positive periodic solutions of singular systems with a parameter abstract the existence and multiplicity of positive periodic solutions for second order non autonomous singular dynamical systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter our results provide a unified treatment for the problem and significantly improve several results in the literature the proof of our results is based on the krasnoselskii fixed point theorem in a cone

platonic polyhedra topological constraints and periodic solutions of the classical n body problem we prove the existence of a number of smooth periodic motions u of the classical newtonian n body problem which up to a relabeling of the n particles are invariant under the rotation group mathcal r of one of the five platonic polyhedra the number n coincides with the order mathcal r of mathcal r and the particles have all the same mass our approach is variational and u is a minimizer of the lagrangian action mathcal a on a suitable subset mathcal k of the h1t periodic maps u 3n the set mathcal k is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group mathcal r there exist infinitely many such cones mathcal k all with the property that mathcal a mathcal k is coercive for a certain number of them using level estimates and local deformations we show that minimizers are free of collisions and therefore classical solutions of the n body problem with a rich geometric kinematic structure

index and stability of symmetric periodic orbits in hamiltonian systems with application to figure eight orbit in this paper using the maslov index theory in symplectic geometry we build up some stability criteria for symmetric periodic orbits in a hamiltonian system which is motivated by the recent discoveries in the n body problem the key ingredient is a generalized bott type iteration formula for periodic solution in the presence of finite group action on the orbit for second order system we prove under general boundary conditions the close formula for the relationship between the morse index of an orbit in a lagrangian system and the maslov index of the fundamental solution for the corresponding orbit in its hamiltonian system counterpart and the boundary conditions cover the cases which appeared in the n body problem as an application we consider the stability problem of the celebrated figure eight orbit due to chenciner and montgomery in the planar three body problem with equal masses and we clarify the relationship between linear stability and its variational nature on various loop spaces the basic idea is as follows the variational characterization of the figure eight orbit provides information about its morse index based on its relation to the maslov index our stability criteria come into play

global regularity for the 2d mhd equations with mixed partial dissipation and magnetic diffusion abstract whether or not classical solutions of the 2d incompressible mhd equations without full dissipation and magnetic diffusion can develop finite time singularities is a difficult issue a major result of this paper establishes the global regularity of classical solutions for the mhd equations with mixed partial dissipation and magnetic diffusion in addition the global existence conditional regularity and uniqueness of a weak solution is obtained for the 2d mhd equations with only magnetic diffusion

a non homogeneous boundary value problem for the korteweg de vries equation posed on a finite domain ii abstract studied here is an initial and boundary value problem for the korteweg de vries equation posed on a bounded interval with nonhomogeneous boundary conditions this particular problem arises naturally in certain circumstances when the equation is used as a model for waves and a numerical scheme is needed it is shown here that this initial boundary value problem is globally well posed in the l 2 based sobolev space h s 0 1 for any s 0 in addition the mapping that associates to appropriate initial and boundary data the corresponding solution is shown to be analytic as a function between appropriate banach spaces

bmo and the regularity criterion for weak solutions to the magnetohydrodynamic equations in this paper we study the regularity criterion for weak solutions to the incompressible magnetohydrodynamic equations we derive the regularity of weak solutions in the marginal class moreover our result demonstrates that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solutions to the magnetohydrodynamic equations

star pentagon and many stable choreographic solutions of the newtonian 4 body problem abstract in this paper we give a rigorous proof of the existence of infinitely many simple choreographic solutions in the classical newtonian 4 body problem these orbits are discovered by a variational method with structural prescribed boundary conditions spbc this method provides an initial path that is obtained by minimizing the lagrangian action functional over the spbc we prove that the initial path can be extended to a periodic or quasi periodic solution with computer assistance a family of choreographic orbits of this type is shown to be linearly stable among the many linearly stable simple choreographic orbits the most extraordinary one is the stable star pentagon choreographic solution we also prove the existence of infinitely many double choreographic periodic solutions infinitely many non choreographic periodic solutions and uncountably many quasi periodic solutions each type of periodic solutions has many stable solutions and possibly infinitely many stable solutions our results with spbc largely complement the current results by minimizing the action on a loop space

on two dimensional magnetohydrodynamic equations with fractional diffusion abstract we consider the existence of the smooth solution for two dimensional magnetohydrodynamic equations with fractional diffusion in the fluid equations in particular we prove the global in time existence of fractional diffusion with power greater than 1 2

