random pullback attractor of a non autonomous local modified stochastic swift hohenberg equation with multiplicative noise in this paper we study the existence of the random d pullback attractor of a non autonomous local modified stochastic swift hohenberg equation with multiplicative noise in the stratonovich sense we show that a random d pullback attractor exists in h02 d when its external force has exponential growth due to the stochastic term a priori estimates are delicate we used the ornstein uhlenbeck transformation and its properties to overcome the difficulty that the stochastic term brings to us

long time dynamics of the swift hohenberg equations abstract we consider the initial boundary value problems for the swift hohenberg equation and its hyperbolic relaxation under the optimal conditions on the nonlinearity we prove the well posedness of the both problems and uniform with respect to the initial data global boundedness of the solutions then we show that the both problems possess the global attractors of optimal regularity

a center manifold for second order semilinear differential equations on the real line and applications to the existence of wave trains for the gurtin mccamy equation

polynomial growth rates abstract we consider linear equations v a t v with a polynomial asymptotic behavior that can be stable unstable and central we show that this behavior is exhibited by a large class of differential equations by giving necessary and sufficient conditions in terms of generalized polynomial lyapunov exponents for the existence of polynomial behavior in particular any linear equation in block form in a finite dimensional space with three blocks having polynomial lyapunov exponents respectively negative positive and zero has a nonuniform version of polynomial trichotomy which corresponds to the usual notion of trichotomy but now with polynomial growth rates we also obtain sharp bounds for the constants in the notion of polynomial trichotomy in addition we establish the persistence under sufficiently small nonlinear perturbations of the stability of a nonuniform polynomial contraction

smoothness of bounded solutions of nonlinear evolution equations abstract it is shown that in many cases globally defined bounded solutions of evolution equations are as smooth in time as the corresponding operator even if a general solution of the initial value problem is much less smooth i e initial values for bounded solutions are selected in such a way that optimal smoothness is attained in particular solutions which bifurcate from certain steady states such as periodic orbits almost periodic orbits and also homo and heteroclinic orbits have this property as examples a neutral functional differential equation a slightly damped non linear wave equation and a heat equation are considered in the latter case the space variable is included into the discussion of smoothness finally generalized hopf bifurcation in infinite dimensions is considered here smoothness of the bifurcation function is discussed and known results on the order of a focus are generalized

on the plane problem of the flow around a submerged beam abstract we consider the problem of the steady flow of an ideal heavy fluid around a submerged beam the problem is obtained from the free boundary problem of the flow past a submerged obstacle in the limit of bodies of vanishing thickness we introduce a special sobolev space formulation of the problem in term of a perturbed stream function and prove its unique solvability for every value of the unperturbed flow velocity with the possible exception of a discrete set depending on the geometry of the domain the asymptotic properties of the solutions are discussed

pullback attractors of nonautonomous reaction diffusion equations abstract in this paper firstly we introduce the concept of norm to weak continuous cocycle in banach space and give a technical method to verify this kind of continuity then we obtain some abstract results for the existence of pullback attractors about this kind of cocycle using the measure of noncompactness as an application we prove the existence of pullback attractors in h 0 1 of the cocycle associated with the solutions for some nonlinear nonautonomous reaction diffusion equations the attractor pullback attracts all bounded subsets of h 0 1 in the norm of h 0 1

centre manifold reduction and the stuart landau equation for fluid motions the centre manifold reduction to derive the stuart landau equation is examined a double expansion in terms of the fourier series and linear eigenfunctions is introduced in hydrodynamic equations a centre manifold reduction scheme is then applied to reduce the resultant system of ordinary differential equations to the stuart landau equation through a formal expansion in linear eigenfunctions the latter equation is shown to be equivalent with the one derived by the method of multiple scales numerical coefficients involved in the quintic stuart landau equation are evaluated for plane poiseuille flow convection in a vertical slot and rayleigh benard convection in all the cases the coefficients converge as a dimension of the ode system increases and approach the numerical values obtained by the method of multiple scales

