morse theory for periodic solutions of hamiltonian systems and the maslov index in this paper we prove morse type inequalities for the contractible 1 periodic solutions of time dependent hamiltonian differential equations on those compact symplectic manifolds m for which the symplectic form and the first chern class of the tangent bundle vanish over q m the proof is based on a version of infinite dimensional morse theory which is due to floer the key point is an index theorem for the fredholm operator which plays a central role in floer homology the index formula involves the maslov index of nondegenerate contractible periodic solutions this maslov index plays the same role as the morse index of a nondegenerate critical point does in finite dimensional morse theory we shall use this connection between floer homology and maslov index to establish the existence of infinitely many periodic solutions having integer periods provided that every i periodic solution has at least one floquet multiplier which is not equal to 1

poincare birkhoff theorems in random dynamics we propose a generalization of the poincare birkhoff theorem on area preserving twist maps to area preserving twist maps f that are random with respect to an ergodic probability measure in this direction we will prove several theorems concerning existence density and type of the fixed points to this end first we introduce a randomized version of generalized generating functions and verify the correspondence between its critical points and the fixed points of f a fact which we successively exploit in order to prove the theorems the study we carry out needs to combine probabilistic techniques with methods from nonlinear pde and from differential geometry notably moser s method and conley zehnder theory our stochastic model in the periodic case coincides with the classical setting of the poincare birkhoff theorem

finite volume flows and morse theory in this paper we present a new approach to morse theory based on the de rham federer theory of currents the full classical theory is derived in a transparent way the methods carry over uniformly to the equivariant and the holomorphic settings moreover the methods are substantially stronger than the classical ones and have interesting applications to geometry they lead for example to formulas relating characteristic forms and singularities of bundle maps

compact families of piecewise constant functions in lp 0 t b abstract a strong compactness result in the spirit of the lions aubin simon lemma is proven for piecewise constant functions in time u with values in a banach space the main feature of our result is that it is sufficient to verify one uniform estimate for the time shifts u u instead of all time shifts u u h for h 0 as required in simon s compactness theorem this simplifies significantly the application of the rothe method in the existence analysis of parabolic problems

on equivalence of floer s and quantum cohomology we show that the floer cohomology and quantum cohomology rings of the almost kahler manifoldm both defined over the novikov ring of the loop space m are isomorphic we do it using a brst trivial deformation of the topological a model the relevant aspect of noncompactness of the moduli of pseudoholomorphic instantons is discussed it is shown nonperturbatively that any brst trivial deformation of a model which does not change the dimensions of brst cohomology does not change the topological correlation functions either

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

how strong a logistic damping can prevent blow up for the minimal keller segel chemotaxis system in this paper we study the minimal keller segel model with a logistic source and obtain quantitative and qualitative descriptions of the competition between logistic damping and other ingredient especially chemotactic aggregation to guarantee boundedness and convergence more specifically we establish how precisely strong a logistic source can prevent blow up and then we obtain an explicit relationship between logistic damping and other ingredient especially chemotactic aggregation so that convergences are ensured and their respective convergence rates are explicitly calculated out known results in the literature are completed and refined furthermore our findings provide clues on how to produce blowup solutions for ks chemotaxis models with logistic sources

aggregating pattern dynamics in a chemotaxis model including growth abstract a population model including diffusion chemotaxis and growth is studied assuming that the diffusion rate and the chemotactic rate are both very small compared with the growth rate we derive a new equation to describe the time evolution of the aggregating region of biological individuals and show the conditions for the existence and stability of radially symmetric equilibrium solutions of the equation which indicate the aggregation of individuals

an improved local well posedness result for the one dimensional zakharov system abstract the 1d cauchy problem for the zakharov system is shown to be locally well posed for low regularity schrodinger data u 0 h k p and wave data n 0 n 1 h l p h l 1 p under certain assumptions on the parameters k l and 1 p 2 where u 0 h k p k u 0 l p generalizing the results for p 2 by ginibre tsutsumi and velo especially we are able to improve the results from the scaling point of view and also allow suitable k 0 l 1 2 i e data u 0 l 2 and n 0 n 1 h 1 2 h 3 2 which was excluded in the case p 2

