associativity of quantum multiplication we proved the associativity of the multiplication of quantum cohomology for a monotone compact symplectic manifold v for which c 1 a 1 for any effective class the same proof also works for any positive compact symplectic manifold with c 1 a 1

from taubes currents to almost kahler forms the taubes current is introduced by taubes in j symplectic geom 9 161 250 2011 as an intermediate step to construct almost kahler forms in this paper we prove that an almost complex structures being almost kahler structure in dimension four is the equivalent of the existence of taubes currents precisely we show that taubes currents could be regularized to almost kahler forms up to small perturbations of cohomology classes on any 4 dimensional almost complex manifold m j a similar result is established for higher dimensions under the assumption of almost kahler an application to donaldson s tamed to compatible question is provided

the limit behavior of solutions for the cauchy problem of the complex ginzburg landau equation this paper is devoted to the study of the limit behavior as e 0 or e 0 and a 0 for the solutions of the cauchy problem of the complex ginzburg landau equation ut e u i u a i u u 0 u 0 x u0 x 4 n 4 n 2 0 0 s 0 1 and s n 2 2 moreover the sharp convergence rate in such spaces is also given 2002 john wiley sons inc

contact structures equivariant spin bordism and periodic fundamental groups using contact surgery and equivariant bordism theory we prove the existence of contact structures on all 5 dimensional spin manifolds with certain finite fundamental groups

normal crossings singularities for symplectic topology abstract we introduce topological notions of normal crossings symplectic divisor and variety and establish that they are equivalent in a suitable sense to the desired geometric notions our proposed concept of equivalence of associated topological and geometric notions fits ideally with important constructions in symplectic topology this partially answers gromov s question on the feasibility of defining singular symplectic sub varieties and lays foundation for rich developments in the future in subsequent papers we establish a smoothability criterion for symplectic normal crossings varieties in the process providing the multifold symplectic sum envisioned by gromov and introduce symplectic analogues of logarithmic structures in the context of normal crossings symplectic divisors

on some examples and constructions of contact manifolds the first goal of this paper is to construct examples of higher dimensional contact manifolds with specific properties our main results in this direction are the existence of tight virtually overtwisted closed contact manifolds in all dimensions and the fact that every closed contact 3 manifold which is not smoothly a rational homology sphere contact embeds with trivial normal bundle inside a hypertight closed contact 5 manifold this uses known construction procedures by bourgeois on products with tori and geiges on branched covering spaces we pass from these procedures to definitions this allows to prove a uniqueness statement in the case of contact branched coverings and to study the global properties such as tightness and fillability of the results of both constructions without relying on any auxiliary choice in the procedures a second goal allowed by these definitions is to study relations between these constructions and the notions of supporting open book as introduced by giroux and of contact fiber bundle as introduced by lerman for instance we give a definition of bourgeois contact structures on flat contact fiber bundles which is local strictly includes the results of the bourgeois construction and allows to recover an isotopy class of supporting open books on the fibers this last point relies on a reinterpretation inspired by an idea by giroux of supporting open books in terms of pairs of contact vector fields

asymptotic behavior of solutions to schrodinger equations with a subcritical dissipative nonlinearity abstract we study the asymptotic behavior in time of solutions to the initial value problem of the nonlinear schrodinger equation with a subcritical dissipative nonlinearity u p 1 u where 1 p 1 2 n n is the space dimension and is a complex constant satisfying im 0 we show the time decay estimates and the large time asymptotics of the solution when the space dimension n 3 p is sufficiently close to 1 2 n and the initial data is sufficiently small

