on a conjecture of anosov abstract in this paper we prove that for every bumpy finsler n sphere s n f with reversibility and flag curvature k satisfying 1 2 k 1 there exist 2 n 1 2 prime closed geodesics this gives a confirmed answer to a conjecture of anosov 1 in 1974 for a generic case

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

multiple closed geodesics on bumpy finsler n spheres abstract in this paper we prove that for every bumpy finsler metric f on every rationally homological n dimensional sphere s n with n 2 there exist always at least two distinct prime closed geodesics

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

on the average indices of closed geodesics on positively curved finsler spheres in this paper we prove that on every finsler n sphere s n f for n 6 with reversibility and flag curvature k satisfying frac lambda lambda 1 2 k le 1 either there exist infinitely many prime closed geodesics or there exist frac n 2 2 closed geodesics possessing irrational average indices if in addition the metric is bumpy then there exist n 3 closed geodesics possessing irrational average indices provided the number of prime closed geodesics is finite

the index growth and multiplicity of closed geodesics abstract in the recent paper 31 of long and duan 2009 we classified closed geodesics on finsler manifolds into rational and irrational two families and gave a complete understanding on the index growth properties of iterates of rational closed geodesics this study yields that a rational closed geodesic cannot be the only closed geodesic on every irreversible or reversible including riemannian finsler sphere and that there exist at least two distinct closed geodesics on every compact simply connected irreversible or reversible including riemannian finsler 3 dimensional manifold in this paper we study the index growth properties of irrational closed geodesics on finsler manifolds this study allows us to extend results in 31 of long and duan 2009 on rational and in 12 of duan and long 2007 39 of rademacher 2010 and 40 of rademacher 2008 on completely non degenerate closed geodesics on spheres and cp 2 to every compact simply connected finsler manifold then we prove the existence of at least two distinct closed geodesics on every compact simply connected irreversible or reversible including riemannian finsler 4 dimensional manifold

sharp bounds for the resolvent of linearized navier stokes equations in the half space around a shear profile abstract in this paper we derive sharp bounds on the semigroup of the linearized incompressible navier stokes equations near a stationary shear layer in the half space r 2 or r 3 with dirichlet boundary conditions assuming that this shear layer in spectrally unstable for euler equations in the inviscid limit due to the prescribed no slip boundary conditions vorticity becomes unbounded near the boundary the novelty of this paper is to introduce boundary layer norms that capture the unbounded vorticity and to derive sharp estimates on this vorticity that are uniform in the inviscid limit

loop products and closed geodesics the critical points of the length function on the free loop space m of a compact riemannian manifold m are the closed geodesics on m the length function gives a filtration of the homology of m and we show that the chas sullivan product hi hj hi j n is compatible with this filtration we obtain a very simple expression for the associated graded homology ring grh m when all geodesics are closed or when all geodesics are nondegenerate we also interpret sullivan s coproduct see su1 su2 on c as a product in cohomology h 0 h 0 hi j n 1 0 where 0 m is the constant loop we show that is also compatible with the length filtration and we obtain a similar expression for the ring grh 0 the nonvanishing of products n and n is shown to be determined by the rate at which the morse index grows when a geodesic is iterated we determine the full ring structure h 0 for spheres m s n 3

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

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

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

two elliptic closed geodesics on positively curved finsler spheres abstract in this paper we prove that for every finsler n dimensional sphere s n f with reversibility and flag curvature k satisfying 1 2 k 1 either there exist infinitely many closed geodesics or there exist at least two elliptic closed geodesics and each linearized poincare map has at least one eigenvalue of the form e 1 with being an irrational multiple of

on the number of closed geodesics on projective spaces 1 let pro k be the m dimensional projective space over the complex numbers k 2 the quaternions k 4 or the cayley plane k 8 m 2 endowed with the standard riemannian metric and m be a compact riemannian manifold and f pm k m a differentiable homotopy equivalence a circle on pro k is mapped under f onto a prime closed curve of length 2 hingston proved in 5 w 5 that under these conditions there are g k m distinct closed geodesics without self intersections and of length g k m in most cases it is g l k m g k m s k 2 against that on the standard sphere s the cup length of the space of unoriented parametrized great circles is always smaller than the cup length of the space of unparametrized great circles for the case of the sphere see 8 ch 3 6

