boundary regularity of rotating vortex patches we show that the boundary of a rotating vortex patch or v state in the terminology of deem and zabusky is c provided the patch is close to the bifurcation circle in the lipschitz norm the rotating patch is also convex if it is close to the bifurcation circle in the c2 norm our proof is based on burbea s approach to v states

generalized surface quasi geostrophic equations with singular velocities this paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals the first family is a generalized surface quasigeostrophic sqg equation with the velocity field u related to the scalar by where and is the zygmund operator the borderline case 1 corresponds to the sqg equation and the situation is more singular for 1 we obtain the local existence and uniqueness of classical solutions the global existence of weak solutions and the local existence of patch type solutions the second family is a dissipative active scalar equation with which is at least logarithmically more singular than the velocity in the first family we prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data this result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by k ohkitani 2012 wiley periodicals inc

efficient calculation of the green function for acoustic propagation above a homogeneous impedance plane this paper is concerned with the problem of propagation from a monofrequency coherent line source above a plane of homogeneous surface impedance the solution of this problem occurs in the kernel of certain boundary integral equation formulations of acoustic propagation above an impedance boundary and the discussion of the paper is motivated by this application the paper starts by deriving representations as laplace type integrals of the solution and its first partial derivatives the evaluation of these integral representations by gauss laguerre quadrature is discussed and theoretical bounds on the truncation error are obtained specific approximations are proposed which are shown to be accurate except in the very near field for all angles of incidence and a wide range of values of surface impedance the paper finishes with derivations of partial results and analogous laplace type integral representations for the case of a point source

determinant bundles manifolds with boundary and surgery we define determinant bundles associated to the following data i a family of generalized dirac operators on even dimensional manifolds with boundary ii the choice of a spectral section for the family of dirac operators induced on the boundary under the assumption that the operators of the boundary family have null spaces of constant dimension we define through the notion ofb zeta function a quillen metric we also introduce the analogue of the bismut freed connection we prove that the curvature of a natural perturbation of the bismut freed connection equals the 2 form piece in the right hand side of the family index formula thus extending to manifolds with boundary results of quillen bismut and freed given a closed fibration we investigate the behaviour of the quillen metric and of the bismut freed connection under the operation of surgery along a fibering hypersurface we prove in particular additivity formulae for the curvature and for the logarithm of the holonomy

a fast direct solver for quasi periodic scattering problems abstract we consider the numerical solution of the scattering of time harmonic plane waves from an infinite periodic array of reflection or transmission obstacles in a homogeneous background medium in two dimensions boundary integral formulations are ideal since they reduce the problem to n unknowns on the obstacle boundary however for complex geometries and or higher frequencies the resulting dense linear system becomes large ruling out dense direct methods and often ill conditioned despite being 2nd kind rendering fast multipole based iterative schemes also inefficient we present an integral equation based solver with o n complexity which handles such ill conditioning using recent advances in fast direct linear algebra to invert hierarchically the isolated obstacle matrix this is combined with a recent periodizing scheme that is robust for all incident angles including wood s anomalies based upon the free space green s function kernel the resulting solver is extremely efficient when multiple incident angles are needed as occurs in many applications our numerical tests include a complicated obstacle several wavelengths in size with n 10 5 and solution error of 10 10 where the solver is 66 times faster per incident angle than a fast multipole based iterative solution and 600 times faster when incident angles are chosen to share bloch phases

on the evolution of nearly circular vortex patches recently the classical problem of the evolution of patches of constant vorticity was reformulated as an evolution equation for the boundary of the patch we study this equation in the neighborhood of the circular vortex patch and introduce a hierarchy of area preserving nonlinear approximate equations the first of these equations is shown to have a rich rigid structure it possesses an exhaustive increasing sequence of linear invariant manifolds of arbitrarily large finite dimensions on each of these manifolds the equation can be written as an explicit finite system of ordinary differential equations solutions of these odes starting from arbitrarily small neighborhoods of the circular vortex patch are shown to blow up

