gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces abstract let m n g e d v be a smooth metric measure space in this paper we derive a series of gradient estimates and a harnack inequality for positive solutions of a nonlinear parabolic partial differential equation t u q u a u ln u in m n where q c 2 1 m n and a r

bifurcations of corank greater than two abstract we study the solution branches of stable and unstable bifurcations in certain three dimensional semilinear elliptic eigenvalue problems with dirichlet boundary conditions numerically by the continuation methods recent fast linear solvers for large sparse continuation problems are briefly reviewed sample numerical results on corank 3 bifurcation points with computer graphic output are reported

parallel adaptive gmres implementations for homotopy methods the success of probability one homotopy methods in solving large scale optimization problems and nonlinear systems of equations on parallel architectures may be significantly enhanced by the accurate parallel solution of large sparse nonsymmetric linear systems iterative solution techniques such as gmres k favor parallel implementations however their straightforward parallelization usually leads to a poor parallel performance because of global communication incurred by processors one variation of gmres k considered here is to adapt the restart value k for any given problem and use householder reflections in the orthogonalization phase coupled with graph based matrix partitioning to achieve high accuracy and reduce the communication overhead this particular gmres implementation is tailored to the uniquely stringent requirements imposed on a linear system solver by probability one homotopy algorithms occasionally unusually high accuracy ability to adapt to problems of widely varying difficulty and parallelism

elliptic gradient estimates for a nonlinear heat equation and applications abstract in this paper we study elliptic gradient estimates for a nonlinear f heat equation which is related to the gradient ricci soliton and the weighted log sobolev constant of smooth metric measure spaces precisely we obtain hamilton s and souplet zhang s gradient estimates for positive solutions to the nonlinear f heat equation only assuming the bakry emery ricci tensor is bounded below as applications we prove parabolic liouville properties for some kind of ancient solutions to the nonlinear f heat equation some special cases are also discussed

li yau type estimates for a nonlinear parabolic equation on complete manifolds abstract let m g be a complete noncompact riemannian manifold with the m dimensional bakry emery ricci curvature bounded below in this paper we give a local li yau type gradient estimate for the positive solutions to a general nonlinear parabolic equation u t u u a u log u q u in m 0 where a r is a c 2 smooth function and q q x t is a function which generalizes many previous well known gradient estimate results

on the numerical analysis of the von karman equations mixed finite element approximation and continuation techniques the purpose of this paper is to study the approximation of the von karman equations by the mixed finite element scheme of miyoshi and to follow the solutions arcs at a neighbourhood of the first eigenvalue of the linearized problem this last problem is solved by a continuation method

a parallel continuation algorithm for nonlinear eigenvalue problems abstract we describe a parallel continuation algorithm for the numerical treatment of certain semilinear eigenvalue problems with dirichlet boundary conditions the computations were executed on a transputer a distributed memory multiprocessor the numerical results show that our algorithm is quite competitive and may be applied to various nonlinear eigenvalue problems

cusp development in hele shaw flow with a free surface free surface flow in a hele shaw cell is considered it has been known for some time that when the fluid region is contracting a finite time blow up can occur in which a cusp forms in the free surface the solution does not exist beyond the time of blow up examples are presented of this blow up and also of a new process in which a cusp of a different type forms in the free surface but after which the solution continues to exist after a transformation of the dependent variable the pressure the new cusps are related to permissible singularities in the free boundary of the obstacle problem

a new augmented singular transform and its partial newton correction method for finding more solutions to nonvariational quasilinear elliptic pdes abstract in this paper in order to find more solutions to a nonvariational quasilinear pde a new augmented singular transform ast is developed to form a barrier surrounding previously found solutions so that an algorithm search from outside cannot pass the barrier and penetrate into the inside to reach a previously found solution thus a solution found by the algorithm must be new mathematical justifications of ast formulation are established a partial newton correction method is designed accordingly to solve the augmented problem and to satisfy a constraint in ast the new method is applied to numerically investigate bifurcation symmetry breaking phenomena to a non variational quasilinear elliptic equation through finding multiple solutions such phenomena are numerically captured and visualized for the first time and still open for theoretical verification since the formulation is general and simple it opens a door to solve other multiple solution problems