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

stability for line solitary waves of zakharov kuznetsov equation abstract in this paper we consider the stability for line solitary waves of the two dimensional zakharov kuznetsov equation on r t l which is one of a high dimensional generalization of korteweg de vries equation where t l is the torus with the 2 l period the orbital and asymptotic stability of the one soliton of korteweg de vries equation on the energy space was proved by benjamin 2 pego and weinstein 41 and martel and merle 30 we regard the one soliton of korteweg de vries equation as a line solitary wave of zakharov kuznetsov equation on r t l we prove the stability and the transverse instability of the line solitary waves of zakharov kuznetsov equation by applying the method of evans function and the argument of rousset and tzvetkov 44 moreover we prove the asymptotic stability for orbitally stable line solitary waves of zakharov kuznetsov equation by using the argument of martel and merle 30 31 32 and a liouville type theorem if l is the critical period with respect to a line solitary wave the line solitary wave is orbitally stable however since this line solitary wave is a bifurcation point of the stationary equation the linearized operator of the stationary equation is degenerate because of the degeneracy of the linearized operator we can not show the liouville type theorem for the line solitary wave by using the usual virial type estimate to show the liouville type theorem for the line solitary wave we modify a virial type estimate

existence of closed geodesics on finsler n spheres abstract in this paper we prove that on every finsler n sphere s n f with reversibility satisfying f 2 1 2 g 0 and l s n f 1 1 there always exist at least n prime closed geodesics without self intersections where g 0 is the standard riemannian metric on s n with constant curvature 1 and l s n f is the length of a shortest geodesic loop on s n f we also study the stability of these closed geodesics

recent developments in inverse acoustic scattering theory we survey some of the highlights of inverse scattering theory as it has developed over the last 15 years with emphasis on uniqueness theorems and reconstruction algorithms for time harmonic acoustic waves included in our presentation are numerical experiments using real data and numerical examples of the use of inverse scattering methods to detect buried objects

on a generic property of geodesic flows in this paper we are concerned with a sufficient condition for a riemannian metric on a compact simply connected manifold to have infinitely many geometrically distinct closed geodesics 1969 gromoll meyer proved in gm that for every riemannian metric on a compact simply connected manifold m there are infinitely many geometrically distinct closed geodesics if the sequence bi m f i of betti numbers of the free loop space m is unbounded for some field f using the theory of minimal models vigue poirrier sullivan proved that the rational cohomology algebra h m i q of m is generated by a single element if and only if the sequence bi m i q i of betti numbers of the free loop space m of m is bounded it is a conjecture that the same statement holds for all fields of prime characteristic partial results are due to mccleary ziller mz and halperin vigue poirrier hv now we turn to the manifolds for which the hypothesis of gromoll meyer s theorem does not hold for example spheres and projective spaces then stability properties of the closed geodesics become important a closed geodesic is hyperbolic if the linearized poincare map has no eigenvalue of norm 1 from the bumpy metrics theorem due to abraham ab and anosov an2 and from a pertubation result due to klingenberg takens kt one can conclude a c4 generic metric on a compact manifold has either a non hyperbolic closed geodesic of twist type or all closed geodesics are hyperbolic in the first case there are infinitely many geometrically distinct closed geodesics in every tubular neighborhood of the closed geodesic of twist type due to a theorem by moser mo in the second case there are infinitely many closed geodesics if m is simply connected due to results by hingston hi and the author ra1 hence it follows from these results that a c4 generic metric on a compact simply connected manifold has infinitely many geometrically distinct closed geodesics we remark that it is an open question whether there is a riemannian metric on a simply connected compact manifold all of whose closed geodesics are hyperbolic in ra2 we show that the examples of metrics on the 2

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

spectral instability of characteristic boundary layer flows in this paper we construct growing modes of the linearized navier stokes equations about generic stationary shear flows of the boundary layer type in a regime of a sufficiently large reynolds number r notably the shear profiles are allowed to be linearly stable at the infinite reynolds number limit and so the instability presented is purely due to the presence of viscosity the formal construction of approximate modes is well documented in physics literature going back to the work of heisenberg c c lin tollmien drazin and reid but a rigorous construction requires delicate mathematical details involving for instance a treatment of primitive airy functions and singular solutions our analysis gives exact unstable eigenvalues and eigenfunctions showing that the solution could grow slowly at the rate of et r the proof follows the general iterative approach introduced in our companion paper avoiding having to deal with matching inner and outer asymptotic expansions but instead involving a careful study of singularity in the critical layers by deriving pointwise bounds on the green function of the corresponding rayleigh and airy operators unlike in the channel flows the spatial domain in the boundary layers is unbounded and the iterative scheme is likely to diverge due to the linear growth in the vertical variable we introduce a new iterative scheme to simultaneously treat the singularity near critical layers and the asymptotic behavior of solutions at infinity the instability of generic boundary layers obtained in this paper is linked to the emergence of tollmien schlichting waves in describing the early stage of the transition from laminar to turbulent flows

the second closed geodesic on finsler spheres of dimension n 2 we show the existence of at least two geometrically distinct closed geodesics on an n dimensional sphere with a bumpy and non reversible finsler metric for n 2 2000 msc classification 53c22 53c60 58e10

symmetrical prandtl boundary layer expansions of steady navier stokes equations on bounded domain abstract this paper is concerned with the validity of the prandtl boundary layer theory in the inviscid limit of the steady incompressible navier stokes equations which is an extension of the pioneer paper 13 y guo et al 2017 ann pde from a domain of 0 l r to 0 l 0 2 under the symmetry assumption we establish the validity of the prandtl boundary layer expansions and the error estimates the convergence rate as e 0 is also given