efficient calculation of two dimensional periodic and waveguide acoustic green s functions new representations and efficient calculation methods are derived for the problem of propagation from an infinite regularly spaced array of coherent line sources above a homogeneous impedance plane and for the green s function for sound propagation in the canyon formed by two infinitely high parallel rigid or sound soft walls and an impedance ground surface the infinite sum of source contributions is replaced by a finite sum and the remainder is expressed as a laplace type integral a pole subtraction technique is used to remove poles in the integrand which lie near the path of integration obtaining a smooth integrand more suitable for numerical integration and a specific numerical integration method is proposed numerical experiments show highly accurate results across the frequency spectrum for a range of ground surface types it is expected that the methods proposed will prove useful in boundary element modeling of noise propagation in canyon streets and in ducts and for problems of scattering by periodic surfaces

relative equilibria with holes for the surface quasi geostrophic equations abstract we study the existence of doubly connected rotating patches for the inviscid surface quasi geostrophic equation left open by de la hoz hassainia and hmidi in 2016 in 10 by using the approach proposed in 2016 by castro cordoba and gomez serrano in 4 we also prove that close to the annulus the boundaries are actually analytic curves

coadjoint orbits vortices and clebsch variables for incompressible fluids this paper is a study of incompressible fluids especially their clebsch variables and vortices using symplectic geometry and the lie poisson structure on the dual of a lie algebra following ideas of arnold and others it is shown that euler s equations are lie poisson equations associated to the group of volume preserving diffeomorphisms the dual of the lie algebra is seen to be the space of vortices and kelvin s circulation theorem is interpreted as preservation of coadjoint orbits in this context clebsch variables can be understood as momentum maps the motion of n point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit and the standard canonical variables for them are a special kind of clebsch variables point vortices with cores vortex patches and vortex filaments can be understood in a similar way this leads to an explanation of the geometry behind the hald beale majda convergence theorems for vorticity algorithms symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the kdv and the schrodinger equations respectively

an l a stable fourth order rosenbrock method with error estimator the computation of stiff systems of ordinary differential equations requires highly stable processes and this led to the development of l stable rosenbrock methods sometimes called generalized runge kutta or semi implicit runge kutta methods they are linearly implicit and require one jacobian evaluation and at least one matrix factorization per step in this paper we develop some results regarding minimum process configuration i e minimum work per step for a given order as a consequence we then develop an efficient l a stable a 89 fourth order process fifth order locally with a reference formula error estimator similar to that of fehlberg and england

steady state solutions of the euler equations in two dimensions ii local analysis of limiting v states steady state solutions of the 2d incompressible euler equations which we call v states are studied analytically that is a local expansion is done in the neighborhood of any point on the boundary of a v state which consists of piecewise constant regions of vorticity the analytical results contained in paper i by wu overman and zabusky are proven in particular the limiting v states for the numerically calculated translating and rotating v states have 90 circ corners and not cusps it is also proven that at a point which lies on the boundary of only one region and at which the tangent angle has a jump discontinuity then the difference in tangent angles can only be 90 circ a corner or 180 circ a cusp the analytical behavior of doubly and triply connected rotating v states is also investigated

desingularizations of systems of point vortices abstract consider the evolution of an incompressible inviscid fluid in r 2 zabusky conjectured that given a steady system of point vortices there exists a one parameter family of steady system of vortex patches in which each patch shrinks to the corresponding point vortex while keeping the same circulation as the parameter tends to zero such a family is called a desingularization of the given system of point vortices in this article we establish that 1 a zabusky s conjecture holds under a mild condition on the system of point vortices 2 b the desingularizations of the system of point vortices preserve their type of stability 3 c desingularizations of an unfolding of a system of point vortices exist under a typical situation

scattering by infinite one dimensional rough surfaces we consider the dirichlet boundary value problem for the helmholtz equation in a non locally perturbed half plane this problem models time harmonic electromagnetic scattering by a one dimensional infinite rough perfectly conducting surface the same problem arises in acoustic scattering by a sound soft surface chandler wilde and zhang have suggested a radiation condition for this problem a generalization of the rayleigh expansion condition for diffraction gratings and uniqueness of solution has been established recently an integral equation formulation of the problem has also been proposed and in the special case when the whole boundary is both lyapunov and a small perturbation of a flat boundary the unique solvability of this integral equation has been shown by chandler wilde and ross by operator perturbation arguments in this paper we study the general case with no limit on surface amplitudes or slopes and show that the same integral equation has exactly one solution in the space of bounded and continuous functions for all wavenumbers as an important corollary we prove that for a variety of incident fields including the incident plane wave the dirichlet boundary value problem for the scattered field has a unique solution