pullback attractors for asymptotically compact non autonomous dynamical systems abstract first we introduce the concept of pullback asymptotically compact non autonomous dynamical system as an extension of the similar concept in the autonomous framework our definition is different from that of asymptotic compactness already used in the theory of random and non autonomous dynamical systems as developed by crauel flandoli kloeden schmalfuss amongst others which means the existence of a random or time dependent family of compact attracting sets next we prove a result ensuring the existence of a pullback attractor for a non autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets this attractor is minimal and in most practical applications it is unique finally we illustrate the theory with a 2d navier stokes model in an unbounded domain

small internal waves in two fluid systems this paper treats travelling waves in a heterogeneous inviscid non diffusive fluid bounded between two horizontal boundaries the fluid has two incompressible components of different but constant density and is acted on by gravity the flow is steady when viewed in a moving reference frame and gives rise to a quasilinear elliptic problem with an eigenvalue parameter related to the wave speed the small amplitude solutions are analyzed using a dynamical systems approach a center manifold reduction in combination with a conserved quantity for the flow is used to parametrise all small solutions of the full elliptic system in terms of solutions of an autonomous first order ordinary differential equation for a principal component of the wave amplitude the result is a characterization of all small waves irrotational in each fluid near the critical speed for the system they are solitary waves surges connecting distinct conjugate flows at extreme ends of the channel conjugate flows and periodic waves

lyapunov functions for trichotomies with growth rates abstract we consider linear equations x a t x that may exhibit stable unstable and central behaviors in different directions with respect to arbitrary asymptotic rates e c t determined by a function t for example the usual exponential behavior with t t is included as a very special case and when t log t we obtain a polynomial behavior we emphasize that we also consider the general case of nonuniform exponential behavior which corresponds to the existence of what we call a nonuniform exponential trichotomy this is known to occur in a large class of nonautonomous linear equations our main objective is to give a complete characterization in terms of strict lyapunov functions of the linear equations admitting a nonuniform exponential trichotomy this includes criteria for the existence of a nonuniform exponential trichotomy as well as inverse theorems providing explicit strict lyapunov functions for each given exponential trichotomy in the particular case of quadratic lyapunov functions we show that the existence of strict lyapunov sequences can be deduced from more algebraic relations between the quadratic forms defining the lyapunov functions as an application of the characterization of nonuniform exponential trichotomies in terms of strict lyapunov functions we establish the robustness of nonuniform exponential trichotomies under sufficiently small linear perturbations we emphasize that in comparison with former works our proof of the robustness is much simpler even when t t

pullback attractor for a non autonomous modified swift hohenberg equation this paper is concerned with a non autonomous modified swift hohenberg equation u t d 2u 2 du au b u 2 u 3 g x t it is shown that a pullback attractor exists when its external force has exponential growth due to the nonlinear terms b u 2 and u 3 the estimates are delicate we overcome this difficulty by imposing the exponential growth condition on the external forcing term g x t

characteristic free representation theory of the general linear group on etudie systematiquement les z formes des representations rationnelles du groupe lineaire generale

stable and unstable manifolds for hyperbolic bi semigroups we show the existence of local lipschitzian stable and unstable manifolds for the ill posed problem of perturbations of hyperbolic bi semigroups we do not assume backward nor forward uniqueness of solutions we do not use cut off functions because we do not assume global smallness conditions on the nonlinearities we introduce what we call dichotomous flows which recovers the symmetry between the past and the future thus we need to prove only a stable manifold theorem we modify the conley mcgehee moeckel approach to avoid appealing to wazewski principle and brouwer degree theory hence we allow both the stable and unstable directions to be infinite dimensional we apply our theorem to the elliptic system u u g u u in an infinite cylinder r