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

birkhoff minimax principle for minimal surfaces with a free boundary 1 1 x 0 in b 1 2 xu xv 0 xu xv in b 1 3 x c s 1 4 xn w tx w s w c here n is the outer normal to b subscripts denote partial derivatives and are the euclidean norm and scalar product in rn 1 and ty s for y s denotes the tangent space to s at y for brevity problem 1 1 1 4 will be referred to as problem p s a solution of p s is a critical point of the following dirichlet integral

linear inviscid damping and enhanced viscous dissipation of shear flows by using the conjugate operator method abstract we study the large time behavior of solutions to two dimensional euler and navier stokes equations linearized about shear flows of the mixing layer type in the unbounded channel t r under a simple spectral stability assumption on a self adjoint operator we prove a local form of the linear inviscid damping that is uniform with respect to small viscosity we also prove a local form of the enhanced viscous dissipation that takes place at times of order 1 3 being the small viscosity to prove these results we use a hamiltonian approach following the conjugate operator method developed in the study of schrodinger operators combined with a hypocoercivity argument to handle the viscous case

on the number of minimal two spheres of small area in manifolds with curvature bounded above a beautiful application of the morse theory of closed geodesics is provided by klingenberg s k 1 study of the number of short smooth closed geodesics in sufficiently pinched spheres together with subsequent contributions in this direction by many authors a particularly striking result see ballmann thorbergsson and ziller btz states that a generic metric on a compact simply connected riemannian manifold whose sectional curvatures satisfy 1 4 k a 1

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

newton kantorovitch algorithm applied to an electromagnetic inverse problem an efficient algorithm is presented to compute the shape of perfectly conducting cylinders via knowledge of scattering cross sections this choice of scattering data avoids the difficulties linked with the phase measurements or reconstructions the mathematical analysis is performed in terms of operator and functions and the fundamental instability of the problem is demonstrated then the stability is restored by means of a tikhonov miller regularization the efficiency of the method is outlined by numerical examples references are given which show that the same algorithm applies to other electromagnetic inverse problems especially to gratings

stability and instability of the kdv solitary wave under the kp i flow we consider the kp i and gkp i equations in mathbb r times mathbb r 2 pi mathbb z we prove that the kdv soliton with subcritical speed 0 c in the spirit of the work by duyckaerts and merle gafa 18 1787 1840 2009 we sharpen our previous instability result and construct a global solution which is different from the solitary wave and its translates and which converges to the solitary wave as time goes to infinity this last result also holds for the gkp i equation

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

the reconstruction of shape and impedance exploiting space mapping with inverse difference method the reconstruction of shape and impedance obtained from measurements of scattered electromagnetic fields is considered for the two dimensional case space mapping with inverse difference technique which is newly developed to solve inverse problem is used for this purpose one of the advantages of this technique over other space mapping techniques is that it does not need parameter extraction during optimization process as the contour of the cylinder is periodic it can be represented by fourier series measurements of scattered field and fourier coefficients of the contour are used as inputs and outputs respectively the results are compared with exact shapes and good agreement is obtained it is also observed that reconstructions obtained by space mapping based technique are better than those obtained by artificial neural network for exact and noisy data sharp cornered tetragonal contour is also investigated for both noisy and noiseless case in order to show efficiency of the technique

long time well posdness of the prandtl equations in sobolev space in this paper we study the long time well posedness for the nonlinear prandtl boundary layer equation on the half plane while the initial data are small perturbations of some monotonic shear profile we prove the existence uniqueness and stability of solutions in weighted sobolev space by energy methods the key point is that the life span of the solution could be any large t as long as its initial date is a perturbation around the monotonic shear profile of small size like e t the nonlinear cancellation properties of prandtl equations under the monotonic assumption are the main ingredients to establish a new energy estimate

existence of closed geodesics on positively curved manifolds on applique la theorie de lusternik schnirelmann pour obtenir l existence de plusieurs geodesiques fermees entre certains niveaux de longueur on etudie les proprietes de stabilite de ces geodesiques fermees

non hyperbolic closed geodesics on positively curved finsler spheres abstract in this paper we prove that for every finsler n dimensional sphere s n f n 3 with reversibility and flag curvature k satisfying 1 2 k 1 there exist at least three distinct closed geodesics and at least two of them are elliptic if the number of prime closed geodesics is finite when n 6 these three distinct closed geodesics are non hyperbolic