non planar minimizers and rotational symmetry in the n body problem abstract the newtonian n body problem admits uniformly rotating relative equilibrium solutions in the plane but not in r 3 when the bodies are allowed to interact in r 3 is it preferable in terms of action to leave the plane and follow a non planar trajectory we use the variational techniques of chenciner and venturelli celestial mech dyn astro 77 2000 139 to show that for an open set of masses there is a class of collision free action minimizing orbits of certain rotational symmetry in the four body problem which are non coplanar i e the planar relative equilibrium is not the least action solution among orbits in r 3 both periodic and quasi periodic solutions are constructed in this way we also discuss constructing collision free action minimizing solutions possessing d rotational symmetry along with various other symmetry constraints

a remarkable periodic solution of the three body problem in the case of equal masses using a variational method we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane the orbit has zero angular momentum and a very rich symmetry pattern its most surprising feature is that the three bodies chase each other around a flxed eight shaped curve setting aside collinear motions the only other known motion along a flxed curve in the inertial plane is the lagrange relative equilibrium in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle our orbit visits in turns every euler conflguration in which one of the bodies sits at the midpoint of the segment deflned by the other two figure 1 numerical computations

generalized mhd equations abstract solutions of the d dimensional generalized mhd gmhd equations t u u u p b b u t b u b b u b are studied in this paper we pay special attention to the impact of the parameters and on the regularity of solutions our investigation is divided into three major cases 1 0 and 0 2 0 and 0 and 3 0 and 0 when 0 and 0 the gmhd equations with any 0 and 0 possess a global weak solution corresponding to any l2 initial data furthermore weak solutions associated with 1 2 d 4 and 1 2 d 4 are actually global classical solutions when their initial data are sufficiently smooth as a special consequence smooth solutions of the 3d gmhd equations with 5 4 and 5 4 do not develop finite time singularities the study of the gmhd equations with 0 and 0 is motivated by their potential applications in magnetic reconnection a local existence result of classical solutions and several global regularity conditions are established for this case these conditions are imposed on either the vorticity u or the current density j b but not both and are weaker than some of current existing ones when 0 and 0 the gmhd equations reduce to the ideal mhd equations it is shown here that the ideal mhd equations admit a unique local solution when the prescribed initial data is in a holder space cr with r 1

highly eccentric hip hop solutions of the 2n body problem abstract we show the existence of families of hip hop solutions in the equal mass 2 n body problem which are close to highly eccentric planar elliptic homographic motions of 2 n bodies plus small perpendicular non harmonic oscillations by introducing a parameter the homographic motion and the small amplitude oscillations can be uncoupled into a purely keplerian homographic motion of fixed period and a vertical oscillation described by a hill type equation small changes in the eccentricity induce large variations in the period of the perpendicular oscillation and give rise via a bolzano argument to resonant periodic solutions of the uncoupled system in a rotating frame for small 0 the topological transversality persists and brouwer s fixed point theorem shows the existence of this kind of solutions in the full system

boundary control of the korteweg de vries burgers equation further results on stabilization and well posedness with numerical demonstration we consider the korteweg de vries burgers equation on the interval 0 1 motivated by simulations resulting in modest decay rates with recently proposed control laws by liu and krstic which keeps some of the boundary conditions as homogeneous we propose a strengthened set of feedback boundary conditions we establish stability properties of the closed loop system prove well posedness and illustrate the performance improvement by a simulation example

a continuum of periodic solutions to the planar four body problem with two pairs of equal masses abstract in this paper we apply the variational method with structural prescribed boundary conditions spbc to prove the existence of periodic and quasi periodic solutions for the planar four body problem with two pairs of equal masses m 1 m 3 and m 2 m 4 a path q t on 0 t satisfies the spbc if the boundaries q 0 a and q t b where a and b are two structural configuration spaces in r 2 4 and they depend on a rotation angle 0 2 and the mass ratio m 2 m 1 r we show that there is a region 0 2 r such that there exists at least one local minimizer of the lagrangian action functional on the path space satisfying the spbc q t h 1 0 t r 2 4 q 0 a q t b for any the corresponding minimizing path of the minimizer can be extended to a non homographic periodic solution if is commensurable with or a quasi periodic solution if is not commensurable with in the variational method with the spbc we only impose constraints on the boundary and we do not impose any symmetry constraint on solutions instead we prove that our solutions that are extended from the initial minimizing paths possess certain symmetries the periodic solutions can be further classified as simple choreographic solutions double choreographic solutions and non choreographic solutions among the many stable simple choreographic orbits the most extraordinary one is the stable star pentagon choreographic solution when 4 5 1 remarkably the unequal mass variants of the stable star pentagon are just as stable as the equal mass choreographies