boundedness and finite time collapse in a chemotaxis system with volume filling effect abstract we consider the elliptic parabolic pde system u t u u u v x t 0 0 v m u x t 0 with nonnegative initial data u 0 having mean value m under homogeneous neumann boundary conditions in a smooth bounded domain r n the nonlinearities and are supposed to generalize the prototypes u u 1 p u u u 1 q 1 with p 0 and q r problems of this type arise as simplified models in the theoretical description of chemotaxis phenomena under the influence of the volume filling effect as introduced by painter and hillen k j painter t hillen volume filling and quorum sensing in models for chemosensitive movement can appl math q 10 2002 501 543 it is proved that if p q 2 n then all solutions are global in time and bounded whereas if p q 2 n q 0 and is a ball then there exist solutions that become unbounded in finite time the former result is consistent with the aggregation inhibiting effect of the volume filling mechanism the latter however is shown to imply that if the space dimension is at least three then chemotactic collapse may occur despite the presence of some nonlinearities that supposedly model a volume filling effect in the sense of painter and hillen

a stable category of lagrangian cobordisms abstract given an exact symplectic manifold m and a support lagrangian m we construct an category lag m which we conjecture to be equivalent after specialization of the coefficients to the partially wrapped fukaya category of m relative to roughly speaking the objects of lag m are lagrangian branes inside of m t r n for large n and the morphisms are lagrangian cobordisms that are non characteristic with respect to the main theorem of this paper is that lag m is a stable category and in particular its homotopy category is triangulated with mapping cones given by an elementary construction the shift functor is equivalent to the familiar shift of grading for lagrangian branes

the unregularized gradient flow of the symplectic action the symplectic action can be defined on the space of smooth paths in a symplectic manifold p which join two lagrangian submanifolds of p to pursue a new approach to the variational theory of this function we define on a subset of the path space the flow whose trajectories are given by the solutions of the cauchy riemann equation with respect to a suitable almost complex structure on p in particular we prove compactness and transversality results for the set of bounded trajectories

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

morse theory for lagrangian intersections soit p une variete symplectique compacte et soit l p une sous variete lagrangienne avec 2 p l 0 pour un diffeomorphisme exact de p avec la propriete que l coupe l transversalement on demontre une inegalite de morse reliant l ensemble l l a la cohomologie de l

morse homology for generating functions of lagrangian submanifolds the purpose of the paper is to give an alternative construction and the proof of the main properties of symplectic invariants developed by viterbo our approach is based on morse homology theory this is a st ep towards relating the finite dimensional symplectic invariants constructed via generating functions to the infinite dimensional ones constructed via floer theory in y g oh symplectic topology as the geometry of action functional i j diff geom 46 1997 499 577

on the convergence of the solution for a generalized zakharov system in this paper we study the limit behavior of the smooth solution for the generalized zakharov system with a magnetic field which describes the spontaneous generation of a magnetic field in a cold plasma by showing uniform bounds of solutions for this system with suitable initial data we prove that solution of the generalized zakharov system converges to the solution of the nonlinear schrodinger equation with a magnetic field

witten s complex and infinite dimensional morse theory we investigate the relation between the trajectories of a finite dimensional gradient flow connecting two critical points and the cohomology of the surrounding space the results are applied to an infinite dimensional problem involving the symplect c action function

on a class of keller segel chemotaxis systems with cross diffusion abstract in this paper we study a class of keller segel chemotaxis systems with cross diffusion by using the entropy dissipation method and assuming mainly the chemotactic sensitivity separates the cell density and the chemical signal we first establish the existence of global weak solutions with the effects of cross diffusion included in 3 d then we show there is a critical cross diffusion rate c such that no patterns may be expected for c while patterns are formed for c and their stability is also derived in particular in 1 d patterns are always formed whenever c and the chemotactic coefficient is larger than an expressible bifurcation value and there is another critical cross diffusion rate c c such that cells with cross diffusion rate c c are stable while for cells with c to be stable their degradation rate must be less than a threshold value hence in some sense cross diffusion is harmful to enable pattern formation while it is helpful to stabilize the cells once patterns are formed finally we show that the cross diffusion plays a role in regularizing the cell aggregation phenomenon for large chemotactic coefficient our results provide global dynamics and insights on how the biological parameters especially the cross diffusion affect pattern formations

