a bound on the total variation of the conserved quantities for solutions of a general resonant nonlinear balance law we introduce a new potential interaction functional and use it to define a new glimm type functional that bounds the total variation of the conserved quantities at time t 0 by the total variation at time t 0 in glimm approximate solutions of a general resonant nonlinear balance law

global bv entropy solutions and uniqueness for hyperbolic systems of balance laws we consider the cauchy problem for n n strictly hyperbolic systems of nonresonant balance laws

an extension of glimm s method to inhomogeneous strictly hyperbolic systems of conservation laws by weaker than weak solutions of the riemann problem abstract we construct a generalized solution of the riemann problem for strictly hyperbolic systems of conservation laws with source terms and we use this to show that glimm s method can be used directly to establish the existence of solutions of the cauchy problem the source terms are taken to be of the form a g and this enables us to extend the method introduced by lax to construct general solutions of the riemann problem our generalized solution of the riemann problem is weaker than weak in the sense that it is weaker than a distributional solution thus we prove that a weak solution of the cauchy problem is the limit of a sequence of glimm scheme approximate solutions that are based on weaker than weak solutions of the riemann problem by establishing the convergence of glimm s method it follows that all of the results on time asymptotics and uniqueness for glimm s method in the presence of a linearly degenerate field now apply unchanged to inhomogeneous systems

on a nonlinear hyperbolic variational equation i global existence of weak solutions we study the nonlinear hyperbolic partial differential equation u t uux x 1 2u x 2 this partial differential equation is the canonical asymptotic equation for weakly nonlinear solutions of a class of hyperbolic equations derived from variational principles in particular it describes waves in a massive director field of a nematic liquid crystal global smooth solutions of the partial differential equation do not exist since their derivatives blow up in finite time while weak solutions are not unique we therefore define two distinct classes of admissible weak solutions which we call dissipative and conservative solutions we prove the global existence of each type of admissible weak solution provided that the derivative of the initial data has bounded variation and compact support these solutions remain continuous despite the fact that their derivatives blow up there are no a priori estimates on the second derivatives in any l p space so the existence of weak solutions cannot be deduced by using sobolev type arguments instead we prove existence by establishing detailed estimates on the blowup singularity for explicit approximate solutions of the partial differential equation we also describe the qualitative properties of the partial differential equation including a comparison with the burgers equation for inviscid fluids and a number of illustrative examples of explicit solutions we show that conservative weak solutions are obtained as a limit of solutions obtained by the regularized method of characteristics and we prove that the large time asymptotic behavior of dissipative solutions is a special piecewise linear solution which we call a kink wave

finite energy foliations of tight three spheres and hamiltonian dynamics surfaces of sections are a classical tool in the study of 3 dimensional dynamical systems their use goes back to the work of poincare and birkhoff in the present paper we give a natural generalization of this concept by constructing a system of transversal sections in the complement of finitely many distinguished periodic solutions such a system is established for nondegenerate reeb flows on the tight 3 sphere by means of pseudoholomorphic curves the applications cover the nondegenerate geodesic flows on t1s2 rp3 via its double covering s3 and also nondegenerate hamiltonian systems in ir4 restricted to sphere like energy surfaces of contact type

from polynomial integrals of hamiltonian flows to a model of non linear elasticity abstract we prove non existence of smooth solutions of a quasi linear system suggested by ericksen in a model of non linear elasticity this system is of mixed elliptic hyperbolic type we discuss also a relation of such a system to polynomial integrals of classical hamiltonian systems

the dynamics on three dimensional strictly convex energy surfaces we show that a hamiltonian flow on a three dimensional strictly convex energy surface s c r4 possesses a global surface of section of disc type it follows in particular that the number of its periodic orbits is either 2 or oc by a recent result of j franks on area preserving homeomorphisms of an open annulus in the plane the construction of this surface of section is based on partial differential equations of cauchy riemann type for maps from punctured riemann surfaces into r x s3 equipped with special almost complex structures

a measure for the decentralized assignability of eigenvalues abstract the results of 7 are generalized to produce a continuous measure of how close a mode is to being a decentralized fixed mode 13 implications of this measure on controller design are also explored