heat wave interaction in 2 3 dimensions optimal rational decay rate abstract in this paper we consider a simplified version of a fluid structure pde model which has been of longstanding interest within the mathematical and biological sciences in it a n dimensional heat equation replaces the original stokes system so as to ultimately have a vector valued heat equation and vector valued wave equation compose the coupled pde system under study the coupling between the two disparate pde components occurs across a boundary interface as such the entire pde dynamics manifests features of both hyperbolicity and parabolicity for this heat structure system our main result of uniform stability is as follows given smooth initial data i e data in the domain of the underlying semigroup generator of the coupled pde system the corresponding solutions decay at the rate of o t 1 this establishes the long conjectured optimal rate the problem of obtaining sharp rational decay rates for the heat wave pde under present consideration has been a much considered problem with the modus operandi of earlier efforts taking place within the time domain by contrast we adopt here a frequency domain approach which is based upon a recent resolvent criterion and which was initiated in our prior effort wherein we obtained the rate of decay o 1 t the present optimal improvement o t 1 is achieved by employing an additional tool in our analysis a microlocal analysis argument to estimate a critical term involving two problematic boundary traces

almost contact 5 manifolds are contact the existence of a contact structure is proved in any homotopy class of almost contact structures on a closed 5 dimensional manifold

rational rates of uniform decay for strong solutions to a fluid structure pde system abstract in this work we investigate the uniform stability properties of solutions to a well established partial differential equation pde model for a fluid structure interaction the pde system under consideration comprises a stokes flow which evolves within a three dimensional cavity moreover a kirchhoff plate equation is invoked to describe the displacements along a fixed portion say of the cavity wall contact between the respective fluid and structure dynamics occurs on the boundary interface the main result in the paper is as follows the solutions to the composite pde system corresponding to smooth initial data decay at the rate of o 1 t our method of proof hinges upon the appropriate invocation of a relatively recent resolvent criterion for polynomial decays of c 0 semigroups while the characterization provided by said criterion originates in the context of operator theory and functional analysis the work entailed here is wholly within the realm of pde

homotopy equivalence of nearby lagrangians and the serre spectral sequence we construct using relatively basic techniques a spectral sequence for exact lagrangians in cotangent bundles similar to the one constructed by fukaya seidel and smith that spectral sequence was used to prove that exact relative spin lagrangians in simply connected cotangent bundles with vanishing maslov class are homology equivalent to the base a similar result was also obtained by nadler the ideas in that paper were extended by abouzaid who proved that vanishing maslov class alone implies homotopy equivalence in this paper we present a short proof of the fact that any exact lagrangian with vanishing maslov class is homology equivalent to the base and that the induced map on fundamental groups is an isomorphism when the fundamental group of the base is pro finite this implies homotopy equivalence

the focusing energy critical nonlinear schrodinger equation in dimensions five and higher we consider the focusing energy critical nonlinear schr odinger equation iu t delta u u 4 over d 2 u in dimensions d geq 5 we prove that if a maximal lifespan solution u colon i times bbb r d to bbb c obeys sup t in i nabla u t 2 nabla w 2 then it is global and scatters both forward and backward in time here w denotes the ground state which is a stationary solution of the equation in particular if a solution has both energy and kinetic energy less than those of the ground state w at some point in time then the solution is global and scatters we also show that any solution that blows up with bounded kinetic energy must concentrate at least the kinetic energy of the ground state similar results were obtained by kenig and merle for spherically symmetric initial data and dimensions d 3 4 5

a cotangent fibre generates the fukaya category abstract we prove that the algebra of chains on the based loop space recovers the derived wrapped fukaya category of the cotangent bundle of a closed smooth oriented manifold the main new idea is the proof that a cotangent fibre generates the fukaya category using a version of the map from symplectic cohomology to the homology of the free loop space introduced by cieliebak and latschev

the inviscid limit for the complex ginzburg landau equation abstract we study the inviscid limit of the complex ginzburg landau equation we observe that the solutions for the complex ginzburg landau equation converge to the corresponding solutions for the nonlinear schrodinger equation we give its convergence rate we estimate the integral forms of solutions for two equations