a morse complex for infinite dimensional manifolds part i abstract in this paper and in the forthcoming part ii we introduce a morse complex for a class of functions f defined on an infinite dimensional hilbert manifold m possibly having critical points of infinite morse index and co index the idea is to consider an infinite dimensional subbundle or more generally an essential subbundle of the tangent bundle of m suitably related with the gradient flow of f this part i deals with the following questions about the intersection w of the unstable manifold of a critical point x and the stable manifold of another critical point y finite dimensionality of w possibility that different components of w have different dimension orientability of w and coherence in the choice of an orientation compactness of the closure of w classification up to topological conjugacy of the gradient flow on the closure of w in the case dim w 2

quivers floer cohomology and braid group actions we consider the derived categories of modules over a certain family a m of graded rings and floer cohomology of lagrangian intersections in the symplectic manifolds which are the milnor fibres of simple singularities of type a m we show that each of these two rather different objects encodes the topology of curves on an m 1 punctured disc we prove that the braid group b m 1 acts faithfully on the derived category of a m modules and that it injects into the symplectic mapping class group of milnor fibers the philosophy behind our results is as follows using floer cohomology one should be able to associate to the milnor fibre a triangulated category its construction has not been carried out in detail this triangulated category should contain a full subcategory which is equivalent up to a slight difference in the grading to the derived category of a m modules the full embedding would connect the two occurrences of the braid group thus explaining the similarity between them

transversality in elliptic morse theory for the symplectic action our goal in this paper is to settle some transversality question for the perturbed nonlinear cauchy riemann equations on the cylinder these results play a central role in the denition of symplectic floer homology and hence in the proof of the arnold conjecture there is currently no other reference to these transversality results in the open literature our approach does not require aronszajn s theorem instead we derive the unique continuation theorem from a generalization of the carleman similarity principle

the role of the legendre transform in the study of the floer complex of cotangent bundles consider a classical hamiltonian h on the cotangent bundle t m of a closed orientable manifold m and let l tm r be its legendre dual lagrangian in a previous paper we constructed an isomorphism from the morse complex of the lagrangian action functional that is associated to l to the floer complex that is determined by h in this paper we give an explicit construction of a homotopy inverse of contrary to other previously defined maps going in the same direction is an isomorphism at the chain level and preserves the action filtration its definition is based on counting floer trajectories on the negative half cylinder that on the boundary satisfy half of the hamilton equations albeit not of lagrangian type such a boundary condition defines fredholm operators with good compactness properties we also present a heuristic argument which independently of any fredholm and compactness analysis explains why the spaces of maps that are used in the definition of and are the natural ones the legendre transform plays a crucial role both in our rigorous and in our heuristic arguments we treat with some detail the delicate issue of orientations and show that the homology of the floer complex is isomorphic to the singular homology of the loop space of m with a system of local coefficients which is defined by the pullback of the second stiefel whitney class of tm on 2 tori in m 2015 wiley periodicals inc

morse novikov theory and cohomology with forward supports we present a new approach to morse and novikov theories based on the derham federer theory of currents using the finite volume flow technique of harvey and lawson hl in the morse case we construct a noncompact analogue of the morse complex relating a morse function to the cohomology with compact forward supports of the manifold this complex is then used in novikov theory to obtain a geometric realization of the novikov complex as a complex of currents and a new characterization of novikov homology as cohomology with compact forward supports two natural backward forward dualities are also established a lambda duality over the novikov ring and a topological vector space duality over the reals

unstable geodesics and topological field theory a topological field theory is used to study the cohomology of mapping space the cohomology is identified with the becchi rouet stora tyutin cohomology realizing the physical hilbert space and the coboundary operator given by the calculations of tunneling between the perturbative vacua the method is illustrated by a simple example

low regularity global well posedness for the zakharov and klein gordon schrodinger systems we prove low regularity global well posedness for the id zakharov system and the 3d klein gordon schrodinger system which are systems in two variables u r d x x r t c and n r d x x r t r the zakharov system is known to be locally well posed in u n l 2 x h and the klein gordon schrodinger system is known to be locally well posed in u n l 2 x l 2 here we show that the zakharov and klein gordon schrodinger systems are globally well posed in these spaces respectively by using an available conservation law for the l 2 norm of u and controlling the growth of n via the estimates in the local theory