the littlewood richardson rule for coschur modules in the last 15 years many mathematicians have turned their attention to the problem of extending to more general settings fields of arbitrary characteristic commutative rings the representation theory of the general linear and symmetric groups over fields of characteristic zero in this undertaking the first problem is perhaps that of finding suitable generalizations of those modules that in the classical theory yield the irreducible representations first discovered and classified by schur in his dissertation of 1901 24 over fields of characteristic zero the more popular constructions of these modules in terms of symmetry classes of tensors are those due to schur 25 and weyl 31 as is well known these contructions are equivalent since they give rise to modules that even though they are different as subspaces of the tensors space turn out to be isomorphic with respect to the action of the group recently it has been recognized that such constructions can be adequately adapted in order to make sense over arbitrary commutative rings akin buchsbaum and weyman 1 have therefore introduced the so called schur and coschur functors that turn out to be the natural generalizations to commutative rings of the constructions of schur and weyl respectively these functors are universally free functors and they give rise to modules schur and coschur modules which even though they are isomorphic over fields of characteristic zero are far from being isomorphic over the integers furthermore schur and coschur modules are indecomposable but they are not in general irreducible over commutative rings working from a slightly different point of view carter and lusztig 7 discovered a class of modules which are irreducible over fields of positive characteristic these modules were later elegantly described by clausen s as the images of a natural operator the capelli operator whose domain and codomain are the so called weyl modules of the second and first kind which turn out to be the same as coschur and schur modules respectively alongside the classification of the irreducible modules lies the second 143

exponential dichotomies and homoclinic orbits in functional differential equations abstract suppose an autonomous functional differential equation has an orbit which is homoclinic to a hyperbolic equilibrium point the purpose of this paper is to give a procedure for determining the behavior of the solutions near of a functional differential equation which is a nonautonomous periodic perturbation of the original one the procedure uses exponential dichotomies and the fredholm alternative it is also shown that any smooth function p t defined on the reals which approaches zero monotonically as t is the solution of a scalar functional differential equation and generates an orbit homoclinic to zero examples illustrating the results are also given

multiple solutions of some semilinear elliptic equations in slender cylindrical domains we analyze the semilinear problem ul ul 0 in 0 2l ul 0 at 0 2l where rn 1 is a bounded smooth domain and is a given nonlinearity in the limit l the solutions of that problem are related to those of the n 1 dimensional problem v v 0 in v 0 at in particular necessary and sufficient conditions on a given solution of the latter v are given for the former to have a solution ul such that ul x1 x2 xn v x2 xn as l in most part of the cylinder 0 2l related bifurcation problems and some generalizations are also considered

stable manifolds to bounded solutions in possibly ill posed pdes abstract we prove several results establishing existence and regularity of stable manifolds for different classes of special solutions for evolution equations these equations may be ill posed a single specific solution an invariant torus filled with quasiperiodic orbits or more general manifolds of solutions in the later cases which include several orbits we also establish the invariant manifolds of an orbit depend smoothly on the orbit analytically in the case of quasi periodic orbits and finitely differentiably in the case of more general families we first establish a general abstract theorem which under suitable spectral non degeneracy analyticity assumptions on the linearized equation establishes the existence of the desired manifold related results appear in the literature but our results allow that the nonlinearity is unbounded and we obtain smoothness of the invariant manifolds this makes the results in this paper applicable to some several models of current interest that could not be treated otherwise we discuss in detail the boussinesq equation of water waves similar phenomena happen in other long wave approximations and complex ginzburg landau equation more recently we observed 11 that our results also apply to mean field games since the equations we consider may be ill posed part of the requirements for the stable manifold is that one can define the forward dynamics on them note also that the methods that are based in the existence of dynamics such as graph transform do not apply to ill posed equation we use the methods based on integral equations perron method associated with the partial dynamics but we need to take advantage of smoothing properties of the partial dynamics note that even if the families of solutions we started with are finite dimensional the stable manifolds may be infinite dimensional