periodic approximations of irrational pseudo rotations using pseudoholomorphic curves we prove that every c 1 smooth area preserving dieomorphism of the closed 2 disk having not more than one periodic point is the uniform limit of periodic c 1 smooth dieomorphisms in particular every smooth irrational pseudo rotation can be c 0 approximated by integrable systems this partially answers a long standing question of a katok regarding zero entropy hamiltonian systems in low dimensions our approach uses pseudoholomorphic curve techniques from symplectic geometry

regularity and estimates for j holomorphic discs attached to a maximal totally real submanifold we prove that pseudo holomorphic discs attached to a maximal totally real submanifold inherit their regularity to the boundary from the regularity of the submanifold and of the almost complex structure the proof is based on the computation of an explicit lower bound for the kobayashi metric in almost complex manifolds which also yields explicit estimates of holderian norms of such discs

convergence of yang mills higgs fields in this paper we study the convergence of yang mills higgs ymh fields defined on fiber bundles over riemann surfaces where the fiber is a compact symplectic manifold and the conformal structure of the underlying surface is allowed to vary we show that away from the nodes the ymh fields converges up to gauge to a smooth ymh field modulo finitely many harmonic spheres while near the nodes where the conformal structure degenerates the ymh fields converges to a pair consisting of a flat connection and a twisted geodesic with potential in particular we generalize the recent compactness results on both harmonic maps from surfaces and twisted holomorphic curves to general ymh fields

the limit behavior of solutions for the nonlinear schrodinger equation including nonlinear loss gain with variable coefficient abstract this paper is devoted to the limit behavior as e 0 for the solution of the cauchy problem of the nonlinear schrodinger equation including nonlinear loss gain with variable coefficient i u t u u u i e a t u p u 0 such an equation appears in the recent studies of bose einstein condensates and optical systems under some conditions we show that the solution will locally converge to the solution of the limiting equation i u t u u u 0 with the same initial data in l 0 l w 1 for all admissible pairs where l 0 t we also show that if the limiting solution u is global and has some decay property as t then u e is global if e is sufficiently small and u e converges to u in l 0 w 1 for all admissible pairs in particular our results hold for both subcritical and critical nonlinearities

on the classification of tight contact structures ii we present complete classification results for tight contact structures on two classes of 3 manifolds i torus bundles which fiber over the circle and ii circle bundles which fiber over closed surfaces

asymptotic smoothing effect for a weakly damped nonlinear schrodinger equation in t2 abstract we prove that the global attractor for a weakly damped two dimensional nonlinear schrodinger equation in the usual energy space is in fact included and compact in a more regular energy space the method relies on a suitable approximation when time goes to infinity of the high frequency modes of the solutions

on the cauchy problem for the nonlinear schrodinger equations with time dependent linear loss gain abstract this paper is devoted to the cauchy problem for the nonlinear schrodinger equation with time dependent loss gain which reads i u t u u u i a t u 0 this equation appears in the recent studies of bose einstein condensates and optical systems we obtain some global existence and blow up results which depend on the size of the loss gain coefficient in particular we prove the global existence for the energy critical nonlinearity by scaling and compactness arguments we also discuss asymptotic profiles and concentration properties of blow up solutions

classification of engel knots let m mathcal d be an engel 4 manifold we show that the scanning map from the space of engel knots to the space of formal engel knots is a weak homotopy equivalence when restricted to the complement of the closed ker mathcal d orbits this is a relative parametric and c 0 close h principle

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

the birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

polyfolds cobordisms and the strong weinstein conjecture abstract we prove the strong weinstein conjecture for closed contact manifolds that appear as the concave boundary of a symplectic cobordism admitting an essential local foliation by holomorphic spheres

solvability of dirac type equations abstract this paper develops a weighted l 2 method for the half dirac equation for dirac bundles over closed riemann surfaces we give a sufficient condition for the solvability of the half dirac equation in terms of a curvature integral applying this to the dolbeault dirac operator we establish an automatic transversality criteria for holomorphic curves in kahler manifolds on compact riemannian manifolds we give a new perspective on some well known results about the first eigenvalue of the dirac operator and improve the estimates when the dirac bundle has a z 2 grading on riemannian manifolds with cylindrical ends we obtain solvability in the l 2 spaces with suitable exponential weights while allowing mild negativity of the curvature