stability of poisson equilibria and hamiltonian relative equilibria by energy methods we develop a general stability theory for equilibrium points of poisson dynamical systems and relative equilibria of hamiltonian systems with symmetries including several generalisations of the energy casimir and energy momentum methods using a topological generalisation of lyapunov s result that an extremal critical point of a conserved quantity is stable we show that a poisson equilibrium is stable if it is an isolated point in the intersection of a level set of a conserved function with a subset of the phase space that is related to the topology of the symplectic leaf space at that point this criterion is applied to generalise the energy momentum method to hamiltonian systems which are invariant under non compact symmetry groups for which the coadjoint orbit space is not hausdorff we also show that a g stable relative equilibrium satisfies the stronger condition of being a stable where a is a specific group theoretically defined subset of g which contains the momentum isotropy subgroup of the relative equilibrium the results are illustrated by an application to the stability of a rigid body in an ideal irrotational fluid

run time estimation of the spectral radius of jacobians it is useful for ordinary differential equation ode solvers to include an estimator of the spectral radius of the jacobian matrix of the system of ode s since this determines the numerical stability of the method hence a knowledge of spectral radius enables the run time selection of a more efficient integrator some techniques for estimating spectral radius are described and compared they include methods suitable for use with any ode solvers but which require additional computation other methods are described which are suitable with runge kutta or rosenbrock methods and which require little extra computation

nonlinear stability of circular vortex patches this paper proves that circular vortex patches in the plane are stable for the nonlinear dynamical system generated by the euler equations of incompressible fluids this is achieved by establishing a relative variational principle in terms of either energy or angular momentum thus we exploit and extend arnold s idea in 1965 1969 to a nonsmooth setting as well

pattern formation i the keller segel model abstract we investigate nonlinear dynamics near an unstable constant equilibrium in the classical keller segel model given any general perturbation of magnitude we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of fastest growing modes over a time period of ln 1 our result can be interpreted as a rigorous mathematical characterization for early pattern formation in the keller segel model

existence for the α patch model and the qg sharp front in sobolev spaces abstract we consider a family of contour dynamics equations depending on a parameter with 0 1 the vortex patch problem of the 2 d euler equation is obtained taking 0 and the case 1 corresponds to a sharp front of the qg equation we prove local in time existence for the family of equations in sobolev spaces

on rotating doubly connected vortices abstract in this paper we consider rotating doubly connected vortex patches for the euler equations in the plane when the inner interface is an ellipse we show that the exterior interface must be a confocal ellipse we then discuss some relations first found by flierl and polvani between the parameters of the ellipses the velocity of rotation and the magnitude of the vorticity in the domain enclosed by the inner ellipse

the stability of rotating vortex patches in this paper we examine the nonlinear and linear stability of variousrotating vortex patches these patches include the kirchhoff ellipse the kelvin waves and the co rotating uniformm vortices these are achieved by using relative variational methods and spectral analysis thus we extend arnol d s idea for stability problems in 1965 1969 to a non smooth symmetric setting and also relate that to the usual linear stability analysis

a minimum configuration fourth order nonautonomous explicit rosenbrock method for nonstiff differential equations abstract one class of methods for solving nonstiff ordinary differential equations is the so called explicit rosenbrock method a general formulation for these methods is proposed and a procedure for converting autonomous methods to nonautonomous form is described a fourth order method which requires the minimum computational work per step is described

on the existence of stationary patches abstract in this paper we show the existence of the first nontrivial family of analytic stationary patch solutions of the sqg and gsqg equations this answers an open problem in f de la hoz et al 2016 13

on the dynamical rayleigh taylor instability the classical rayleigh taylor instability is shown to be valid in a nonlinear dynamical sense for smooth density profiles