heteroclinic orbits for retarded functional differential equations abstract suppose is a heteroclinic orbit of a ck functional differential equation x t x i with limit set and limit set being either hyperbolic equilibrium points or periodic orbits necessary and sufficient conditions are given for the existence of an ftf close to in ck with the property that x t tf x t has a heteroclinic orbit gg close to the orbits gg are obtained from the zeros of a finite number of bifurcation functions g f r d r d 1 transversality of is characterized by the nondegeneracy of the derivative d g it is also shown that the tf which have heteroclinic orbits near are on a ck submanifold of finite codimension max 0 ind or on the closure of it where ind is the index of

global attractor of a modified swift hohenberg equation in hk spaces abstract in this paper by using an iteration procedure regularity estimates for the linear semigroups and a classical existence theorem of global attractor we prove that a modified swift hohenberg equation u t 2 u g u possesses a global attractor in sobolev space h k for all k 0 which attracts any bounded subset of h k in the h k norm

robust heteroclinic cycles in the one dimensional complex ginzburg landau equation numerical evidence is presented for the existence of stable heteroclinic cycles in large parameter regions of the one dimensional complex ginzburg landau equation cgl on the unit spatially periodic domain these cycles connect difierent spatially and temporally inhomogeneous time periodic solutions as t 1 a careful analysis of the connections is made using a projection onto 5 complex fourier modes it is shown flrst that the time periodic solutions can be treated as relative equilibria after consideration of the symmetries of the cgl second the cycles are shown to be robust since the individual heteroclinic connections exist in invariant

stability of concatenated traveling waves alternate approaches abstract we consider a reaction diffusion equation in one space dimension whose initial condition is approximately a sequence of widely separated traveling waves with increasing velocity each of which is asymptotically stable as in 14 24 25 we show that the sequence of traveling waves is itself asymptotically stable as t the solution approaches the concatenated wave pattern with different shifts of each wave allowed our proof is similar to that of 14 in that it is based on spatial dynamics laplace transform and exponential dichotomies but it incorporates a number of modifications

small amplitude steady water waves with critical layers non symmetric waves abstract the problem for two dimensional steady water waves with vorticity is considered using methods of spatial dynamics we reduce the problem to a finite dimensional hamiltonian system the reduced system describes all small amplitude solutions of the problem and as an application we give a proof of the existence of non symmetric steady water waves whenever the number of roots of the dispersion equation is greater than one

center manifolds under nonuniform hyperbolicity maximal regularity abstract we establish the existence of invariant center manifolds with maximal c r regularity for a nonautonomous dynamics with discrete time we consider the general case of perturbations of a nonuniform exponential trichotomy our proof uses the fiber contraction principle and allows linear perturbations without any further effort

invariant manifolds for parabolic partial differential equations on unbounded domains in this paper finite dimensional invariant manifolds for nonlinear parabolic partial differential equations of the form

stable and unstable manifolds for the nonlinear wave equation with dissipation on considere la stabilite pour des solutions solitons stationnaires du probleme de cauchy sur r n u tt u t u f u 0 u x 0 u t x 0 donnes

existence of quasiperiodic solutions of elliptic equations on rn 1 via center manifold and kam theorems abstract we consider elliptic equations on r n 1 of the form 1 x u u y y g x u 0 x y r n r where g x u is a sufficiently regular function with g 0 0 we give sufficient conditions for the existence of solutions of 1 which are quasiperiodic in y and decaying as x uniformly in y such solutions are found using a center manifold reduction and results from the kam theory we discuss several classes of nonlinearities g to which our results apply

pullback attractors for the norm to weak continuous process and application to the nonautonomous reaction diffusion equations abstract in this paper we introduce the concept of norm to weak continuous process in a banach space and obtain the existence of pullback attractors for this kind of process then we give a new method for proving the existence of the pullback attractors as an application we obtain the existence of pullback attractors for nonautonomous reaction diffusion equation in h 0 1 with exponential growth of the external force

smooth center manifolds for nonuniformly partially hyperbolic trajectories abstract we establish the existence of unique smooth center manifolds for ordinary differential equations v a t v f t v in banach spaces assuming that v a t v admits a nonuniform exponential trichotomy this allows us to show the existence of unique smooth center manifolds for the nonuniformly partially hyperbolic trajectories in addition we prove that the center manifolds are as regular as the vector field our proof of the c k smoothness of the manifolds uses a single fixed point problem in an appropriate complete metric space to the best of our knowledge we establish in this paper the first smooth center manifold theorem in the nonuniform setting