boundedness and global existence in the higher dimensional parabolic parabolic chemotaxis system with without growth source abstract in this paper we are concerned with a general class of quasilinear parabolic parabolic chemotaxis systems with without growth source under homogeneous neumann boundary conditions in a smooth bounded domain r n with n 2 it is recently known that blowup is possible even in the presence of superlinear growth restrictions here we derive new and interesting characterizations on the growth versus the boundedness we show that the hard task of proving the l boundedness of the cell density can be reduced to proving its l r boundedness in other words we show that the l r boundedness of the cell density can successfully guarantee its l boundedness and hence its global boundedness where r n or n 2 depending on whether the growth restriction is essentially linear including no growth or superlinear hence a blowup solution also blows up in l p norm for any suitably large p more detailed information on how the growth source affects the boundedness of the solution is derived these results reveal deep understandings of blowup mechanism for chemotaxis models then we use these criteria to establish uniform boundedness and hence global existence of the underlying models logistic source in 2 d cubic source as initially proposed by mimura and tsujikawa in 3 d n 1 st source in n d with n 4 as a consequence in a chemotaxis growth model blowup is impossible if the growth effect is suitably strong finally we underline that our results remove the commonly assumed convexity on the domain

lagrangian intersections and the serre spectral sequence for a transversal pair of closed lagrangian submanifolds l lof a sym plectic manifold m such that 1 l 1 l 0 c1 2 m 2 m and for a generic almost complex structure j we construct an invariant with a high homotopical content which consists in the pages of order 2o f as pec tral sequence whose differentials provide an algebraic measure of the high dimensional moduli spaces of pseudo holomorpic strips of finite energy that join l and l when l and lare hamiltonian isotopic we show that the pages of the spectral sequence coincide up to a horizontal translation with the terms of the serre spectral sequence of the path loop fibration l pl l and we deduce some applications

boundedness in a quasilinear parabolic parabolic keller segel system with subcritical sensitivity abstract we consider the quasilinear parabolic parabolic keller segel system u t d u u s u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a convex smooth bounded domain r n with n 1 it is proved that if s u d u c u with 2 n and some constant c 0 for all u 1 then the classical solutions to the above system are uniformly in time bounded provided that d u satisfies some technical conditions such as algebraic upper and lower growth resp decay estimates as u this boundedness result is optimal according to a recent result by the second author winkler 2010 27 which says that if s u d u c u for u 1 with c 0 and some 2 n n 2 then for each mass m 0 there exist blow up solutions with mass u 0 m in addition this paper also proves a general boundedness result for quasilinear non uniformly parabolic equations by modifying the iterative technique of moser alikakos alikakos 1979 1

boundedness and blow up in the higher dimensional attraction repulsion chemotaxis system with nonlinear diffusion abstract this paper deals with the following quasilinear attraction repulsion chemotaxis system u t d u u u v u w x t 0 v t v u v x t 0 0 w u w x t 0 in a bounded domain r n n 2 under homogeneous neumann boundary conditions the parameters and are positive and the diffusion d u is supposed to generalize the prototype d u d 0 u with d 0 0 and r under the assumption 2 n 1 it is proved that the corresponding initial boundary problem possesses a nonnegative globally bounded solution moreover by constructing a lyapunov functional and by assuming that 2 n 1 and n 3 it is proved that there exist radially symmetric solutions which blow up in finite time in b r 0 with r 0 finally it is also shown that blow up may occur if 2 n 1 and 0 when is a ball in r n with n 3

on the cauchy problem for the zakharov system abstract we study the local cauchy problem in time for the zakharov system 1 1 and 1 2 governing langmuir turbulence with initial data u 0 n 0 t n 0 h k h lscr h l 1 in arbitrary space dimension we define a natural notion of criticality according to which the critical values of k l are 2 3 2 2 2 using a method recently developed by bourgain we prove that the zakharov system is locally well posed for a variety of values of k l the results cover the whole subcritical range for 4 for 3 they cover only part of it and the lowest admissible values are k l 1 2 0 for 2 3 and k l 0 1 2 for 1 as a by product of the one dimensional result we prove well posedness of the benney system 1 14 and 1 15 governing the interaction of short and long waves for the same values of k l