eigenvalue problem of sturm liouville systems with separated boundary conditions let lambda j be the jth eigenvalue of sturm liouville systems with separated boundary conditions we build up the hill type formula which represent prod nolimits j 1 lambda j 1 as a determinant of finite matrix consequently we get the krein type trace formula based on the hill type formula which express sum nolimits j 1 over lambda j m as trace of finite matrices the trace formula can be used to estimate the conjugate point alone a geodesic in riemannian manifold and to get some infinite sum identities

global bifurcation of planar and spatial periodic solutions from the polygonal relative equilibria for the n body problem abstract given n point masses turning in a plane at a constant speed this paper deals with the global bifurcation of periodic solutions for the masses in that plane and in space as a special case one has a complete study of n identical masses on a regular polygon and a central mass the symmetries of the problem are used in order to find the irreducible representations the linearization and with the help of the orthogonal degree theory all the symmetries of the bifurcating branches

on the regularity of weak solutions to the magnetohydrodynamic equations abstract in this paper we study the regularity of weak solution to the incompressible magnetohydrodynamic equations we obtain some sufficient conditions for regularity of weak solutions to the magnetohydrodynamic equations which is similar to that of incompressible navier stokes equations moreover our results demonstrate that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solution to the magneto hydrodynamic equations

output feedback stabilization of the linearized korteweg de vries equation with right endpoint controllers abstract in this paper we prove the output feedback stabilization for the linearized korteweg de vries kdv equation posed on a finite domain in the case the full state of the system cannot be measured we assume that there is a sensor at the left end point of the domain capable of measuring the first and second order boundary traces of the solution this allows us to design a suitable observer system whose states can be used for constructing boundary feedbacks acting at the right endpoint so that both the observer and the original plant become exponentially stable stabilization of the original system is proved in the l 2 sense while the convergence of the observer system to the original plant is also proved in higher order sobolev norms the standard backstepping approach used to construct a left endpoint controller fails and presents mathematical challenges when building right endpoint controllers due to the overdetermined nature of the related kernel models in order to deal with this difficulty we use the method of ozsari and batal 2019 which is based on using modified target systems involving extra trace terms in addition we show that the number of controllers and boundary measurements can be reduced to one with the cost of a slightly lower exponential rate of decay we provide numerical simulations illustrating the efficacy of our controllers

trace formula for linear hamiltonian systems with its applications to elliptic lagrangian solutions in the present paper we build up trace formulas for both the linear hamiltonian systems and sturm liouville systems the formula connects the monodromy matrix of a symmetric periodic orbit with the infinite sum of eigenvalues of the hessian of the action functional a natural application is to study the non degeneracy of linear hamiltonian systems precisely by the trace formula we can give an estimation for the upper bound such that the non degeneracy preserves moreover we could estimate the relative morse index by the trace formula consequently a series of new stability criteria for the symmetric periodic orbits is given as a concrete application the trace formula is used to study the linear stability of elliptic lagrangian solutions of the classical planar three body problem which depends on the mass parameter beta in 0 9 and the eccentricity e in 0 1 based on the trace formula we estimate the stable region and hyperbolic region of the elliptic lagrangian solutions

global well posedness and exponential decay for fifth order korteweg de vries equation posed on the finite domain abstract this paper studies the fifth order korteweg de veries equation posed on finite interval 0 1 t u x 5 u e 1 x 4 u 1 x 3 u e 2 x 2 u 2 x u e 3 u u x u 0 x 0 1 t 0 u x 0 x x 0 1 u 0 t u 1 t x u 0 t x u 1 t 0 x 2 u 1 t x 2 u 0 t t 0 firstly we establish smoothing effect of kato type for the fifth order korteweg de veries equation posed on finite interval secondly we show that the closed loop system 0 1 is globally well posed in the space h s 0 1 for s 0 5 lastly we prove that the solution of the closed loop system 0 1 is exponentially decay in the space h s 0 1 for s 0 5

local exact controllability to the trajectories of the korteweg de vries burgers equation on a bounded domain with mixed boundary conditions abstract this paper studies the internal control of the korteweg de vries burgers kdvb equation on a bounded domain the diffusion coefficient is time dependent and the boundary conditions are mixed in the sense that homogeneous dirichlet and periodic neumann boundary conditions are considered the exact controllability to the trajectories is proven for a linearized system by using duality and getting a new carleman estimate then using an inversion theorem we deduce the local exact controllability to the trajectories for the original kdvb equation which is nonlinear

partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations in this paper we study the local behavior of the solutions to the three dimensional magnetohydrodynamic equations we are interested in both the uniform gradient estimates for smooth solutions and regularity of weak solutions it is shown that in some neighborhood of x0 t0 the gradients of the velocity field u and the magnetic field b are locally uniformly bounded in l norm as long as that either the scaled local l2 norm of the gradient or the scaled local total energy of the velocity field is small and the scaled local total energy of the magnetic field is uniformly bounded these estimates indicate that the velocity field plays a more dominant role than that of the magnetic field in the regularity theory as an immediately corollary we can derive an estimates of hausdorff dimension on the possible singular set of a suitable weak solution as in the case of pure fluid various partial regularity results are obtained as consequences of our blow up estimates

on the existence of infinitely many periodic solutions to some problems of n body type we prove the existence of infinitely many periodic solutions with prescribed period to a class of problems of n body type

action minimizing orbits in the parallelogram four body problem with equal masses in this paper we prove the existence of a new periodic solution for the planar newtonian four body problem with equal masses on this orbit two mass points travel on one star shaped closed curve while the other two travel on another and have the opposite orientation the configuration of the masses changes from square to collinear periodically and remains a parallelogram for all time

simple choreographies of the planar newtonian n body problem in the n body problem a simple choreography is a periodic solution where all masses chase each other on a single loop in this paper we prove that for the planar newtonian n body problem with equal masses n 3 there are at least 2n 3 2 n 3 2 different main simple choreographies this confirms a conjecture given by chenciner et al geometry mechanics and dynamics springer new york pp 287 308 2002 all the simple choreoagraphies we prove belong to the linear chain family

variational proof of the existence of the super eight orbit in the four body problem using the variational method chenciner and montgomery ann math 152 881 901 2000 proved the existence of an eight shaped periodic solution of the planar three body problem with equal masses just after the discovery gerver numerically found a similar periodic solution called super eight in the planar four body problem with equal mass in this paper we prove the existence of the super eight orbit by using the variational method the difficulty of the proof is to eliminate the possibility of collisions in order to solve it we apply the scaling technique established by tanaka ann inst h poincare anal non lineaire 10 215 238 1993 proc am math soc 122 275 284 1994 and investigate the asymptotic behavior of a binary collision

rapid stabilization for a korteweg de vries equation from the left dirichlet boundary condition this paper deals with the stabilization problem for the korteweg de vries equation posed on a bounded interval the control acts on the left dirichlet boundary condition at the right end point dirichlet and neumann homogeneous boundary conditions are considered the proposed feedback law forces the exponential decay of the system under a smallness condition on the initial data moreover the decay rate can be tuned to be as large as desired the feedback control law is designed by using the backstepping method

minimizing periodic orbits with regularizable collisions in the n body problem in the newtonian n body problem there are various subsystems with two degrees of freedom such as the collinear three body problem and the isosceles three body problem after we determine a normal form of the lagrangians of these subsystems we prove the existence of periodic solutions with regularizable collisions for these systems our result includes several examples such as schubart s orbit with or without equal masses among others

collisionless periodic solutions to some three body problems we provide sufficient conditions for the existence of periodic solutions to some three body problems periodic solutions are found as minima of the associated action integral and are shown to be free of double and triple collisions

new periodic solutions for 2n body problems in r3 for newtonian 2n body problems with equal masses in r3 we prove the existence of new noncollision periodic solution such that 2n bodies move on two different closed curves

symmetry groups and non planar collisionless action minimizing solutions of the three body problem in three dimensional space periodic and quasi periodic solutions of the n body problem can be found as minimizers of the lagrangian action functional restricted to suitable spaces of symmetric paths the main purpose of this paper is to develop a systematic approach to the equivariant minimization for the three body problem in three dimensional space first we give a finite complete list of symmetry groups fitting to the minimization of the action with the property that any other symmetry group can be reduced to be isomorphic to one of these representatives a second step is to prove that the resulting local and global symmetric action minimizers are always collisionless when they are not already bound to collisions furthermore we prove some results which address the question of whether minimizers are planar or non planar as a consequence of our theory we will give general criteria for a symmetry group to yield planar or homographic minimizers either homographic or not as in the chenciner montgomery eight solution on the other hand we will provide a rigorous proof of the existence of some interesting one parameter families of periodic and quasi periodic non planar orbits these include the choreographic marchal s p12 family with equal masses together with a less symmetric choreographic family which anyway probably coincides with the p12 family