exact solution of the wave resistance problem for a submerged cylinder ii the non linear problem we study the problem of wave resistance for a slender cylinder submerged in a heavy fluid of finite depth with the cylinder moving at uniform supercritical speed in the direction orthogonal to its generators we look for a divergence free irrotational flow the boundary of the region occupied by the fluid consisting of the free surface the bottom and the obstacle profile is assumed to belong to streamlines and the bernoulli condition is taken on the free surface the problem is transformed via the hodograph map into a problem set in a strip with a cut by using a hard version of the inverse function theorem and by taking account of the results obtained in part i which we recall here we prove the existence of a complex velocity function satisfying all the requirements of the problem in particular this function is continuous up to the surface of the obstacle and the only possible singularities appear at the end points where the boundary is not smooth moreover two stagnation points appear near to the extremities of the submerged body

center manifolds depending on a parameter abstract for differential equations u a t u f t u obtained from sufficiently small c 1 perturbations of a nonuniform exponential trichotomy we establish the c 1 dependence of the center manifolds on the parameter our proof uses the fiber contraction principle to establish the regularity property we note that our argument also applies to linear perturbations without further changes

lyapunov sequences for exponential trichotomies abstract for a linear cocycle with discrete time we give a complete characterization of nonuniform exponential trichotomies in terms of strict lyapunov sequences we also obtain inverse theorems by constructing explicitly strict lyapunov sequences for each nonuniform exponential trichotomy these are constructed in terms of lyapunov norms with respect to which the nonuniform behavior of the trichotomies becomes uniform we also obtain a corresponding version of the results for cocyles over measure preserving transformations

the dichotomy theorem for evolution bi families we prove that the operator g the closure of the rst order dierential operator d dt d t on l2 r x is fredholm if and only if the not well posed equation u 0 t d t u t t 2 r has exponential dichotomies on r and r and the ranges of the dichotomy projections form a fredholm pair moreover the index of this pair is equal to the fredholm index of g here x is a hilbert space d t a b t a is the generator of a bi semigroup b is a bounded piecewise strongly continuous operator valued function also we prove some perturbations results and consider various examples of not well posed problems

steady flows of inviscid fluids under localized perturbations on essaie d obtenir tous les ecoulements stationnaires dans un canal avec un petit obstacle pour des nombres de froude proches de la valeur critique fo

abundance of entire solutions to nonlinear elliptic equations by the variational method abstract we study entire bounded solutions to the equation u u u 3 0 in r 2 our approach is purely variational and is based on concentration arguments and symmetry considerations this method allows us to construct in an unified way several types of solutions with various symmetries radial breather type rectangular triangular hexagonal etc both positive and sign changing it is also applicable for more general equations in any dimension

a center stable manifold theorem for differential equations in banach spaces we prove a center stable manifold theorem for a class of differential equations in infinite dimensional banach spaces

robustness of nonuniform exponential trichotomies in banach spaces abstract we establish the robustness under sufficiently small linear perturbations of nonuniform exponential trichotomies defined by linear equations x a t x in banach spaces we also establish the continuous dependence on the perturbation of the constants in the notion of trichotomy we consider both trichotomies in semi infinite intervals and trichotomies in r

numerical computation of solitary waves in infinite cylindrical domains the numerical computation of solitary waves to semilinear elliptic equations in infinite cylindrical domains is investigated rather than solving on the infinite cylinder the equation is approximated by a boundary value problem on a finite cylinder convergence and stability results for this approach are given it is also shown that galerkin approximations can be used to compute solitary waves of the elliptic problem on the finite cylinder in addition it is demonstrated that the aforementioned procedures simplify in cases where the elliptic equation admits an additional reversibility structure finally the theoretical predictions are compared with numerical computations in particular post buckling of an infinitely long cylindrical shell under axial compression is considered it is shown numerically that for a fixed spatial truncation the error in the truncation scales with the length of the cylinder as predicted theoretically