on parametric gevrey asymptotics for some nonlinear initial value cauchy problems abstract we study a nonlinear initial value cauchy problem depending upon a complex perturbation parameter with vanishing initial data at complex time t 0 and whose coefficients depend analytically on t near the origin in c 2 and are bounded holomorphic on some horizontal strip in c w r t the space variable this problem is assumed to be non kowalevskian in time t therefore analytic solutions at t 0 cannot be expected in general nevertheless we are able to construct a family of actual holomorphic solutions defined on a common bounded open sector with vertex at 0 in time and on the given strip above in space when the complex parameter belongs to a suitably chosen set of open bounded sectors whose union form a covering of some neighborhood of 0 in c these solutions are achieved by means of laplace and fourier inverse transforms of some common depending function on c r analytic near the origin and with exponential growth on some unbounded sectors with appropriate bisecting directions in the first variable and exponential decay in the second when the perturbation parameter belongs to moreover these solutions satisfy the remarkable property that the difference between any two of them is exponentially flat for some integer order w r t with the help of the classical ramis sibuya theorem we obtain the existence of a formal series generally divergent in which is the common gevrey asymptotic expansion of the built up actual solutions considered above

long time existence for a slightly perturbed vortex sheet consider a flat two dimensional vortex sheet perturbed initially by a small analytic disturbance by a formal perturbation analysis moore derived an approximate differential equation for the evolution of the vortex sheet we present a simplified derivation of moore s approximate equation and analyze errors in the approximation the result is used to prove existence of smooth solutions for long time if the initial perturbation is of size and is analytic in a strip m existence of a smooth solution of birkhoff s equation is shown for time t k2p if is sufficiently small with 1 as 0 for the particular case of sinusoidal data of wave length and amplitude e moore s analysis and independent numerical results show singularity development at time tc log o log log our results prove existence for t log if is sufficiently small with k 1 as 0 thus our existence results are nearly optimal

existence and uniqueness for a class of nonlinear higher order partial differential equations in the complex plane we prove existence and uniqueness results for nonlinear third order partial differential equations of the form

finite dimensional approximation of non linear problems 3 simple bifurcation points note univ pavia cnr ist analisi numer i 27100 pavia italy ecole polytech f 91128 palaiseau france ecole polytech ctr math appl cnrs era 747 f 91128 palaiseau france brezzi f univ pavia ist mat appl i 27100 pavia italy isi document delivery no mn015times cited 75cited reference count 20 reference asn article 1981 001 record created on 2006 08 24 modified on 2017 05 12

secondary bifurcations in the buckling problem abstract the secondary bifurcation points and secondary states of a thin plate buckling problem are numerically studied in this paper the continuation methods and local perturbation techniques are successfully implemented to trace the first few secondary branch solutions our computer graphic output shows that the secondary branch process is an unsymmetric wrinkling

zero viscosity limit for analytic solutions of the navier stokes equation on a half space i existence for euler and prandtl equations this is the first of two papers on the zero viscosity limit for the incompressible navier stokes equations in a half space in this paper we prove short time existence theorems for the euler and prandtl equations with analytic initial data in either two or three spatial dimensions the main technical tool in this analysis is the abstract cauchy kowalewski theorem for the euler equations the projection method is used in the primitive variables to which the cauchy kowalewski theorem is directly applicable for the prandtl equations cauchy kowalewski is applicable once the diffusion operator in the vertical direction is inverted

resonance theory for schrodinger operators resonances which result from perturbation of embedded eigenvalues are studied by time dependent methods a general theory is developed with new and weaker conditions allowing for perturbations of threshold eigenvalues and relaxed fermi golden rule the exponential decay rate of resonances is addressed its uniqueness in the time dependent picture is shown in certain cases the relation to the existence of meromorphic continuation of the properly weighted green s function to time dependent resonance is further elucidated by giving an equivalent time dependent asymptotic expansion of the solutions of the schrodinger equation

on steplength algorithms for a class of continuation methods the continuation methods considered here are algorithms for the computational analysis of the regular parts of the solution field of equations of the form fx b f d subset r n 1 to r n for given b in r n while these methods are similar in structure to those used for ode solvers their errors are independent of the history of the process and are solely determined by the termination criterion of the corrector at the current step this suggests the use of a posteriors estimates of the convergence radii of the corrector it is proved here that such estimates cannot be obtained from the sequence of corrector iterates alone but that they require some global information about f however it is shown that a finite sequence of corrector iterates does allow for the computation of effective estimates of the convergence quality of certain types of correctors this is used for the design of various step algorithms for continuation processes two of them are based on a newton corrector while the third o

