian manifold there are two special cases the exceptional holonomy groups g2 in 7 dimensions and spin 7 in 8 dimensions bryant 6 proved that such metrics exist locally using the theory of exterior differential systems and gave some explicit examples later bryant and salamon 7 constructed explicit complete metrics with holonomy g2 and spin 7 in two previous papers 12 13 the author constructed many examples of compact riemannian 7 manifolds with holonomy g2 this paper will construct examples of compact riemannian 8 manifolds with holonomy spin 7 using similar methods we believe that these are the first examples known since metrics with holonomy spin 7 are ricci flat these are also new examples of compact ricci flat riemannian 8 manifolds let m be an 8 manifold a spin 7 structure on m can be encoded in a 4 form on m a special 4 form satisfying the condition that the stabilizer of at each point should be isomorphic to spin 7 by an abuse of notation we usually identify the spin 7 structure with its associated 4 form since spin 7 so 8 the spin 7 structure also induces a riemannian metric g and an orientation on m it turns out that the holonomy group hol g of g is a subgroup of spin 7 with spin 7 structure if and only if d 0 the quantity d is called the torsion of the spin 7 structure and is called torsion free if d 0 very briefly the plan of the paper splits into four steps as follows firstly a compact 8 manifold m is given secondly a spin 7 structure on m is found with small torsion i e d is small thirdly is deformed to a nearby spin 7 structure with d 0 thus is torsion free and if g is the associated metric then hol g spin 7 fourthly it is shown that hol g is spin 7 and not some proper subgroup the structure of the paper is designed around this division of the construction into four steps there are six chapters this first chapter is of introductory material the second chapter has the full statements of the main results in

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

zero dissipation limit of the 1d linearized navier stokes equations for a compressible fluid abstract in this paper we study the asymptotic limiting behavior of the solutions to the initial boundary value problem for linearized one dimensional compressible navier stokes equations we consider the characteristic boundary conditions that is we assume that an eigenvalue of the associated inviscid euler system vanishes uniformly on the boundary the aim of this paper is to understand the evolution of the boundary layer to construct the asymptotic ansatz which is uniformly valid up to the boundary and to obtain rigorously the uniform convergence to the solution of the euler equations without the weakness assumption on the boundary layer

initial boundary layer associated with the nonlinear darcy brinkman oberbeck boussinesq system abstract in this paper we study the vanishing darcy number limit of the nonlinear darcy brinkman oberbeck boussinesq system dbob this singular perturbation problem involves singular structures both in time and in space giving rise to initial layers boundary layers and initial boundary layers we construct an approximate solution to the dbob system by the method of multiple scale expansions the convergence with optimal convergence rates in certain sobolev norms is established rigorously via the energy method

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 inviscid limit for an inflow problem of compressible viscous gas in presence of both shocks and boundary layers in this paper we study the inviscid limit problem for the navier stokes equations of one dimensional compressible viscous gas on half plane we prove that if the solution of the inviscid euler system on half plane is piecewise smooth with a single shock satisfying the entropy condition then there exist solutions to navier stokes equations which converge to the inviscid solution away from the shock discontinuity and the boundary at an optimal rate of e1 as the viscosity e tends to zero

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

vanishing viscosity limit for the 3d magnetohydrodynamic system with a slip boundary condition abstract this work investigates the solvability regularity and vanishing viscosity limit of the 3d viscous magnetohydrodynamic system in a class of bounded domains with a slip boundary condition

boundary layers for compressible navier stokes equations with outflow boundary condition abstract in this paper we study the asymptotic relation between the solutions to the initial boundary value problem of the one dimensional compressible full navier stokes equations with outflow boundary condition and the associated euler equations we assume all the three characteristics to the corresponding euler equations are all negative up to some small time then we prove the existence and the stability of the boundary layers as long as the strength of the boundary layers is suitably small