tight not semi fillable contact circle bundles extending our earlier results we prove that certain tight contact structures on circle bundles over surfaces are not symplectically semi fillable thus confirming a conjecture of ko honda

holomorphic shadows in the eyes of model theory we define a subset of an almost complex manifold m j to be a holomorphic shadow if it is the image of a j holomorphic map from a compact complex manifold notice that a j holomorphic curve is a holomorphic shadow and so is a complex subvariety of a compact complex manifold we show that under some conditions on an almost complex structure j on a manifold m the holomorphic shadows in the cartesian products of m j form a zariski type structure checking this leads to non trivial geometric questions and results we then apply the work of hrushovski and zilber on zariski type structures we also restate results of gromov and mcduff on j holomorphic curves in symplectic geometry in the language of shadows structures

open gromov witten disk invariants in the presence of an anti symplectic involution abstract for a symplectic manifold with an anti symplectic involution having non empty fixed locus we construct a model of the moduli space of real sphere maps out of moduli spaces of decorated disk maps and give an explicit expression for its first stiefel whitney class as a corollary we obtain a large number of examples which include all odd dimensional projective spaces and many complete intersections for which many types of real moduli spaces are orientable for these manifolds we define open gromov witten invariants with no restriction on the dimension of the manifolds or the type of the constraints if there are no boundary marked points a wdvv type recursion obtained in a sequel computes these invariants for many real symplectic manifolds if there are boundary marked points we define the invariants under some restrictions on the allowed boundary constraints even though the moduli spaces are not orientable in these cases

weak and strong fillability of higher dimensional contact manifolds for contact manifolds in dimension three the notions of weak and strong symplectic fillability and tightness are all known to be inequivalent we extend these facts to higher dimensions in particular we define a natural generalization of weak fillings and prove that it is indeed weaker at least in dimension five while also being obstructed by all known manifestations of overtwistedness we also find the first examples of contact manifolds in all dimensions that are not symplectically fillable but also cannot be called overtwisted in any reasonable sense these depend on a higher dimensional analogue of giroux torsion which we define via the existence in all dimensions of exact symplectic manifolds with disconnected contact boundary

a note on optimal regularity and regularizing effects of point mass coupling for a heat wave system abstract we consider a coupled 1d heat wave system which serves as a simplified fluid structure interaction problem the system is coupled in two different ways the first when the interface does not have mass and the second when the interface does have mass we prove an optimal regularity result in sobolev spaces for both cases the main idea behind the proof is to reduce the coupled problem to a single nonlocal equation on the interface by using neummann to diriclet operator furthermore we show that point mass coupling regularizes the problem and quantify this regularization in the sense of sobolev spaces

motion of an elastic solid inside an incompressible viscous fluid the motion of an elastic solid inside an incompressible viscous fluid is ubiquitous in nature mathematically such motion is described by a pde system that couples the parabolic and hyperbolic phases the latter inducing a loss of regularity which has left the basic question of existence open until now in this paper we prove the existence and uniqueness of such motions locally in time when the elastic solid is the linear kirchhoff elastic material the solution is found using a topological fixed point theorem that requires the analysis of a linear problem consisting of the coupling between the time dependent navier stokes equations set in lagrangian variables and the linear equations of elastodynamics for which we prove the existence of a unique weak solution we then establish the regularity of the weak solution this regularity is obtained in function spaces that scale in a hyperbolic fashion in both the fluid and solid phases our functional framework is optimal and provides the a priori estimates necessary for us to employ our fixed point procedure

