vessiot structure for manifolds of hyperbolic type darboux integrability and symmetry it is well known that if a scalar second order hyperbolic partial differential equation in two independent variables is darboux integrable then its local cauchy problem may be solved by ordinary differential equations in addition such an equation has infinitely many non trivial conservation laws moreover darboux integrable equations have properties in common with infinite dimensional completely integrable systems in this paper we employ a geometric object intrinsically associated with any hyperbolic partial differential equation its hyperbolic structure to study the darboux integrability of the class e of semilinear second order hyperbolic partial differential equations in one dependent and two independent variables it is shown that the problem of classifying the darboux integrable equations in e contains as a subproblem that of classifying the manifolds of p q hyperbolic type of rank 4 and dimension 2k 3 k 2 p 2 q 2 in turn it is shown that the problem of classifying these manifolds in the two lowest cases p q 2 2 2 3 contains as a subproblem the classification problem for lie groups this generalizes classical results of e vessiot the main result is that if an equation in e is 2 2 or 2 3 darboux integrable on the k jets k 2 then its intrinsic hyperbolic structure admits a lie group of symmetries of dimension 2k 1 or 2k 2 respectively it follows that part of the moduli space for the darboux integrable equations in e is determined by isomorphism classes of lie groups the lie group in question is the group of automorphisms of the characteristic systems of the given equation which leaves invariant the foliation induced by the characteristic or riemann invariants of the equation the tangential characteristic symmetries the isomorphism class of the tangential characteristic symmetries is a contact invariant of the corresponding darboux integrable partial differential equation

almost periodic weak solutions of neutral delay differential equations with piecewise constant argument abstract in this paper we give some theorems on the almost periodic weak solutions of second order neutral delay differential equations with piecewise constant argument of the form x t px t 1 qx 2 t 1 2 f t where denotes the greatest integer function p q are nonzero constants and f t is almost periodic

nonlocal cahn hilliard navier stokes systems with shear dependent viscosity abstract we consider a diffuse interface model for the phase separation of an incompressible and isothermal non newtonian binary fluid mixture in three dimensions the averaged velocity u is governed by a navier stokes system with a shear dependent viscosity controlled by a power p 2 this system is nonlinearly coupled through the korteweg force with a convective nonlocal cahn hilliard equation for the order parameter that is the relative concentration difference of the two components the resulting equations are endowed with the no slip boundary condition for u and the no flux boundary condition for the chemical potential the latter variable is the functional derivative of a nonlocal and nonconvex ginzburg landau type functional which accounts for the presence of two phases we first prove the existence of a weak solution in the case p 11 5 then we extend some previous results on time regularity and uniqueness if p 11 5

classification of second order ordinary differential equations admitting lie groups of fibre preserving point symmetries we use elie cartan s method of equivalence to give a complete classification in terms of differential invariants of second order ordinary differential equations admitting lie groups of fibre preserving point symmetries we then apply our results to the determination of all second order equations which are equivalent under fibre preserving transformations to the free particle equation in addition we present those equations of painleve type which admit a transitive symmetry group finally we determine the symmetry group of some equations of physical interest such as the duffing and holmes rand equations which arise as models of non linear oscillators

on the solvability of models for two phase flows of viscous incompressible fluid with shear dependent viscosity in this paper we consider a model describing the motion of a mixture of two incompressible and partially immiscible fluids in d dimension email protected 4 space domain whose viscosities have a p structure depending on the shear rate or called with shear dependent viscosity the existence of weak solutions is established for the exponent p p email protected d 3d 2 4dd 2 under the no slip boundary condition to velocity and non flux boundary condition to relative density this improves the known results proved for p 3d 2d 2 in kim et al 2006

weighted pseudo almost periodic solutions of second order neutral differential equations with piecewise constant argument abstract in this work we establish a new existence and uniqueness theorem of weighted pseudo almost periodic solution for second order neutral differential equations with piecewise constant argument of the form d 2 d t 2 x t p x t 1 q x 2 t 1 2 f t where denotes the greatest integer function p q are nonzero constants with p 1 and f t is discontinuous weighted pseudo almost periodic comparing with the known results the condition of our result is given explicitly in terms of p q which seems simpler and easier to check even in the special case of pseudo almost periodicity