a note on boundary layer of a nonlinear evolution system with damping and diffusions abstract in this paper we consider an initial boundary value problem for a nonlinear evolution system with damping and diffusions on the strip our main purpose is to investigate zero diffusion limits which formally yield parabolic hyperbolic coupled equations resulting from parabolic parabolic coupled equations it is shown that the boundary layer thickness is of the order o with 0 3 4 as the diffusion parameter goes to zero

boundary layer problem on a hyperbolic system arising from chemotaxis abstract this paper is concerned with the boundary layer problem for a hyperbolic system transformed via a cole hopf type transformation from a repulsive chemotaxis model with logarithmic sensitivity proposed in 23 34 modeling the biological movement of reinforced random walkers which deposit a non diffusible or slowly moving signal that modifies the local environment for succeeding passages by prescribing the dirichlet boundary conditions to the transformed hyperbolic system in an interval 0 1 we show that the system has the boundary layer solutions as the chemical diffusion coefficient e 0 and further use the formal asymptotic analysis to show that the boundary layer thickness is e 1 2 our work justifies the boundary layer phenomenon that was numerically found in the recent work 25 however we find that the original chemotaxis system does not possess boundary layer solutions when the results are reverted to the pre transformed system

characteristic boundary layers for parabolic perturbations of quasilinear hyperbolic problems abstract in this paper we study the existence and nonlinear stability of the totally characteristic boundary layer for the quasilinear equations with positive definite viscosity matrix under the assumption that the boundary matrix vanishes identically on the boundary x 0 we carry out a series of weighted estimates to the boundary layer equations prandtl type equations to get the regularity and the far field behavior of the solutions this allows us to perform a weighted energy estimate for the error equation to prove the stability of the boundary layers the stability result finally implies the asymptotic limit of the viscous solutions

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

stability of multiple boundary layers for 2d quasilinear parabolic equations abstract a parabolic system with small viscosity is considered in a two dimensional channel with the help of discussions on suitable boundary conditions of the corresponding hyperbolic equations on both sides of the channel the existence and stability of multiple boundary layers are proved by using the matched asymptotic analysis and energy estimates the results therefore consequently justify the zero viscosity limit when e tends to 0

boundary layer associated with the darcy brinkman boussinesq model for convection in porous media abstract we study the asymptotic behavior of the infinite darcy prandtl number darcy brinkman boussinesq system for convection in porous media at a small brinkman darcy number the existence of a boundary layer with thickness proportional to the square root of the brinkman darcy number for the velocity field is established in both the l h 1 norm in 2 and 3 d and the l l norm in 2d this improves in several respects an earlier result of payne and straughan 1998 60 where the vanishing brinkman darcy number limit is studied without resolving the boundary layer

zero viscosity limit of the linearized navier stokes equations for a compressible viscous fluid in the half plane the zero viscosity limit for an initial boundary value problem of the linearized navier stokes equations of a compressible viscous fluid in the half plane is studied by means of the asymptotic analysis with multiple scales we first construct an approximate solution of the linearized problem of the navier stokes equations as the combination of inner and boundary expansions next by carefully using the technique on energy methods we show the pointwise estimates of the error term of the approximate solution which readily yield the uniform stability result for the linearized navier stokes solution in the zero viscosity limit 1999 john wiley sons inc

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

on certain kahler quotients of quaternionic kahler manifolds we prove that given a certain isometric action of a two dimensional abelian group a on a quaternionic kahler manifold m which preserves a submanifold n m the quotient m n a has a natural kahler structure we verify that the assumptions on the group action and on the submanifold n m are satisfied for a large class of examples obtained from the supergravity c map in particular we find that all quaternionic kahler manifolds m in the image of the c map admit an integrable complex structure compatible with the quaternionic structure such that n m is a complex submanifold finally we discuss how the existence of the kahler structure on m is required by the consistency of spontaneous mathcal n 2 to mathcal n 1 supersymmetry breaking

the compressible euler equations in a bounded domain existence of solutions and the incompressible limit a short time existence theorem is proven for the euler equations for nonisentropic compressible fluid flow in a bounded domain and solutions with low mach number and almost incompressible initial data are shown to be close to corresponding solutions of the equations for incompressible flow