riccati theory and singular estimates for a bolza control problem arising in linearized fluid structure interaction abstract we consider a bolza boundary control problem involving a fluid structure interaction model the aim of this paper is to develop an optimal feedback control synthesis based on riccati theory the model considered consists of a linearized navier stokes equation coupled on the interface with a dynamic wave equation the model incorporates convective terms resulting from the linearization of the navier stokes equation around equilibrium the existence of the optimal control and its feedback characterization via a solution to a riccati equation is established the main mathematical difficulty of the problem is caused by unbounded action of control forces which in turn result in riccati equations with unbounded coefficients and in singular behavior of the gain operator this class of problems has been recently studied via the so called singular estimate control systems secs theory which is based on the validity of the singular estimate se g avalos differential riccati equations for the active control of a problem in structural acoustic j optim theory appl 91 1996 695 728 i lasiecka mathematical control theory of coupled pde s in nsf cmbs lecture notes siam 2002 with unbounded controls i lasiecka a tuffaha riccati equations for the bolza problem arising in boundary point control problems governed by c0 semigroups satisfying a singular estimate j optim theory appl 136 2008 229 246 it is shown that the fluid structure interaction does satisfy the singular estimate se condition this is accomplished by showing that the maximal abstract parabolic regularity is transported via hidden hyperbolic regularity of the boundary traces on the interface thus the established singular estimate allows for the application of recently developed general theory which in turn implies well posedness of the feedback synthesis and of the associated riccati equation moreover the singularities in the optimal control and in the feedback operator at the terminal time are quantitatively described

a c 2 smooth counterexample to the hamiltonian seifert conjecture in r 4 we construct a proper c 2 smooth function on r 4 such that its hamiltonian flow has no periodic orbits on at least one regular level set this result can be viewed as a c 2 smooth counterexample to the hamiltonian seifert conjecture in dimension four

wrinkling of smooth mappings and its applications i this paper together with its sequel papers em1 and em2 contains a reexposition of the authors theory of mappings with simple singularities see e1 e2 m1 m2 in addition we give here new applications of the theory the method of wrinkling of smooth mappings described in these papers provides an easier path to theorems from e1 and e2 about construction of mappings with prescribed singularities it gives an alternative proof of thurston s theorem about foliations of codimension 1 see th and me the method gives a new and we think simpler proof of k igusa s theorem on mappings without higher singularities see ig moreover it allows us to remove all dimensional restrictions in this theorem the idea of wrinkling rst appeared in a somewhat implicit form in s smale s work on regular homotopy see sm and later was used by v poenaru see po it was then transformed by m gromov see g1 into a powerful general tool for solving partial di erential relations in the context of singularities the wrinkling was rst explicitly described by one of the authors see m1 the three papers the present one em1 and em2 have the following organization we begin by introducing the notion of a wrinkled map this is a map which can have fold and cusp singularities of very simple topology see fig 1 and 1 2 1 3 below for the exact de nition then we formulate the main technical result of the paper see theorem 1 5a below and its globalized versions 1 6a and 1 6b which give an h principle type description of invent math 130 345 369 1997

propagation in hamiltonian dynamics and relative symplectic homology the main result asserts the existence of noncontractible periodic orbits for compactly supported time dependent hamiltonian systems on the unit cotangent bundle of the torus or of a negatively curved manifold whenever the generating hamiltonian is sufficiently large over the zero section the proof is based on floer homology and on the notion of a relative symplectic capacity applications include results about propagation properties of sequential hamiltonian systems periodic orbits on hypersurfaces hamiltonian circle actions and smooth lagrangian skeletons in stein manifolds

the maslov class of the lagrange surfaces and gromov s pseudo holomorphic curves for an immersed lagrange submanifold w c t x one can define a nonnegative integer topologic invariant m w such that the image of h1 w z under the maslov class is equal to m w z in this paper the value of m w is calculated for the case of a two dimensional oriented manifold x with the universal cover homeomorphic to r2 and an embedded lagrange torus w it is proved that if x t2 and w is homologic to the zero section then m w 0 in all the other cases m w 2 the last result is true also for a wide class of oriented properly embedded lagrange surfaces in t r the proof is based on the gromov s theory of pseudo holomorphic curves some applications to the hamiltonian mechanics are mentioned