comparison geometry for the bakry emery ricci tensor for riemannian manifolds with a measure m g edvolg we prove mean curvature and volume comparison results when the bakry emery ricci tensor is bounded from below and f is bounded or rf is bounded from below generalizing the classical ones i e when f is constant this leads to extensions of many theorems for ricci curvature bounded below to the bakry emery ricci tensor in particular we give extensions of all of the major comparison theorems when f is bounded simple examples show the bound on f is necessary for these results

finite dimensional approximation of nonlinear problems part ii limit points note univ paris 6 f 75230 paris 05 france ecole polytech ctr math appl cnrs equipe rech 747 f 91128 palaiseau france univ pavia cnr ist anal numer i 27100 pavia italy brezzi f univ pavia cnr ist math appl i 27100 pavia italy isi document delivery no lr761times cited 54cited reference count 13 reference asn article 1981 002doi 10 1007 bf01396184 record created on 2006 08 24 modified on 2017 05 12

a nonlinear approximation for vortex sheet evolution and singularity formation abstract the evolution of a vortex sheet in two dimensional incompressible inviscid flow is governed by the integro differential equation of birkhoff rott we derive a simple approximation for vortex sheet evolution consisting of a system of four first order differential equations this approximate system has the advantage of involving only local operators the errors in the approximation are shown to be relatively small even if the sheet has infinite curvature at a point for the approximate equations exact similarity solutions exhibiting singularity formation are constructed

gradient estimates for the heat equation under the ricci flow the paper considers a manifold m evolving under the ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on m

gradient estimates for a simple elliptic equation on complete non compact riemannian manifolds in this paper we study the local gradient estimate for the positive solution to the following equation u aulog u bu 0i nm where a 0 b are real constants m is a complete non compact riemannian manifold our result is optimal in the sense when m g is a complete non compact expanding gradient ricci soliton by definition m g is called an expanding gradient ricci soliton if for some constant c 0 it satisfies that rc cg d 2 f

generalized conjugate directions abstract this paper presents a simple unifying framework for a wide class of conjugate directions algorithms whose iterates minimize some quadratic functional over a subspace our approach is motivated by its advantages for nonlinear minimization but the purpose of this paper is to present the greatly simplified convergence analysis that results for the linear case

existence uniqueness analyticity and borel summability for boussinesq equations through borel summation methods we analyze the boussinesq equations for coupled fluid velocity and temperature fields 1 ut u p u u ae2 f t u we prove that an equivalent system of integral equations in the borel variable p r dual to 1 t has a unique solution in a class of exponentially bounded functions implying the existence of a classical solution to 1 in a complex t region that includes a real positive time axis segment for analytic initial data and forcing it is shown that the solution is borel summable implying that formal series in powers of t is gevrey 1 asymptotic and within the time interval of existence the solution remains analytic with the same analyticity strip width as the initial data and forcing we also determine conditions on the integral equation solution that improve the estimate for existence time

zero viscosity limit for analytic solutions of the navier stokes equation on a half space ii construction of the navier stokes solution this is the second of two papers on the zero viscosity limit for the incompressible navier stokes equations in a half space in either 2d or 3d under the assumption of analytic initial data we construct solutions of navier stokes for a short time which is independent of the viscosity the navier stokes solution is constructed through a composite asymptotic expansion involving the solutions of the euler and prandtl equations which were constructed in the first paper plus an error term this shows that the navier stokes solution goes to an euler solution outside a boundary layer and to a solution of the prandtl equations within the boundary layer the error term is written as a sum of first order euler and prandtl corrections plus a further error term the equation for the error term is weakly nonlinear its linear part is the time dependent stokes equation this error equation is solved by inversion of the stokes equation through expressing the solution as a regular euler like part plus a boundary layer prandtl like part the main technical tool in this analysis is the abstract cauchy kowalewski theorem

an augmented singular transform and its partial newton method for finding new solutions to find new solutions through the information provided by previously found solution s this paper is first to develop a new singular transform to change the local basin barrier structure of the original problem in an infinite dimensional space and then to propose an augmented partial newton method to solve the transformed problem on the solution set mathematical justifications of the new formulation and method are established details on algorithm implementation are presented examples for two very different variational pde problems are carried out to illustrate the method their numerical results are displayed by profile contour plots and error convergence data the formulation and method developed in this paper can also be applied to solve multiple fixed point problems