resonance identity and multiplicity of non contractible closed geodesics on finsler rpn abstract in this paper we establish first the resonance identity for non contractible homologically visible prime closed geodesics on finsler n dimensional real projective space r p n f when there exist only finitely many distinct non contractible closed geodesics on r p n f where the integer n 2 then as an application of this resonance identity we prove the existence of at least two distinct non contractible closed geodesics on r p n with a bumpy and irreversible finsler metric together with two previous results on bumpy and reversible finsler metrics in 14 and 39 it yields that every r p n with a bumpy finsler metric possesses at least two distinct non contractible closed geodesics

finite time analyticity for the two and three dimensional kelvin helmholtz instability the well posed property for the finite time vortex sheet problem with analytic initial data was first conjectured by birkhoff in two dimensions and is shown here to hold both in two and three dimensions incompressible inviscid and irrotational flow with a velocity jump across an interface is assumed in two dimensions global existence of a weak solution to the euler equation with such initial conditions is established in three dimensions a lagrangian representation of the vortex sheet analogous to the birkhoff equation in two dimensions is presented

genericity of geodesic flows with positive topological entropy on s2 we show that the set of c riemannian metrics on s 2 or rp 2 whose geodesic flow has positive topological entropy is open and dense in the c 2 topology the proof is partially based on an analogue of franks lemma for geodesic flows on surfaces

topological structure of non contractible loop space and closed geodesics on real projective spaces with odd dimensions abstract in this paper we use chas sullivan theory on loop homology and leray serre spectral sequence to investigate the topological structure of the non contractible component of the free loop space on the real projective spaces with odd dimensions then we apply the result to get the resonance identity of non contractible homologically visible prime closed geodesics on such spaces provided the total number of distinct prime closed geodesics is finite

index iteration theories for periodic orbits old and new abstract in this paper we give a survey of the index iteration theories developed for periodic solution orbits of hamiltonian systems in treating various problems from 1950s up to now including some recent developments and some of their applications to periodic solution problems

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

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

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

convex hamiltonian energy surfaces and their periodic trajectories in this paper we introduce symplectic invariants for convex hamiltonian energy surfaces and their periodic trajectories and show that these quentities satisfy several nontrivial relations in particular we show that they can be used to prove multiplicity results for the number of periodic trajectories

bumpy metrics and closed parametrized minimal surfaces in riemannian manifolds the purpose of this article is to study conformal harmonic maps f m where is a closed riemann surface and m is a compact riemannian manifold of dimension at least four such maps define parametrized minimal surfaces possibly with branch points we show that when the ambient manifold m is given a generic metric all prime closed parametrized minimal surfaces are free of branch points and are as morse nondegenerate as allowed by the group of automorphisms of e they are morse nondegenerate in the usual sense if has genus at least two lie on two dimensional nondegenerate critical submanifolds if s has genus one and on six dimensional nondegenerate critical submanifolds if has genus zero

resonance identities for closed characteristics on compact star shaped hypersurfaces in r2n resonance relations among periodic orbits on given energy hypersurfaces are very important for getting deeper understanding of the dynamics of the corresponding hamiltonian systems in this paper we establish two new resonance identities for closed characteristics on every compact star shaped hypersurface in r2n when the number of geometrically distinct closed characteristics on is finite which extend those identities established by c viterbo in 1989 for star shaped hypersurfaces assuming in addition that all the closed characteristics and their iterates are non degenerate and that by w wang x hu and y long in 2007 for strictly convex hypersurfaces in r2n

on the berger conjecture for manifolds all of whose geodesics are closed we define a besse manifold as a riemannian manifold m g all of whose geodesics are closed a conjecture of berger states that all prime geodesics have the same length for any simply connected besse manifold we firstly show that the energy function on the free loop space of a simply connected besse manifold is a perfect morse bott function with respect to a suitable cohomology secondly we explain when the negative bundles along the critical manifolds are orientable these two general results then lead to a solution of berger s conjecture when the underlying manifold is a sphere of dimension at least four

minimal two spheres of low index in manifolds with positive complex sectional curvature suppose that s n is given a generic riemannian metric with sectional curvatures which satisfy a suitable pinching condition formulated in terms of complex sectional curvatures this pinching condition is satisfied by manifolds whose real sectional curvatures k r sigma satisfy begin aligned 1 2 k r sigma le 1 end aligned then the number of minimal two spheres of morse index lambda for n 2 le lambda le 2n 5 is at least p 3 lambda n 2 where p 3 k is the number of k cells in the schubert cell decomposition for g 3 mathbb r n 1

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

ill posedness of the hydrostatic euler and navier stokes equations we prove that the linearization of the hydrostatic euler equations at certain parallel shear flows is ill posed the result also extends to the hydrostatic navier stokes equations with a small viscosity