irreversibility from staircases in symplectic embeddings abstract we present an argument whose goal is to trace the origin of the macroscopically irreversible behavior of hamiltonian systems of many degrees of freedom we use recent flexibility and rigidity results of symplectic embeddings quantified via the stabilized fibonacci and pell staircases to encode the underlying breadth of the possible initial conditions which alongside the multitude of degrees of freedom of the underlying system give rise to time irreversibility

existence and classification of overtwisted contact structures in all dimensions we establish a parametric extension h principle for overtwisted contact structures on manifolds of all dimensions which is the direct generalization of the 3 dimensional result from 12 it implies in particular that any closed manifold admits a contact structure in any given homotopy class of almost contact structures

strong solutions to a nonlinear fluid structure interaction system abstract in this paper we prove the existence of local in time smooth solutions to the nonlinear fluid structure interaction model first introduced in j l lions quelques methodes de resolution des problemes aux limites non lineaires dunod 1969 and considered in v barbu z grujic i lasiecka a tuffaha existence of the energy level weak solutions for a nonlinear fluid structure interaction model in fluids and waves in contemp math vol 440 amer math soc providence ri 2007 pp 55 82 v barbu z grujic i lasiecka a tuffaha smoothness of weak solutions to a nonlinear fluid structure interaction model indiana univ math j 57 3 2008 1173 1207 in particular the strong solutions here are obtained given initial datum for the navier stokes equation in the space h 1 and initial data for the wave equation w 0 and w 1 in the spaces h 2 e and h 1 e respectively

the interaction between quasilinear elastodynamics and the navier stokes equations the interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations coupled to one another along the moving material interface through the continuity of the velocity and traction vectors we prove the existence and uniqueness locally in time of strong solutions in sobolev spaces for quasilinear elastodynamics coupled to the incompressible navier stokes equations unlike our approach in 5 for the case of linear elastodynamics we cannot employ a fixed point argument on the nonlinear system itself and are instead forced to regularize it by a particular parabolic artificial viscosity term we proceed to show that with this specific regularization we obtain a time interval of existence which is independent of the artificial viscosity together with a priori estimates we identify the global solution in both phases as well as the interface motion as a weak limit in strong norms of our sequence of regularized problems

existence of a weak solution to a fluid elastic structure interaction problem with the navier slip boundary condition abstract we study a nonlinear moving boundary fluid structure interaction fsi problem between an incompressible viscous newtonian fluid modeled by the 2d navier stokes equations and an elastic structure modeled by the shell or plate equations the fluid and structure are coupled via the navier slip boundary condition and balance of contact forces at the fluid structure interface the slip boundary condition might be more realistic than the classical no slip boundary condition in situations e g when the structure is rough and in modeling fsi dynamics near or at a contact cardiovascular tissue and cell seeded tissue constructs which consist of grooves in tissue scaffolds that are lined with cells are examples of rough elastic interfaces interacting with an incompressible viscous fluid the problem of heart valve closure is an example of a fsi problem with a contact involving elastic interfaces we prove the existence of a weak solution to this class of problems by designing a constructive proof based on the time discretization via operator splitting this is the first existence result for fluid structure interaction problems involving elastic structures satisfying the navier slip boundary condition

existence h principle for engel structures in this article we prove that the inclusion of the space of engel structures of a smooth 4 manifold into the space of full flags of its tangent bundle induces surjections in all homotopy groups in particular we construct engel structures representing any given full flag

examples of symplectic structures in this paper we construct symplectic forms tilde omega k k geqq 0 on a compact manifold tilde y which have the same homotopy theoretic invariants but which are not diffeomorphic