ricci deformation of the metric on complete noncompact riemannian manifolds soit m une variete de riemann complete non compacte a n dimensions n 3 pour c1 c2 0 et 0 il existe une constante e e n c1 c2 0 telle que si la courbure de m satisfait a vol b x y c 1 n x m 0 et b rm 2 er 2 0 r c 2 x 0 x 2 x m alors l equation d evolution g ij t t 2r ij t g ij 0 g ij a une solution pour tout temps 0 t et la metrique g ij t converge vers une metrique lisse g ij quand t telle que r ijkl 0 sur m

finite dimensional approximation of nonlinear problems in the first two papers of this series 4 5 we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a banach space we derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point the abstract theory is applied to the galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von karman equations

continuation minimization methods for stability problems abstract we study the solution branches of stable and unstable bifurcations in certain semilinear elliptic eigenvalue problems with dirichlet boundary conditions a secant predictor line search backtrack corrector continuation method is described to trace the solution curves numerically sample numerical results with computer graphic output are reported

new li yau hamilton inequalities for the ricci flow via the space time approach we prove li yau hamilton inequalties that extend hamilton s matrix inequality for solutions of the ricci flow with nonnegative curvature operators to obtain our extensions we apply the space time formalism of s c chu and one of the authors to solutions of the ricci flow modified by a cosmological constant then we adjoin to the ricci flow the evolution of a 1 form and a 2 form flowing by a system of heat type equations by a rescaling argument the inequalities we obtain in this manner yield new inequalities which are reminiscent of the linear trace inequality of hamilton and one of the authors

large sparse continuation problems we study and develop efficient and versatile predictor corrector continuation methods for large sparse problems the first object is to show how special solving methods for a large sparse linear systems can be incorporated into the basic steps of a continuation method next we describe how to use a special nonlinear conjugate gradient method to perform the corrector phase it is shown how such methods can be used to detect bifurcation points and how to trace bifurcating solution branches by using local perturbations finally a numerical example involving bifurcating branches of a nonlinear eigenvalue problem is given

conjugate gradient methods for continuation problems ii we study linear conjugate gradient cg methods for large sparse continuation problems first we show how the cg methods can be incorporated as linear solvers in the context of a continuation method next we indicate how the preconditioned generalized minimal residual algorithm may be used to test for simple and multiple bifurcations without solving the bordered linear systems sample numerical results are reported some concluding remarks concerning the performance of three term cg methods are given

borel summability of the heat equation with variable coefficients abstract we prove borel summability in nonsingular directions of solutions of partial differential equations of the form u t a z u z z where a z is a quartic polynomial and the initial condition is analytic in the special case a z z we obtain the detailed resurgent structure even in such a simple case the structure of the singular manifolds is quite intricate

on gevrey solutions of threefold singular nonlinear partial differential equations abstract we study gevrey asymptotics of the solutions to a family of threefold singular nonlinear partial differential equations in the complex domain we deal with both fuchsian and irregular singularities and allow the presence of a singular perturbation parameter by means of the borel laplace summation method we construct sectorial actual holomorphic solutions which turn out to share a same formal power series as their gevrey asymptotic expansion in the perturbation parameter this result rests on the malgrange sibuya theorem and it requires to prove that the difference between two neighboring solutions is exponentially small what in this case involves an asymptotic estimate for a particular dirichlet like series

large scale continuation and numerical bifurcation for partial differential equations in this paper the problem of computing bifurcation diagrams for large scale nonlinear parameter dependent steady state systems which arise following the spatial discretization of semilinear pdes is investigated a continuation algorithm which employs a preconditioned version of the recursive projection method rpm is presented the rpm is often expensive when it is used in conjunction with the numerical method of lines preconditioning the jacobian of the underlying fixed point operator results in an algorithm the preconditioned recursive projection method prpm which is capable of efficiently computing equilibrium solution diagrams of large stiff systems for many pde problems the prpm is a fast and effective means of detecting both steady state and hopf bifurcation along a branch of solutions a description of the performance of the prpm when applied to two numerical examples is given