semi classical limit of the bottom of spectrum of a schrodinger operator on a path space over a compact riemannian manifold abstract we determine the limit of the bottom of spectrum of schrodinger operators with variable coefficients on wiener spaces and path spaces over finite dimensional compact riemannian manifolds in the semi classical limit these are extensions of the results in s aida semiclassical limit of the lowest eigenvalue of a schrodinger operator on a wiener space j funct anal 203 2 2003 401 424 the problem on path spaces over riemannian manifolds is considered as a problem on wiener spaces by using ito s map however the coefficient operator is not a bounded linear operator and the dependence on the path is not continuous in the uniform convergence topology if the riemannian curvature tensor on the underling manifold is not equal to 0 the difficulties are solved by using unitary transformations of the schrodinger operators by approximate ground state functions and estimates in the rough path analysis

blow up and nonlinear instability for the magnetic zakharov system abstract this study deals with the generalized zakharov system with magnetic field first of all we construct a kind of blow up solutions and establish the existence of blow up solutions to the system through considering an elliptic system next we show the nonlinear instability for a kind of periodic solutions in addition we consider the concentration properties of blow up solutions for the system under study at the end of this paper we establish the global existence of weak solutions to the cauchy problem of the system under consideration

on global smooth solution for generalized zakharov equations this paper considers the existence and uniqueness of the global smooth solution for the initial value problem of generalized zakharov equations in dimension two by means of a priori integral estimates galerkin method and compactness theory one has the existence and uniqueness for small initial data

boundedness vs blow up in a chemotaxis system abstract we determine the critical blow up exponent for a keller segel type chemotaxis model where the chemotactic sensitivity equals some nonlinear function of the particle density assuming some growth conditions for the chemotactic sensitivity function we establish an a priori estimate for the solution of the problem considered and conclude the global existence and boundedness of the solution furthermore we prove the existence of solutions that become unbounded in finite or infinite time in that situation where this a priori estimate fails

morse homotopy and chern simons perturbation theory we define and invariant of a three manifold equipped with a flat bundle with vanishing homology the construction is based on morse theory using several morse functions simultaneously and is regarded as a higher loop analogue of various product operations in algebraic topology there is a heuristic argument that this invariant is related to perturbative chern simons gauge theory by axelrod singer etc there is also a theorem which gives a relation of the construction to open string theory on the cotangent bundle

kirwan novikov inequalities on a manifold with boundary we extend the novikov morse type inequalities for closed 1 forms in 2 directions first we consider manifolds with boundary second we allow a very degenerate structure of the critical set of the form assuming only that the form is non degenerated in the sense of kirwan in particular we obtain a generalization of a result of floer about the usual morse inequalities on a manifold with boundary we also obtain an equivariant version of our inequalities our proof is based on an application of the witten deformation technique the main novelty here is that we consider the neighborhood of the critical set as a manifold with a cylindrical end this leads to a considerable simplification of the local analysis in particular we obtain a new analytic proof of the morse bott inequalities on a closed manifold

on global bifurcation for quasilinear elliptic systems on bounded domains general second order quasilinear elliptic systems with nonlinear boundary conditions on bounded domains are formulated into nonlinear mappings between sobolev spaces it is shown that the linearized mapping is a fredholm operator of index zero this and the abstract global bifurcation theorem of jacobo pejsachowicz patrick j rabier degree theory for c1 fredholm mappings of index 0 j anal math 76 1998 289 319 allow us to carry out bifurcation analysis directly on these elliptic systems at the abstract level we establish a unilateral global bifurcation result that is needed when studying positive solutions finally we supply two examples of cross diffusion population model and chemotaxis model to demonstrate how the theory can be applied

the conley conjecture we prove the conley conjecture for a closed symplectically aspherical symplectic manifold a hamiltonian diffeomorphism of such a manifold has infinitely many periodic points more precisely we show that a hamiltonian diffeomorphism with finitely many fixed points has simple periodic points of arbitrarily large period this theorem generalizes for instance a recent result of hingston establishing the conley conjecture for tori