decentralized stabilization and pole assignment for general proper systems considered is the problem of finding existence conditions and a controller synthesis procedure using decentralized control for assigning the poles of a linear time invariant proper system described by a state space model c a b d where no assumption is made regarding the structure of d this problem has direct application to the decentralized stabilization problem decentralized robust servomechanism problem etc and is a nontrivial extension to the standard decentralized problem where it is assumed that the direct feedthrough terms either are absent or have a block diagonal structure

the generalized buckley leverett and the regularized buckley leverett equations this paper studies solutions of the generalized buckley leverett bl equation with variable porosity and solutions of the regularized bl equation with the burgers benjamin bona mahony type regularization we construct global in time weak solutions to the cauchy problem and to the initial and boundary value problem for the generalized bl equation and global classical solutions for the regularized bl equation solutions of the regularized bl equation are shown to converge to the corresponding solution of the generalized bl equation when the coefficient of the bbm term and the coefficient of the burgers term obey o 2

globally lipschitz continuous solutions to a class of quasilinear wave equations abstract this work investigates the existence of globally lipschitz continuous solutions to a class of cauchy problem of quasilinear wave equations applying lax s method and generalized glimm s method we construct the approximate solutions of the corresponding perturbed riemann problem and establish the global existence for the derivatives of solutions then the existence of global lipschitz continuous solutions can be carried out by showing the weak convergence of residuals for the source term of equation

symplectic manifolds with contact type boundaries an example of a 4 dimensional symplectic manifold with disconnected boundary of contact type is constructed a collection of other results about symplectic manifolds with contact type boundaries are derived using the theory ofj holomorphic spheres in particular the following theorem of eliashberg floer mcduff is proved if a neighbourhood of the boundary of v is symplectomorphic to a neighbourhood ofs2n 1 in standard euclidean space and if vanishes on all 2 spheres inv thenv is diffeomorphic to the ballb2n

integration theory on the zero sets of polyfold fredholm sections we construct an integration theory for sc differential forms on oriented branched ep subgroupoid for which stokes theorem holds true the construction is compatible with equivalences between ep groupoids and so gives rise to an integration theory for branched suborbifolds of polyfolds examples are the solutions sets of proper oriented fredholm sections of strong polyfold bundles for which we obtain invariants this way

pseudoholomorphic curves in symplectizations with applications to the weinstein conjecture in dimension three l in t roduct ion 515 1 1 the weinstein conjecture 515 1 2 the main results 517 1 3 sketch of the proof 522 2 local f redho lm theory 525 2 1 similar i ty principle and consequences 525 2 2 n m as a lmost complex manifold and local fillings 529 2 3 local propert ies of ho lomorphic curves 531 3 bubbl ing off analysis and global fredholm theory 533 3 l grad ien t bounds imply ca bounds 533 3 2 bubbing off analysis of ho lomorphic planes 534 3 3 grad ien t bounds near the boundary 537 4 proof of the main results 544 4 1 set up for a global fredholm theory 544 4 2 bishop s theorem 545 4 3 proof of theorem 5 546 4 4 proof of theorems 8 and 11 547 4 5 uniqueness results for foliations 547 5 appendix 550 5 1 per turbing surfaces 550 5 2 ki l l ing crit ical points of hypersurfaces 552 5 3 adding a thin tube to a sphere 560

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

the initial boundary value problem of hyperbolic integro differential systems of nonlinear balance laws abstract this research explores the initial boundary value problem for the 2 2 hyperbolic systems of balance laws whose sources are the time dependent and contain the integral of unknowns perturbed riemann and boundary riemann problems are provided to account for the time dependence of sources their approximation solutions are constructed by modified lax s method in addition we introduce a new version of glimm scheme ggs and study its stability which is proved by the wave interaction estimates in a dissipativity assumption with the consistency of ggs the existence of a global weak solution satisfying the entropy inequality is then achieved finally the lipschitz continuous solution to the problem is established by the weak convergence of the residual

weak solutions to a nonlinear variational wave equation we establish the global existence of weak solutions to the cauchy problem for a nonlinear variational wave equation where the wave speed is a given monotone function of the wave amplitude the equation arises in modeling wave motions of nematic liquid crystals long waves on a dipole chain and a few other fields we use the young measure method in the setting of lp spaces we overcome the difficulty that oscillations get amplified by the growth terms of the equation