almost complex structures on quaternion kahler manifolds and inner symmetric spaces we prove that compact quaternionic kahler manifolds of positive scalar curvature admit no almost complex structure even in the weak sense except for the complex grassmannians gr2 n 2 we also prove that irreducible inner symmetric spaces m 4n of compact type are not weakly complex except for spheres and hermitian symmetric spaces

fixed points of automorphisms of free groups abstract the set of fixed points of an automorphism of a finitely generated free group is a finitely generated group settling a conjecture of g p scott s

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

boundary layer effects for the 2 d boussinesq equations with vanishing diffusivity limit in the half plane nsfc 40890150 10801111 10971171 national basic research program 2005cb321700 natural science foundation of fujian province of china 2010j05011 fundamental research funds for the central universities 2010121006

asymptotic analysis for singularly perturbed convection diffusion equations with a turning point turning points occur in many circumstances in fluid mechanics when the viscosity is small very complex phenomena can occur near turning points which are not yet well understood a model problem corresponding to a linear convection diffusion equation e g suitable linearization of the navier stokes or benard convection equations is considered our analysis shows the diversity and complexity of behaviors and boundary or interior layers which already appear for our equations simpler than the navier stokes or benard convection equations of course the diversity and complexity of these structures will have to be taken into consideration for the study of the nonlinear problems in our case at this stage the full theoretical asymptotic analysis is provided this study is totally new to the best of our knowledge numerical treatment and more complex problems will be considered elsewhere

convergence rates of vanishing diffusion limit on nonlinear hyperbolic system with damping and diffusion the aim of this paper is to study the global unique solvability on c solution to the cauchy problem of a special 2 2 nonlinear hyperbolic system with damping and diffusion furthermore we also investigate the convergence rates as the diffusion parameter goes to zero it is shown that the convergence rates in c norm is of the order o the novelty of analysis taken in the present article is to obtain the uniform in a priori estimates on hs norm for any s n which are implemented by introducing induction we focus on two cases in this paper both and since the uniform in a priori estimates obtained are distinctly different for different cases

special metrics and triality abstract we investigate a new 8 dimensional riemannian geometry defined by a generic closed and coclosed 3 form with stabiliser psu 3 and which arises as a critical point of hitchin s variational principle we give a riemannian characterisation of this structure in terms of invariant spinor valued 1 forms which are harmonic with respect to the twisted dirac operator on 1 we establish various obstructions to the existence of topological reductions to psu 3 for compact manifolds we also give sufficient conditions for topological psu 3 structures that can be lifted to topological su 3 structures we also construct the first known compact example of an integrable non symmetric psu 3 structure in the same vein we give a new riemannian characterisation for topological quaternionic kahler structures which are defined by an sp 1 sp 2 invariant self dual 4 form again we show that this form is closed if and only if the corresponding spinor valued 1 form is harmonic for and that these equivalent conditions produce constraints on the ricci tensor

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

clifford structures on riemannian manifolds we introduce the notion of even clifford structures on riemannian manifolds which for rank r 2 and r 3 reduce to almost hermitian and quaternion hermitian structures respectively we give the complete classification of manifolds carrying parallel rank r even clifford structures kahler quaternion kahler and riemannian products of quaternion kahler manifolds for r 2 3 and 4 respectively several classes of 8 dimensional manifolds for 5 r 8 families of real complex and quaternionic grassmannians for r 8 6 and 5 respectively and rosenfeld s elliptic projective planes op 2 c o p 2 h o p 2 and o o p 2 which are symmetric spaces associated to the exceptional simple lie groups f4 e6 e7 and e8 for r 9 10 12 and 16 respectively as an application we classify all riemannian manifolds whose metric is bundle like along the curvature constancy distribution generalizing well known results in sasakian and 3 sasakian geometry 2000 mathematics subject classification primary 53c26 53c35 53c10 53c15