contact homology and virtual fundamental cycles we give a construction of contact homology in the sense of eliashberg givental hofer specifically we construct coherent virtual fundamental cycles on the relevant compactified moduli spaces of pseudo holomorphic curves the aim of this work is to provide a rigorous construction of contact homology an invariant of contact manifolds and exact symplectic cobordisms due to eliashberg givental hofer eli98 egh00 the contact homology of y is defined in terms of pseudo holomorphic curve counts in the sense of gromov gro85 in the symplectization r y unfortunately the moduli spaces of such curves often suffer from a severe lack of transversality and for this reason it has until now remained an important open problem to define the relevant curve counts in full generality we will use the framework developed in par15 to construct coherent virtual fundamental cycles on the moduli spaces in question thus giving rise to the desired curve counts our methods are quite general and should apply equally well to many other moduli spaces of interest previously a number of important special cases of contact homology and closely related invariants have been constructed rigorously using generic and or automatic transversality techniques cylindrical contact homology of some three manifolds was constructed by bao

lagrangian non squeezing and a geometric inequality we prove that if the unit codisc bundle of a closed riemannian manifold embeds symplectically into a symplectic cylinder of radius one then the length of the shortest non trivial closed geodesic is at most half the area of the unit disc

minimizing drag in a moving boundary fluid elasticity interaction abstract our goal is to minimize the fluid vorticity in the case of an elastic body moving and deforming inside the fluid using a distributed control this translates into analyzing an optimal control problem subject to a moving boundary fluid structure interaction fsi the fsi is described by the coupling of navier stokes and wave equations the control is inherently a nonlinear control acting as feedback on the moving frame its action depends on the flow map of the domain which is itself defined through the dynamics of the problem a key ingredient in the optimal control problem is represented by the long time behavior of the forced dynamics which was an open problem in the field our main results include existence of solutions for all times with small distributed sources and small initial data as well as existence of optimal control for the problem of minimization of drag in the fluid

periodic fourth order cubic nls local well posedness and non squeezing property abstract in this paper we consider the cubic fourth order nonlinear schrodinger equation 4nls under the periodic boundary condition we prove two results one is the local well posedness in h s t with 1 3 s 0 for the cauchy problem of the wick ordered 4nls the other one is the non squeezing property for the flow map of 4nls in the symplectic phase space l 2 t to prove the former we used the ideas introduced in 36 and 27 and to prove the latter we used the ideas in 8

on well posedness for a free boundary fluid structure model we address a fluid structure interaction model describing the motion of an elastic body immersed in an incompressible fluid we establish a priori estimates for the local existence of solutions for a class of initial data which also guarantees uniqueness

hamiltonian collapsing of irrational lagrangian submanifolds with small first betti number among the main symplectic invariants of a closed lagrange submanifoldl of the cotangent of n is the tubular radiusr l defined as the smallest tube d r n 1 of n t n in whichl can be pushed by an hamiltonian diffeotopy of n we show here using pseudo holomorphic techniques that such a submanifold cannot collapse if the first betti number ofl is smaller than 3 and if the maslov class ofl does not vanish in other words r l is then strictly positive and one can actually give an explicit lower bound in terms of the liouville and maslov classes ofl

stable loosely coupled type algorithm for fluid structure interaction in blood flow we introduce a novel loosely coupled type algorithm for fluid structure interaction between blood flow and thin vascular walls this algorithm successfully deals with the difficulties associated with the added mass effect which is known to be the cause of numerical instabilities in fluid structure interaction problems involving fluid and structure of comparable densities our algorithm is based on a time discretization via operator splitting which is applied in a novel way to separate the fluid sub problem from the structure elastodynamics sub problem in contrast with traditional loosely coupled schemes no iterations are necessary between the fluid and structure sub problems this is due to the fact that our novel splitting strategy uses the added mass effect to stabilize rather than to destabilize the numerical algorithm this stabilizing effect is obtained by employing the kinematic lateral boundary condition to establish a tight link between the velocities of the fluid and of the structure in each sub problem the stability of the scheme is discussed on a simplified benchmark problem and we use energy arguments to show that the proposed scheme is unconditionally stable due to the crucial role played by the kinematic lateral boundary condition the proposed algorithm is named the kinematically coupled scheme