contour dynamics for the euler equations in two dimensions we present a contour dynamics algorithm for the euler equations of fluid dynamics in two dimensions this is applied to regions of piecewise constant vorticity withinfinite area vortexregions favrs essentially this reduces the dimensionality by one and we are treating the interaction of closed polygonal contours whose nodes are advected by the total fluid motion computed self consistently a leapfrog centered scheme is used for temporal advancement computer simulation results are given for two and four like signed interacting favrs in all cases wavelike surface deformations are observed if the distance between favrs is comparable to their extent diameter these surface deformations are large they play an essential role in the observed coalescence of favrs

pade approximants for the acoustical characteristics of rigid frame porous media in this paper it is shown that a number of theoretical models of the acoustical properties of rigid frame porous media especially those involving ratios of bessel functions of complex argument can be accurately approximated and greatly simplified by the use of pade approximation techniques in the case of the model of attenborough j acoust soc am 81 93 102 1987 rational approximations are produced for the characteristic impedance propagation constant dynamic compressibility and dynamic density as a function of frequency and the material parameters the model proposed by stinson and champoux j acoust soc am 91 685 695 1992 is approximated via rational approximation for the viscosity correction function

a comparison of some ode solvers which require jacobian evaluations a variety of third order ode solvers which have a minimum configuration i e minimum work per step have been numerically tested and the results compared they include implicit and explicit processes and share the property that a jacobian matrix must be evaluated at least once during the integration some of these processes have not been previously described in the literature

steady state solutions of the euler equations in two dimensions rotating and translating v states with limiting cases i numerical algorithms and results new second and third order algorithms are presented for calculating translating and rotating steady state solutions of the 2d incompressible euler equations which we call v states these are piecewise constant regions of vorticity and the contours bounding them are obtained by solving iteratively a nonlinear integro differential equation new limiting contours with corners are obtained and compared with local analytical solutions the precise results correct mistakes for limiting contours that were previously given

nonlinear stability of vortex patches to establish the nonlinear liapunov stability of both circular and elliptical vortex patches in the plane for the nonlinear dynamical system generated by the two dimensional euler equations of incompressible inviscid hydrodynamics this is accomplished by using a relative variational principle in terms of energy function a counterexample shows that our result in the case of an elliptical vortex patch is the best one that can be attained by applying the energy estimate

doubly connected v states for the generalized surface quasi geostrophic equations in this paper we prove the existence of doubly connected v states for the generalized sqg equations with 0 1 they can be described by countable branches bifurcating from the annulus at some explicit eigenvalues related to bessel functions of the first kind contrary to euler equations hmidi et al doubly connected v states for the planar euler equations arxiv 1409 7096 2015 we find v states rotating with positive and negative angular velocities at the end of the paper we discuss some numerical experiments concerning the limiting v states

global regularity for vortex patches we present a proof of chemin s 4 result which states that the boundary of a vortex patch remains smooth for all time if it is initially smooth

on nonlinear stability of stationary euler flows on a rotating sphere abstract we consider euler fluids on a rotating sphere and prove the liapounov stability for a class of stationary eventually nonsmooth flows in particular in the smooth case we adapt to our context the classical arnold method

computational synergetics and mathematical innovation it is demonstrated how the computer used in a heuristic mode has greatly augmented our understanding of the mathematics of nonlinear dynamical process examples are given of recent work in soliton mathematics mainly but also nonlinear iterative mappings and low order nonlinear ordinary differential equation systems the role of good graphics in enhancing the discovery and retention of new mathematical properties of equations is illustrated several examples are given of the present frontier of research in two space dimensional nonlinear evolutionary problems

on the evolution of sharp fronts for the quasi geostrophic equation we consider the problem of the evolution of sharp fronts for the surface quasi geostrophic qg equation this problem is the analogue to the vortex patch problem for the two dimensional euler equation the special interest of the quasi geostrophic equation lies in its strong similarities with the three dimensional euler equation while being a two dimen sional model in particular an analogue of the problem considered here the evolution of sharp fronts for qg is the evolution of a vortex line for the three dimensional euler equation the rigorous derivation of an equation for the evolution of a vortex line is still an open problem the influence of the singularity appearing in the velocity when using the biot savart law still needs to be understood we present two derivations for the evolution of a periodic sharp front the first one heuristic shows the presence of a logarithmic singularity in the velocity while the second making use of weak solutions obtains a rigorous equation for the evolution explaining the influence of that term in the evolution of the curve finally using a nash moser argument as the main tool we obtain local existence and uniqueness of a solution for the derived equation in the c case 2004 wiley periodicals inc