global existence of weak solutions to a nonlocal cahn hilliard navier stokes system abstract a well known diffuse interface model consists of the navier stokes equations nonlinearly coupled with a convective cahn hilliard type equation this system describes the evolution of an incompressible isothermal mixture of binary fluids and it has been investigated by many authors here we consider a variant of this model where the standard cahn hilliard equation is replaced by its nonlocal version more precisely the gradient term in the free energy functional is replaced by a spatial convolution operator acting on the order parameter while the potential f may have any polynomial growth therefore the coupling with the navier stokes equations is difficult to handle even in two spatial dimensions because of the lack of regularity of we establish the global existence of a weak solution in the two dimensional case we also prove that such a solution satisfies the energy identity and a dissipative estimate provided that f fulfills a suitable coercivity condition

on a diffuse interface model for two phase flows of viscous incompressible fluids with matched densities we study a diffuse interface model for the flow of two viscous incompressible newtonian fluids of the same density in a bounded domain the fluids are assumed to be macroscopically immiscible but a partial mixing in a small interfacial region is assumed in the model moreover diffusion of both components is taken into account this leads to a coupled navier stokes cahn hilliard system which is capable of describing the evolution of droplet formation and collision during the flow we prove the existence of weak solutions of the non stationary system in two and three space dimensions for a class of physical relevant and singular free energy densities which ensures in contrast to the usual case of a smooth free energy density that the concentration stays in the physical reasonable interval furthermore we find that unique strong solutions exist in two dimensions globally in time and in three dimensions locally in time moreover we show that for any weak solution the concentration is uniformly continuous in space and time because of this regularity we are able to show that any weak solution becomes regular for large times and converges as t to a solution of the stationary system these results are based on a regularity theory for the cahn hilliard equation with convection and singular potentials in spaces of fractional time regularity as well as on maximal regularity of a stokes system with variable viscosity and forces in l2 0 hs s in 0 frac12 which are new themselves

existence of weak solutions for a diffuse interface model for viscous incompressible fluids with general densities we study a diffuse interface model for the flow of two viscous incompressible newtonian fluids in a bounded domain the fluids are assumed to be macroscopically immiscible but a partial mixing in a small interfacial region is assumed in the model moreover diffusion of both components is taken into account in contrast to previous works we study the general case that the fluids have different densities this leads to an inhomogeneous navier stokes system coupled to a cahn hilliard system where the density of the mixture depends on the concentration the velocity field is no longer divergence free and the pressure enters the equation for the chemical potential we prove existence of weak solutions for the non stationary system in two and three space dimensions

periodic solution and almost periodic solution for a nonautonomous lotka volterra dispersal system with infinite delay abstract this paper studies a nonautonomous lotka volterra dispersal systems with infinite time delay which models the diffusion of a single species into n patches by discrete dispersal our results show that the system is uniformly persistent under an appropriate condition the sufficient condition for the global asymptotical stability of the system is also given by using mawhin continuation theorem of coincidence degree we prove that the periodic system has at least one positive periodic solution further obtain the uniqueness and globally asymptotical stability for periodic system by using functional hull theory and directly analyzing the right functional of almost periodic system we show that the almost periodic system has a unique globally asymptotical stable positive almost periodic solution we also show that the delays have very important effects on the dynamic behaviors of the system

characteristics and the geometry of hyperbolic equations in the plane abstract this paper focuses on the implementation and application of elie cartan s ideas on characteristics for second order non linear hyperbolic equations in the plane the topics include a discussion of both intrinsic and extrinsic contact equivalence a description of the theory of backlund and laplace transformations based on characteristics a geometric approach to the notion of riemann invariants and a study of the various possibilities for their existence resulting in a new simple proof of lie s description of the contact orbit of the wave equation the initial value problem is analyzed from the viewpoint of exterior differential systems and applied to establish a smooth existence theorem with an application to a shock wave problem