well posedness for chemotaxis dynamics with nonlinear cell diffusion abstract this paper investigates the well posedness of a reaction diffusion system of chemotaxis type with a nonlinear diffusion coefficient and a dynamics growth death of the cell population b and a stationary equation for the chemoattractant c with respect to other works in which a nonlinear diffusion for cells has been considered we treat here two distinct cases for this diffusion coefficient the first in which it is a positive bounded function on r and the other in which it may display a singularity at a finite value of the cell density essentially the latter model is new and describes the saturation of the cell population in the neighborhood of a critical value for its diffusion coefficient the chemotactic sensitivity function is supposed to depend both on the cell and chemoattractant densities for homogeneous neumann boundary conditions for the cell population and chemoattractant we prove the existence of a local in time solution when the l 2 norm of the initial datum b 0 is sufficiently small and compute the maximum time interval for which the solution is bounded and smooth under a stronger assumption related to the chemotactic sensitivity we show that there exists a global in time solution for arbitrarily large initial data in a case when the diffusion coefficient is singular we focus on a model expressed by a variational inequality describing the saturation of the cell population at the blowing up diffusion value here the proof requires the study of an intermediate problem with robin boundary conditions which may be interesting by itself in all situations uniqueness follows on a time interval included not necessarily strictly in that of the solution existence under sufficient conditions

symplectic fixed points and holomorphic spheres letp be a symplectic manifold whose symplectic form integrated over the spheres inp is proportional to its first chern class on the loop space ofp we consider the variational theory of the symplectic action function perturbed by a hamiltonian term in particular we associate to each isolated invariant set of its gradient flow an abelian group with a cyclic grading it is shown to have properties similar to the homology of the conley index in locally compact spaces as an application we show that if the fixed point set of an exact diffeomorphism onp is nondegenerate then it satisfies the morse inequalities onp we also discuss fixed point estimates for general exact diffeomorphisms

new topological field theories in two dimensions it is shown that two 1 1 dimensional 2d free abelian and self interacting non abelian gauge theories without any interaction with matter fields belong to a new class of topological field theories tfts these new theories capture together some of the key features of witten and schwarz types of tft because they are endowed with symmetries that are reminiscent of the schwarz type theories but their lagrangian density has the appearance of the witten type theories the topological invariants for these theories are computed on a 2d compact manifold and their recursion relations are obtained these new theories are shown to provide a class of tractable field theoretical models for the hodge theory in two dimensions of flat minkowski spacetime where there are no propagating degrees of freedom associated with the 2d gauge boson

equations of langmuir turbulence and nonlinear schrodinger equation smoothness and approximation we consider the following family of systems parametrized by e 0 which describe langmuir s turbulence in the case where initial data are sufficiently small we study the asymptotic behaviour of the solutions ee ne when e goes to zero for d 1 2 3 namely we state convergence results of ee ne to the couple e e 2 where e is the solution of the nonlinear schrodinger equation moreover in three dimensions we show that the smooth solution ee ne is defined on 0 tmax e with tmax e going to infinity as e goes to 0

energy convergence for singular limits of zakharov type systems we prove existence and uniqueness of solutions to the klein gordon zakharov system in the energy space h1 l2 on some time interval which is uniform with respect to two large parameters c and these two parameters correspond to the plasma frequency and the sound speed in the simultaneous high frequency and subsonic limit we recover the nonlinear schrodinger system at the limit we are also able to say more when we take the limits separately

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely

morse homology for the heat flow we use the heat flow on the loop space of a closed riemannian manifold viewed as a parabolic boundary value problem for infinite cylinders to construct an algebraic chain complex the chain groups are generated by perturbed closed geodesics the boundary operator is defined by counting modulo time shift heat flow trajectories between geodesics of morse index difference one by salamon and weber gafa 16 1050 138 2006 this heat flow homology is naturally isomorphic to floer homology of the cotangent bundle for hamiltonians given by kinetic plus potential energy

global smooth solution for the klein gordon zakharov equations in this paper the initial value problem to the klein gordon zakharov equations in two dimensions is discussed without assuming that the cauchy data are small we prove the existence and uniqueness of the global smooth solution for the problem via the so called continuous method and delicate a priori estimates also the asymptotic behaviour of the solution to the k g z equations with a small parameter approaching zero is studied

on the existence and uniqueness of smooth solution for a generalized zakharov equation abstract the paper deals with the existence and uniqueness of smooth solution for a generalized zakharov equation we establish local in time existence and uniqueness in the case of dimension d 2 3 moreover by using the conservation laws and brezis gallouet inequality the solution can be extended globally in time in two dimensional case for small initial data besides we also prove global existence of smooth solution in one spatial dimension without any small assumption for initial data