degenerate bifurcation of the rotating patches abstract in this paper we study the existence of doubly connected rotating patches for euler equations when the classical non degeneracy conditions are not satisfied we prove the bifurcation of the v states with two fold symmetry however for higher m fold symmetry with m 3 the bifurcation does not occur this answers to a problem left open in 10 note that contrary to the known results for simply connected and doubly connected cases where the bifurcation is pitchfork we show that the degenerate bifurcation is actually transcritical these results are in agreement with the numerical observations recently discussed in 10 the proofs stem from the local structure of the quadratic form associated to the reduced bifurcation equation

the steady motion of a symmetric finite core size counterrotating vortex pair the steady motion of a symmetric finite core size counterrotating vortex pair is characterized by circulation r a velocity v and a spacing 2x in the classical limit of a point vortex the normalized velocity vx r is 1 4 the effect of finite core size is to reduce the normalized velocity below the value for a point vortex the flow is governed by a single geometrical parameter r x the ratio of effective vortex size to vortex half spacing perturbation analysis is used to derive general closed form analytical solutions for the complete velocity field the vortex pair velocity and the boundary shape for a continuum of values of r x both uniform and piecewise constant density cases are treated these solutions illustrate the different orders at which the solution deviates from the point vortex pair for example the vortex shape becomes noncircular at order r x 2 but the normalized velocity does not change until order r x 5 for the uniform density case calculation of specific values of vortex pair velocity aspect ratio and gap ratio shows good agreement with previous numerical results

dynamics near an unstable kirchhoff ellipse we describe the dynamics of the kirchhoff ellipse by formulating a nonlinear equation for the boundary of a perturbed vortex patch in elliptical coordinates we demonstrate that in the regime for which the linearized equation of motion is unstable the nonlinear dynamics of a rather general initial perturbation of the kirchhoff ellipse are determined by the fastest growing mode for the corresponding linearized equation on a time scale when the nonlinear instability occurs in particular we resolve a question suggested by love s results and prove that such elliptical patches are indeed unstable in the full nonlinear sense

instability of periodic bgk equilibria a collisionless plasma is described by the vlasov poisson equations the bgk equilibria were proposed in 1957 as the simplest spatially dependent equilibria since that time the question of their instability has been an important open problem we prove that certain periodic bgk equilibria are unstable 1995 john wiley sons inc

spatial pattern formation in the keller segel model with a logistic source this paper deals with a neumann boundary value problem in a d dimensional box t d 0 p d d 1 2 3 for the chemotaxis diffusion growth model u t d u u gu v ru 1 u k v t d v 2v au bv which describes the movement of cells in response to the presence of a chemical signal substance it is proved that given any general perturbation of magnitude d its nonlinear evolution is dominated by the corresponding linear dynamics along a finite number of fixed fastest growing modes over a time period of the order ln1 d each initial perturbation certainly can behave drastically differently from another which gives rise to the richness of patterns our results provide a mathematical characterization for the early stage pattern formation in the keller segel model

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

corotating steady vortex flows with n fold symmetry in this paper we study certain vertical flows of an ideal fluid in two dimensions which are steady relative to a uniformly rotating frame in particular we confine our analysis to those flows for which the vorticity relative to the rotating frame possesses n fold rotational symmetry for some given n 2 3 that is we require that the relative vorticity be invariant under rotation through arbitrary multiples of 2zjn as in our earlier related work 19 20 these relative steady solutions of the euler fluid dynamical equations are obtained as extremals for a certain variational problem the solutions then represent steady vortex flows in the sense defined in 19 namely each such flow has a prescribed distribution of relative vorticity and is separated into regions of zero and of positive relative vorticity in 1883 thomson 18 considered the flow due to a configuration of n unit point vortices spaced equally around the circumference of a circle the relative vorticity of such a configuration on a circle of radius a is given by