pullback attractors for a non autonomous generalized 2d parabolic system in an unbounded domain abstract the existence of a pullback attractor is proven for a non autonomous generalized 2d parabolic system in an unbounded domain the asymptotic compactness of the solution operator is obtained by the uniform estimates on the tails of solutions

two dimensional travelling waves over a moving bottom abstract two dimensional travelling waves on an ideal fluid with gravity and surface tension over a periodically moving bottom with a small amplitude are studied the bottom and the wave travel with a same speed the exact euler equations are formulated as a spatial dynamic system by using the stream function a manifold reduction technique is applied to reduce the system into one of ordinary differential equations with finite dimensions a homoclinic solution to the normal form of this reduced system persists when higher order terms are added which gives a generalized solitary wave the homoclinic solution connecting a periodic solution

normal hyperbolicity of center manifolds and saint venant s principle the concept of normal hyperbolicity of center manifolds is generalized to infinite dimensional differential equations in particular to elliptic problems in cylindrical domains it is shown that all solutions u staying close to the center manifold for t l l satisfy an estimate of the form left u t tilde u t right leqslant ce alpha l t where c and are independent of l and is a solution on the center manifold these results are applied to saint venant s principle for the static deformation of nonlinearly elastic prismatic bodies the use of the center manifold permits the effective treatment of the general case of non zero resultant forces and moments acting on each cross section

multi hump pulses in systems with reflection and phase invariance abstract we investigate the existence and stability of standing and travelling multi hump waves in partial differential equations with reflection and phase symmetries we focus on 2 and 3 pulse solutions that arise near bi foci and apply our results to the complex cubic quintic ginzburg landau equation

schur modules weyl modules and capelli operators much work has been done in recent years towards presenting explicitly and with a minimum of machinery the irreducible representations of the general linear group gl n over a field k the representations obtained from the modules named after schur and weyl have undergone a tortuous refining process in the hands of notable authors akin buchsbaum and weyman 1 2 carter and lusztig 7 clausen 8 9 de concini eisenbud and procesi 10 green 16 james and kerber 19 towber 26 27 in the wake of the pioneering work of schur 23 and weyl 28 one objective of such simplifications has been that of obtaining information as to the reducibility of representations when the field k if of positive characteristic an objective which seems as yet out of reach the purpose of this work is that of organically presenting the main results on schur and weyl modules our techniques are at variance with most of those previously used and we should like to believe our proofs signify a return to the primordial combinatorial underpinning of the topic we hope thereby to have attained a maximum of characteristic freeness our two warhorses are 1 the well known straightening technique first used by doubilet rota and stein 15 and recently generalized to the supersymmetric case by grosshans rota and stein 17 an altogether different approach from which we have also benefitted has been developed by akin buchsbaum and weyman 1 both techniques has been skillfully displayed in the recent work of kouwenhoven 21 2 the use of polarization operators combined with the striking power of kostant s z form 20 in the guise of de sarme nien kung and rota s capelli operators 13 superalgebraic methods are used in obtaining two distinct standard basis theorems for the case of the exterior letterplace algebra such an algebra with its straightening possibilities was first considered by doubilet and rota 14 and then brought to fruish by akin buchsbaum and weyman 1 by these methods we derive what is perhaps the main novelty of the doi 10 1006 aima 1999 1872 available online at http www idealibrary com on

polynomial contractions and lyapunov regularity abstract the purpose of this note is twofold to introduce the notion of polynomial contraction for a linear nonautonomous dynamics with discrete time and to show that it persists under sufficiently small linear and nonlinear perturbations the notion of polynomial contraction mimics the notion of exponential contraction but with the exponential decay replaced by a polynomial decay we show that this behavior is exhibited by a large class of dynamics by giving necessary conditions in terms of polynomial lyapunov exponents finally we establish the persistence of the asymptotic stability of a polynomial contraction under sufficiently small linear and nonlinear perturbations we also consider the case of nonuniform polynomial contractions for which the lyapunov stability is not uniform