binary decompositions for planar n body problems and symmetric periodic solutions a binary decomposition for a system of n masses is a way of treating the system as big n atop 2 big binaries with the total action exactly the same as that of the original system by considering binary decompositions we are able to provide effective lower bound estimates for the action of collision paths in several spaces of symmetric loops as applications we use our estimates to prove the existence of some new classes of symmetric periodic solutions for the n body problem

symmetric trajectories for the 2n body problem with equal masses we consider the problem of 2n bodies of equal masses in mathbb r 3 for the newtonian like weak force potential r and we prove the existence of a family of collision free nonplanar and nonhomographic symmetric solutions that are periodic modulo rotations in addition the rotation number with respect to the vertical axis ranges in a suitable interval these solutions have the hip hop symmetry a generalization of that introduced in 19 for the case of many bodies and taking account of a topological constraint the argument exploits the variational structure of the problem and is based on the minimization of lagrangian action on a given class of paths

entire minimal parabolic trajectories the planar anisotropic kepler problem we continue the variational approach to parabolic trajectories introduced in our previous paper barutello et al entire parabolic trajectories as minimal phase transitions arxiv 1105 3358v1 2011 which sees parabolic orbits as minimal phase transitions we deepen and complete the analysis in the planar case for homogeneous singular potentials we characterize all parabolic orbits connecting two minimal central configurations as free time morse minimizers in a given homotopy class of paths these may occur for at most one value of the homogeneity exponent in addition we link this threshold of existence of parabolic trajectories with the absence of collisions for all the minimizers of fixed end problems and also with the existence of action minimizing periodic trajectories with nontrivial homotopy type

existence and minimizing properties of retrograde orbits to the three body problem with various choices of masses poincare made the first attempt in 1896 on applying variational calculus to the three body problem and observed that collision orbits do not necessarily have higher values of action than classical solutions little progress had been made on resolving this difficulty until a recent breakthrough by chenciner and montgomery afterward variational methods were successfully applied to the jv body problem to construct new classes of solutions in order to avoid collisions the problem is confined to symmetric path spaces and all new planar solutions were constructed under the assumption that some masses are equal a question for the variational approach on planar problems naturally arises are minimizing methods useful only when some masses are identical this article addresses this question for the three body problem for various choices of masses it is proved that there exist infinitely many solutions with a certain topological type called retrograde orbits that minimize the action functional on certain path spaces cases covered in our work include triple stars in retrograde motions double stars with one outer planet and some double stars with one planet orbiting around one primary mass our results largely complement the classical results by the poincare continuation method and conley s geometric approach

controllability aspects of the korteweg de vries burgers equation on unbounded domains abstract the aim of this work is to consider the controllability problem of the linear system associated to korteweg de vries burgers equation posed in the whole real line we obtain a sort of exact controllability for solutions in l l o c 2 r 2 by deriving an internal observability inequality and a global carleman estimate following the ideas contained in 25 the problem is reduced to prove an approximate theorem

boosting the decay of solutions of the linearised korteweg de vries burgers equation to a predetermined rate from the boundary abstractthe aim of this article is to extend recent results on the boundary feedback controllability of the korteweg de vries equation to the korteweg de vries burgers equation which is posed on a bounded domain in the first part of the paper it is proven that all the sufficiently small solutions can be steered to zero at any desired exponential rate by means of a suitably constructed boundary feedback controller in the second part an observer is proposed when a type of boundary measurement is available while there is no full access to the medium

on the regularity criteria for weak solutions to the magnetohydrodynamic equations abstract we study the regularity criteria for weak solutions to the incompressible magnetohydrodynamic equations some regularity criteria are obtained for weak solutions to the magnetohydrodynamic equations which generalize the results in c he z xin on the regularity of solutions to the magneto hydrodynamic equations j differential equations 213 2 2005 235 254 our results reveal that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solutions to the magnetohydrodynamic equations

regularity criteria for the 3d generalized mhd equations in terms of vorticity abstract we prove regularity criteria for the 3d generalized mhd equations these criteria impose assumptions on the vorticity only in addition we also prove a result of global existence for smooth solution under some special conditions