second order neutral delay differential equations with piecewise constant time dependence abstract we find conditions for the existence and uniqueness of almost periodic solutions of second order neutral delay differential equations with almost periodic time dependence of the form x t px t 1 qx t f t here is the greatest integer function p and q are nonzero constants and f t is bohr almost periodic

superposition formulas for exterior differential systems abstract in this article we solve an inverse problem in the theory of quotients for differential equations we characterize a family of exterior differential systems that can be written as a quotient of a direct sum of two associated systems that are constructed from the original the fact that a system can be written as a quotient can be used to find the general solution to these equations some examples are given to demonstrate the theory

almost periodic solutions of second order neutral delay differential equations with piecewise constant arguments in this paper we present an existence theorem of almost periodic solutions of second order neutral delay differential equations with piecewise constant arguments of the form x t x t 1 qx t f t where denotes the greatest integer function q is a nonzero constant and f t is almost periodic

internal external and generalized symmetries backlund s theorem which characterizes contact transformations is generalized to give an analogous characterization of internal symmetries of systems of differential equations for a wide class of systems of differential equations we prove that every internal symmetry comes from a first order generalized symmetry and conversely every first order generalized symmetry satisfying certain explicit contact conditions determines an internal symmetry we analyze the contact conditions in detail deducing powerful necessary conditions for a system of differential equations that admit genuine internal symmetries i e ones which do not come from classical external symmetries applications include a direct proof that both the internal symmetry group and the first order generalized symmetries of a remarkable differential equation due to hilbert and cartan are the noncompact real form of the exceptional simple lie group g2

integration of pdes by differential geometric means we use vessiot theory and exterior calculus to solve partial differential equations pdes of the type uyy f x y u ux uy uxx uxy and associated evolution equations these equations are represented by the vessiot distribution of vector fields we develop and apply an algorithm to find the largest integrable sub distributions and hence solutions of the pdes we then apply the integrating factor technique sherring and prince 1992 trans am math soc 433 453 to integrate this integrable vessiot sub distribution the method is successfully applied to a large class of linear and nonlinear pdes

analysis of a cahn hilliard ladyzhenskaya system with singular potential abstract the evolution of a mixture of two incompressible and partially immiscible fluids is here described by ladyzhenskaya navier stokes type equations for the average fluid velocity coupled with a convective cahn hilliard equation with a singular e g logarithmic potential the former is endowed with no slip boundary conditions while the latter is subject to no flux boundary conditions so that the total mass is conserved here we first prove the existence of a weak solution in three dimensions and some regularity properties then we establish the existence of a weak trajectory attractor for a sufficiently general time dependent external force finally taking advantage of the validity of the energy identity we show that the trajectory attractor actually attracts with respect to the strong topology

the symmetries of solitons in this article we will retrace one of the great mathematical adventures of this century the discovery of the soliton and the gradual explanation of its remarkable properties in terms of hidden symmetries we will take an historical approach starting with a famous numerical experiment carried out by fermi pasta and ulam on one of the first electronic computers and with zabusky and kruskal s insightful explanation of the surprising results of that experiment and of a follow up experiment of their own in terms of a new concept they called solitons solitons however raised even more questions than they answered in particular the evolution equations that govern solitons were found to be hamiltonian and have infinitely many conserved quantities pointing to the existence of many non obvious symmetries we will cover next the elegant approach to solitons in terms of the inverse scattering transform and lax pairs and finally explain how those ideas led step by step to the discovery that loop groups acting by dressing transformations give a conceptually satisfying explanation of the secret soliton symmetries

boundary layers on sobolev besov spaces and poisson s equation for the laplacian in lipschitz domains we study inhomogeneous boundary value problems for the laplacian in arbitrary lipschitz domains with data in sobolev besov spaces as such this is a natural continuation of work in jerison and kenig j funct anal 1995 16 219 where the inhomogeneous dirichlet problem is treated via harmonic measure techniques the novelty of our approach resides in the systematic use of boundary integral methods in this regard the key results are establishing the invertibility of the classical layer potential operators on scales of sobolev besov spaces on lipschitz boundaries for optimal ranges of indices applications tolp based helmholtz type decompositions of vector fields in lipschitz domains are also presented