symplectic hypersurfaces in the complement of an isotropic submanifold using donaldson s approximately holomorphic techniques we construct symplectic hypersurfaces lying in the complement of any given compact isotropic submanifold of a compact symplectic manifold we discuss the connection with rational convexity results in the kahler case and various applications

smooth solutions to a class of quasilinear wave equations this article investigates the existence nonexistence of smooth solutions of nonlinear vibration equations which arise from the one dimensional motion of polytropic gas without external forces contained in a finite interval for any fixed arbitrarily long time we show that there are smooth small amplitude solutions of the nonlinear equations for which the periodic solutions of the linearized equation are the first order approximations on the other hand when the nonlinearity is strictly convex or concave there exists no time periodic solutions which are twice continuously differentiable an example of possible singularities which occur at the second derivatives is illustrated we also give another kind of exact solutions with singularity such that shocks occur after a finite time furthermore we get an estimate of the life span of smooth solutions to the initial boundary value problem

conservative solutions to a nonlinear variational wave equation we establish the existence of a conservative weak solution to the cauchy problem for the nonlinear variational wave equation utt c u c u ux x 0 for initial data of finite energy here c is any smooth function with uniformly positive bounded values

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

global solutions for initial boundary value problem of quasilinear wave equations abstract this work investigates the existence of globally lipschitz continuous solutions to a class of initial boundary value problem of quasilinear wave equations applying the lax s method and generalized glimm s method we construct the approximate solutions of initial boundary riemann problem near the boundary layer and perturbed riemann problem away from the boundary layer by showing the weak convergence of residuals for the approximate solutions we establish the global existence for the derivatives of solutions and obtain the existence of global lipschitz continuous solutions of the problem

on algebraic properties of general proper decentralized abstract the new concepts of the decentralized output feedback variable polynomial the decentralized output feedback cycle index of general proper systems and the geometric multiplicities of decentralized fixed modes are introduced their computational methods and some algebraic properties are presented it is shown that the decentralized output feedback cycle index of a general proper system is equal to one when the system has no fixed modes or equal to the maximum of the geometric multiplicities of its decentralized fixed modes it is also shown that almost all decentralized output feedback can be used to make the zeros of the decentralized variable polynomial distinct and disjoint from any given finite set of points on the complex plane

compressible euler equations with general pressure law we study the hyperbolic system of euler equations for an isentropic compressible fluid governed by a general pressure law the existence and regularity of the entropy kernel that generates the family of weak entropies is established by solving a new euler poisson darboux equation which is highly singular when the density of the fluid vanishes new properties of cancellation of singularities in combinations of the entropy kernel and the associated entropy flux kernel are found

decay rates for nonconvex systems of conservation laws we study decay of solutions for hyperbolic systems of conservation laws which are not genuinely nonlinear for a generic class of such systems we determine sharp algebraic rates of decay in the total variation of the wave speed for solutions with compact initial support our analysis involves generalized characteristic arguments and the random choice difference scheme of glimm 1993 john wiley sons inc

algebraic properties of decentralized control systems the new concepts of the decentralized output feedback cycle index the geometric multiplicities of decentralized fixed modes dfm and the decentralized output feedback variable polynomial dvp of a decentralized control system dcs are introduced their computational methods and some properties are given it is shown that the decentralized output feedback cycle index is equal to one when the dcs has no decentralized fixed modes dfm or equal to the maximum of the geometric multiplicities of dfm otherwise almost every decentralized output feedback law can be used to make the zeros of the dvp distinct it is also shown that almost any decentralized output feedback law will make the zeros of the dvp disjoint from any given finite set of points on the complex plane it is then shown that if a dcs has no dfm then it can be made both controllable and observable with respect to any non trivial input and output by applying almost any decentralized output feedback law

concentration of source terms in generalized glimm scheme for global entropy solutions of nonlinear hyperbolic balance laws abstract in this paper we investigate the cauchy problem for a nonlinear hyperbolic system of balance laws with sources a x g and a t h to get the approximate solutions of our problem we consider a version of generalized riemann problem that concentrates the variation of a on a thin t shaped region of each grid a new version of glimm scheme is introduced to construct the approximate solutions and its stability is proved by considering two types of conditions on a finally we verify the consistency of the scheme and the entropy inequality to establish the global existence of entropy solutions