nonconvergent bounded solutions of semilinear heat equations on arbitrary domains abstract we consider the dirichlet problem for the semilinear heat equation 0 1 u t u g x u x where is an arbitrary bounded domain in r n n 2 with c 2 boundary we find a c function g x u such that 0 1 has a bounded solution whose limit set is a continuum of equilibria this extends and improves an earlier result of the first author with rybakowski in which is a disk in r 2 and g is of finite differentiability class we also show that 0 1 can have an infinite dimensional manifold of nonconvergent bounded trajectories

the neumann kelvin problem for a beam abstract we discuss existence uniqueness and regularity of the solution of the plane neumann kelvin problem in a strip in the case of a beam the problem is obtained by linearization of the equations of the wave resistance problem for a slender cylinder semisubmerged in a heavy fluid and moving at uniform supercritical speed in the direction orthogonal to its generators the results obtained may be relevant for the resolution of the nonlinear wave resistance problem

complements of tableaux and straightening bideterminants abstract in this paper we study the combinatorics related to complements of tableaux using the representation theory of general linear groups and a known construction of the dual weyl modules we obtain a new straightening algorithm for bideterminants and prove that it is compatible with the complement construction of tableaux as an application we get a purely combinatorial construction of the isomorphisms between generalized classical schur algebras introduced first by henke koenig and the author

smooth conjugacy of centre manifolds in this paper we prove that for a system of ordinary differential equations of class c r 1 1 r 0 and two arbitrary c r 1 1 local centre manifolds of a given equilibrium point the equations when restricted to the centre manifolds are c r conjugate the same result is proved for similinear parabolic equations the method is based on the geometric theory of invariant foliations for centre stable and centre unstable manifolds

symbolic method in invariant theory abstract a symbolic method based on an extension of the straightening algorithm is developed for the representation of joint invariants of symmetric and skew symmetric tensors for skew symmetric tensors the method holds over infinite fields of arbitrary characteristic

exact theory of secondary supercritical solutions for steady surface waves over a bump abstract two dimensional steady surface waves on an ideal fluid over a small and symmetric bump are studied when the froude number f of the uniform flow far upstream is greater than unity recent numerical results for a symmetric bump indicate that corresponding to f 1 there may exist two solutions for the problem one approaches the uniform flow and the other a solitary wave as the bump size goes to zero in this paper we use a two parameter asymptotic expansion to derive an explicit approximate expression for the later solution and show that the expansion is indeed an asymptotic approximation to the solution of the exact nonlinear equations

on the stability of periodic travelling waves with large spatial period abstract in many circumstances a pulse to a partial differential equation pde on the real line is accompanied by periodic wave trains that have arbitrarily large period it is then interesting to investigate the pde stability of the periodic wave trains given that the pulse is stable using the evans function gardner has demonstrated that every isolated eigenvalue of the linearization about the pulse generates a small circle of eigenvalues for the linearization about the periodic waves in this article the precise location of these circles is determined it is demonstrated that the stability properties of the periodic waves depend on certain decay and oscillation properties of the tails of the pulse as a consequence periodic waves with long wavelength typically destabilize at homoclinic bifurcation points at which multi hump pulses are created that is in contrast to the situation for the underlying pulses whose stability properties are not affected by these bifurcations the proof uses lyapunov schmidt reduction and relies on the existence of exponential dichotomies the approach is also applicable to periodic waves with large spatial period of elliptic problems on r n or on unbounded cylinders r with bounded