on the existence of collisionless equivariant minimizers for the classical n body problem we show that the minimization of the lagrangian action functional on suitable classes of symmetric loops yields collisionless periodic orbits of the n body problem provided that some simple conditions on the symmetry group are satisfied more precisely we give a fairly general condition on symmetry groups g of the loop space for the n body problem with potential of homogeneous degree with 0 which ensures that the restriction of the lagrangian action mathcal a to the space g of g equivariant loops is coercive and its minimizers are collisionless without any strong force assumption in proving that local minima of g are free of collisions we develop an averaging technique based on marchal s idea of replacing some of the point masses with suitable shapes see 10 as an application several new orbits can be found with some appropriate choice of g furthermore the result can be used to give a simplified and unitary proof of the existence of many already known minimizing periodic orbits

boundary feedback stabilization of the korteweg de vries burgers equation posed on a finite interval abstract this paper studies the boundary feedback stabilization problem of the korteweg de vries burgers equation posed on a finite interval a linear boundary feedback control law is proposed then the resulting closed loop nonlinear system is shown not only to be globally well posed but also to be exponentially stable in the space h s 0 l with s 0 3 the sharp kato smoothing property of the associated linear problem is revealed through the boundary integral operator which plays a key role in our analysis

symmetry groups of the planar three body problem and action minimizing trajectories we consider periodic and quasi periodic solutions of the three body problem with homogeneous potential from the point of view of equivariant calculus of variations first we show that symmetry groups of the lagrangian action functional can be reduced to groups in a finite explicitly given list after a suitable change of coordinates then we show that local symmetric minimizers are always collisionless without any assumption on the group other than the fact that collisions are not forced by the group itself moreover we describe some properties of the resulting symmetric collisionless minimizers lagrange euler hill type orbits and chenciner montgomery figure eights

a new branch of mountain pass solutions for the choreographical 3 body problem we prove the existence of a new branch of solutions of mountain pass type for the periodic 3 body problem with choreographical constraint at first we describe the variational structure of the action functional associated to the choreographical three body problem in mathbb r 3 in the second part using a bisection algorithm we provide a numerical non rigorous solution of mountain pass type for this problem in a rotating frame with angular velocity 1 5 the last step consists in the rigorous computer assisted proof of the existence of a full branch of solutions for the problem starting from the mountain pass solution detected numerically

variational constructions for some satellite orbits in periodic gravitational force fields in a gravitational force field a bounded orbit of an infinitesimal point mass is called a satellite orbit the purpose of this paper is to establish a variational theory for the existence of some relative periodic satellite orbits in periodic gravitational force fields the gravitation field can be generated by a relative equilibrium or a uniformly rotating asteroid our approach is to regard the aggregate of primaries together with the small satellite as a restricted full two body problem and then look for direct and retrograde relative periodic satellite orbits by direct methods of calculus of variations regularity of the action minimizing satellite orbit is obtained by providing satisfactory lower bound estimates for the distance between the satellite and the mass center an upper bound estimate for this distance is also provided as a contrast to classical existence proofs by continuation from infinity

subharmonic pl solutions of first order hamiltonian systems abstract in this paper for any symplectic matrix p the existence of subharmonic p l solutions of the first order non autonomous superquadratic hamiltonian systems is considered under the convex condition the existence of infinitely many geometrically distinct p l solutions is proved

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references

periodic solutions of second order non autonomous singular dynamical systems abstract in this paper we establish two different existence results of positive periodic solutions for second order non autonomous singular dynamical systems the first one is based on a nonlinear alternative principle of leray schauder and the result is applicable to the case of a strong singularity as well as the case of a weak singularity the second one is based on schauder s fixed point theorem and the result sheds some new light on problems with weak singularities and proves that in some situations weak singularities may help create periodic solutions recent results in the literature are generalized and significantly improved

removing collision singularities from action minimizers for the n body problem with free boundaries for the planar and spatial n body problems it has been proved by marchal and chenciner that solutions for the minimizing problem with fixed ends are free from interior collisions this important result has been extended by ferrario terracini to newtonian type problems and equivariant problems it has also been used to construct many symmetric solutions for the n body problem

on the minimizing triple collision orbits in the planar newtonian three body problem in this paper we study the minimizing triple collision orbits in the planar newtonian three body problem with arbitrary masses we show that for a given non collinear initial configuration the minimizing triple collision orbit is collision free until a simultaneous collision and its limiting configuration is the lagrangian configuration with the same orientation as the initial configuration for the collinear initial configuration under a certain technical assumption there exist two minimizing orbits the limiting configurations of these orbits are the two opposite lagrangian configurations