convergence of the 2 2 godunov method for a general resonant nonlinear balance law we introduce a generalized solution of the riemann problem for a general resonant nonlinear balance law and we prove the convergence of the 2 x 2 godunov numerical method based on these solutions in particular we obtain generic conditions that guarantee a canonical structure for the elementary waves in the solution of the riemann problem and an interesting multiplicity of time asymptotic wave patterns is observed and characterized

global entropy solutions to a class of quasi linear wave equations with large time oscillating sources abstract this research explores the cauchy problem for a class of quasi linear wave equations with time dependent sources it can be transformed into the cauchy problem of hyperbolic integro differential systems of nonlinear balance laws we introduce the generalized glimm scheme in new version and study its stability which is proved by glimm type interaction estimates in a dissipativity assumption the generalized solutions to the perturbed riemann problems the building blocks of generalized glimm scheme are constructed by riemann problem method modeled on the source free equations the global existence for the lipschitz continuous solutions and weak solutions to the systems is established by the consistency of scheme and the weak convergence of source finally the weak solutions are also the entropy solutions which satisfy the entropy inequality

smooth solutions of the one dimensional compressible euler equation with gravity abstract we study one dimensional motions of polytropic gas governed by the compressible euler equations the problem on the half space under a constant gravity gives an equilibrium which has free boundary touching the vacuum and the linearized approximation at this equilibrium gives time periodic solutions but it is difficult to justify the existence of long time true solutions for which this time periodic solution is the first approximation the situation is in contrast to the problem of free motions without gravity the reason is that the usual iteration method for quasilinear hyperbolic problem cannot be used because of the loss of regularities which causes from the touch with the vacuum due to this reason we try to find a family of solutions expanded by a small parameter and apply the nash moser theorem to justify this expansion note that the application of nash moser theorem is necessary for the sake of conquest of the trouble with loss of regularities and the justification of the applicability requires a very delicate analysis of the problem

decay of entropy solutions of nonlinear conservation laws we are concerned with the asymptotic behavior of entropy solutions of nonlinear conservation laws the main objective of this paper is to present an analytical approach and to explore its applications to studying the large time behavior of periodic entropy solutions of hyperbolic conservation laws the asymptotic decay of periodic solutions of nonlinear hyperbolic conservation laws is an important nonlinear phenomenon it is observed that the genuine nonlinearity of equations forces the waves of each characteristic family to interact vigorously and to cancel each other the insightful analysis of glimm lax gl for scalar equations and 2 2 systems has indicated that the resultant mutual cancellation of interacting shock and rarefaction waves of the same family induces the decay of periodic solutions such a result was first shown by lax l1 in 1957 for one dimensional convex scalar conservation laws dafermos d1 applying his uniform processes proved the decay result for the case that the set of inflection points of the flux does not have an accumulation point the glimm lax theory gl indicates that for 2 2 strictly hyperbolic and genuinely nonlinear systems any periodic glimm solution decays like o 1 t this result was proved by using the approximate characteristic method in the glimm difference solutions provided that the oscillation of the corresponding initial data is small recently using the method of generalized characteristics dafermos d3 showed that any periodic solution with local bounded variation and small oscillation for 2 2 systems decays asymptotically with the same detailed structure pas found by lax l1 for the

structurally fixed modes a notion of structurally fixed modes is introduced in the framework of decentralized control systems to identify the modes that cannot be shifted by decentralized feedback regardless of the numerical values of system parameters in this way a qualitative characterization of decentralized fixed modes is provided which is suitable for delineating decentrally stabilizable systems the characterization leads naturally to the notion of fixed modes with respect to arbitrary feedback structure constraints

initial boundary value problems for conservation laws abstract we establish some general concepts concerning boundary conditions for systems of conservation laws defined on one dimensional space intervals and prove some general results among these is an extension of glimm s existence theorem to initial boundary value problems we also prove the convergence of the time averages of the solutions to certain initial boundary value problems for systems arising in fluid flows in porous media isentropic gas dynamics and elasticity theory