some considerations on the nonlinear stability of stationary planar euler flows we give sufficient conditions for the nonlinear stability of possibly nonsmooth stationary solutions of the two dimensional euler equation in symmetric bounded domains we use as lyapunov functions first integrals due to the symmetry of the problem moreover we investigate the stability of smooth solutions under perturbations of the boundary the last result is based on a generalization of the well known arnold approach

application of a theorem of m g krein to singular integrals on donne des estimations holderiennes et de type l 2 pour les integrales singulieres sur des espaces homogenes au sens de coifman weiss on utilise un theoreme de m g krein pour passer des estimations holderiennes aux l 2 estimations

on the v states for the generalized quasi geostrophic equations we prove the existence of the v states for the generalized inviscid sqg equations with alpha in 0 1 these structures are special rotating simply connected patches with m fold symmetry bifurcating from the trivial solution at some explicit values of the angular velocity this produces inter alia an infinite family of non stationary global solutions with uniqueness

a uniformly valid far field asymptotic expansion of the green function for two dimensional propagation above a homogeneous impedance plane abstract a generalized asymptotic expansion in the far field for the problem of cylindrical wave reflection at a homogeneous impedance plane is derived the expansion is shown to be uniformly valid over all angles of incidence and values of surface impedance including the limiting cases of zero and infinite impedance the technique used is a rigorous application of the modified steepest descent method of ott 1

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

green s function for a plane three dimensional fluid layer at small horizontal distances from the source the green s function for a plane three dimensional layer in its common form is usually determined as an infinite series of the waveguide s normal modes the slow convergence of the series close to the source limits the applicability of green s function to solving sound generation and scattering problems in waveguides especially if near field effects are significant in the present work a more convenient form of green s function for such a waveguide is obtained as a sum of two quickly converging series the first series is a difference between the green s functions for helmholtz and laplace equations whereas the second series is green s function for the laplace equation determined as a sum of the mirror images of the source in the waveguide boundaries the behaviour of the obtained function close to the source is investigated numerical experiments show significantly better convergence for the obtained function as compared with the green s function in its common form it is also shown that the obtained function can be easily calculated at the points directly underneath or above the source where the terms of the green s function series in the common form are singular

concentrated steady vorticities of the euler equation on 2 d domains and their linear stability abstract we consider concentrated vorticities for the euler equation on a smooth domain r 2 in the form of j 1 n j j j r j 2 j j d j 0 supported on well separated vortical domains j j 1 n of small diameters o r j a conformal mapping framework is set up to study this free boundary problem with j being part of unknowns for any given vorticities 1 n and small r 1 r n r through a perturbation approach we obtain such piecewise constant steady vortex patches as well as piecewise smooth lipschitz steady vorticities both concentrated near non degenerate critical configurations of the kirchhoff routh hamiltonian function when vortex patch evolution is considered as the boundary dynamics of j through an invariant subspace decomposition it is also proved that the spectral linear stability of such steady vortex patches is largely determined by that of the 2n dimensional linearized point vortex dynamics while the motion is highly oscillatory in the 2n codim directions corresponding to the vortical domain shapes

existence and regularity of rotating global solutions for the generalized surface quasi geostrophic equations motivated by the recent work of hassainia and hmidi z hassainia t hmidi on the vstates for the generalized quasi geostrophic equations arxiv preprint arxiv 1405 0858 we close the question of the existence of convex global rotating solutions for the generalized surface quasigeostrophic equation for 2 1 2 we also show c 1 regularity of their boundary for all 2 0 2

dynamics near unstable interfacial fluids we study three examples of unstable interfacial fluid motions vortex sheets with surface tension hele shaw flows with surface tension and vortex patches in all three cases the nonlinear dynamics of a large class of smooth perturbations is proven to be characterized by the corresponding fastest linear growing mode s up to the time scale of log frac 1 delta where delta is the magnitude of the initial perturbation in all three cases the analysis is based on an unified analytical framework that includes precise bounds on the growth of the linearized operator given by an explicit solution formula as well as a special sharp nonlinear energy growth estimate our main contribution is establishing this nonlinear energy growth estimate for each interface problem in certain high energy norms