dichotomous solutions for semilinear ill posed equations with sectorially dichotomous operator abstract in this paper we study a class of semilinear ill posed equations with sectorially dichotomous operator s on banach space z firstly we give a direct sum decomposition of z z z z corresponding to spectrum of s such that hyperbolic bisectorial operator s can be split into two sectorial operators s z and s z on z and z respectively then we construct the intermediate spaces between whole space z and domain d s of sectorially dichotomous operator s following elbialy s works we propose the dichotomous initial condition for this semilinear ill posed equation and obtain the existence uniqueness continuous dependence on the dichotomous initial value regularity and z estimate of dichotomous solutions as applications of the results we give the existence and uniqueness of local solutions for an elliptic pde in infinite cylindrical domain and an abstract semilinear ill posed equation with non dense domain

exponential dichotomies for solitary wave solutions of semilinear elliptic equations on infinite cylinders in applications solitary wave solutions of semilinear elliptic equations u g u u 0 x y r in infinite cylinders frequently arise as travelling waves of parabolic equations as such their bifurcations are an interesting issue interpreting elliptic equations on infinite cylinders as dynamical systems inxhas proved very useful still there are major obstacles in obtaining for instance bifurcation results similar to those for ordinary differential equations in this article persistence and continuation of exponential dichotomies for linear elliptic equations is proved with this technique at hands lyapunov schmidt reduction near solitary waves can be applied as an example existence of shift dynamics near solitary waves is shown if a perturbation h x u u periodic inxis added

pattern selection of solutions of the swift hohenberg equation abstract we study the large time behaviour of solutions u x t of the swift hohenberg equation on a one dimensional domain 0 l focusing in particular on the role of the eigenvalue parameter and the length l of the domain on the selection of the limiting profiles v we show by means of numerical simulations how different values of these parameters may lead to qualitatively different final profiles and prove the existence of a collection n of disjoint intervals which depend on the value of such that if l n then solutions all converge to the trivial solution v 0 we show that branches of nontrivial solutions bifurcate from the trivial solution at the endpoints of these intervals and we study the local behaviour of these branches we establish global bounds for the solutions u x t and identify the global attractor finally we derive an estimate about the shape of the final state

analytic and algebraic conditions for bifurcations of homoclinic orbits i saddle equilibria abstract we study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in a class of four dimensional systems which may be hamiltonian or not only one parameter is enough to treat these types of bifurcations in hamiltonian systems but two parameters are needed in general systems we apply a version of melnikov s method due to gruendler to obtain saddle node and pitchfork types of bifurcation results for homoclinic orbits furthermore we prove that if these bifurcations occur then the variational equations around the homoclinic orbits are integrable in the meaning of differential galois theory under the assumption that the homoclinic orbits lie on analytic invariant manifolds we illustrate our theories with an example which arises as stationary states of coupled real ginzburg landau partial differential equations and demonstrate the theoretical results by numerical ones

schur functors and schur complexes the study of schur functors has a relatively long history its main impetus derived from representation theory originally in characteristic zero over the years however with the development of modular representations and algebraic geometry over fields of positive characteristic the need for a theory of universal polynomial functors increased and since the mid 1960s approaches to a characteristic free treatment of schur functors have been developing see for instance the recent book of green 1 l in which the treatments by carter and lusztig 6 higman 12 and towber 20 among others are discussed our own interest in such a treatment was awakened by the work of lascoux 141 on resolutions of determinantal ideals although his thesis treated only the characteristic zero case it suggested that a general and elementary theory of schur functors could be developed using only the rudiments of multilinear algebra involving the hopf algebra structures of the symmetric exterior and divided power algebras moreover this elementary development admitted of a natural generalization to the idea of schur complexes whose usefulness was demonstrated for instance in our construction of a universal minimal

invariant manifolds for flows in banach spaces on presente une theorie des varietes invariantes lisses basee sur la methode classique de lyapunov penon pour des semi flots continus dans des espaces de banach

center manifolds for infinite delay we obtain a c1 center manifold theorem for perturbations of delay difference equations in banach spaces with infinite delay our results extend in several directions the existing center manifold theorems besides considering infinite delay equations we consider perturbations of nonuniform exponential trichotomies and generalized trichotomies that may exhibit stable unstable and central behaviors with respect to arbitrary asymptotic rates ec n for some diverging sequence n this includes as a very special case the usual exponential behavior with n n