dissipative quasi geostrophic equation for large initial data in the critical sobolev space the critical and super critical dissipative quasi geostrophic equations are investigated in mathbb r 2 we prove local existence of a unique regular solution for arbitrary initial data in h2 2 which corresponds to the scaling invariant space of the equation we also consider the behavior of the solution near t 0 in the sobolev space

on the quasi geostrophic equations on compact closed surfaces in r3 abstract the quasi geostrophic equations on compact surfaces in r 3 without boundary are studied by means of the theory of quasi linear parabolic evolution equations based on maximal l p regularity it is shown that the problem is globally strongly well posed in l q the solutions regularize instantly and form a global semiflow in the proper state manifold and as time goes to infinity a solution converges exponentially to a constant in the topology of the state space

global well posedness for the critical 2d dissipative quasi geostrophic equation we give an elementary proof of the global well posedness for the critical 2d dissipative quasi geostrophic equation the argument is based on a non local maximum principle involving appropriate moduli of continuity

critical spaces for quasilinear parabolic evolution equations and applications abstract we present a comprehensive theory of critical spaces for the broad class of quasilinear parabolic evolution equations the approach is based on maximal l p regularity in time weighted function spaces it is shown that our notion of critical spaces coincides with the concept of scaling invariant spaces in case that the underlying partial differential equation enjoys a scaling invariance applications to the vorticity equations for the navier stokes problem convection diffusion equations the nernst planck poisson equations in electro chemistry chemotaxis equations the mhd equations and some other well known parabolic equations are given

nonlinear analytic semiflows in this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems x t f x t and quasilinear initial value problems x t a x t x t f x t this theory extends the theory of daprato and grisvard of 1979 and shows how various properties like analyticity of solutions can be derived as a direct corollary of the existence theorem

new thoughts on old results of r t seeley in this paper we show that the lp realization of a vector valued elliptic boundary value problem j admits a bounded calculus on lp g e 1 p provided the top order coefficients of are holder continuous here g denotes a domain in n with compact c2m boundary and e a banach space of class our proof is based on an abstract perturbation result for operators admitting bounded calculus and kernel estimates for the solution of j

new elliptic projections and a priori error estimates of h1 galerkin mixed finite element methods for optimal control problems governed by parabolic integro differential equations in this paper we discuss a priori error estimates of h1 galerkin mixed finite element methods for optimal control problems governed by parabolic integro differential equations the state variables and co state variables are approximated by the lowest order raviart thomas mixed finite element and linear finite element and the control variable is approximated by piecewise constant functions both semidiscrete and fully discrete schemes are considered based on some new elliptic projections we derive a priori error estimates for the control variable the state variables and the adjoint state variables the related a priori error estimates for the new projections error are also established a numerical example is given to demonstrate the theoretical results

on the derivation of the herman kluk propagator following an earlier derivation by miller we give a concise derivation of the herman kluk propagator that fully exploits the symplectic structure of the problem we also show directly that the version of the thawed gaussian semiclassical propagator obtained by baranger and co workers is equivalent to the linearized herman kluk propagator

finite element approximation of parabolic time optimal control problems the numerical approximation of a parabolic time optimal control problem via piecewise linear splines is considered at each stage of the approximation a bang bang approximate control is selected by solving for its switching times as the solution of a constrained nonlinear least squares optimization problem the well posedness of the approximation scheme is shown and the rate of convergence to the exact solution investigated numerical results for some one and two dimensional parabolic control problems are given

adaptive finite element approximation for distributed elliptic optimal control problems in this paper sharp a posteriori error estimators are derived for a class of distributed elliptic optimal control problems these error estimators are shown to be useful in adaptive finite element approximation for the optimal control problems and are implemented in the adaptive approach our numerical results indicate that the sharp error estimators work satisfactorily in guiding the mesh adjustments and can save substantial computational work

semiclassical approximations in phase space with coherent states we present a complete derivation of the semiclassical limit of the coherent state propagator in one dimension starting from path integrals in phase space we show that the arbitrariness in the path integral representation which follows from the overcompleteness of the coherent states results in many different semiclassical limits we explicitly derive two possible semiclassical formulae for the propagator we suggest a third one and we discuss their relationships we also derive an initial value representation for the semiclassical propagator based on an initial gaussian wavepacket it turns out to be related to but different from heller s thawed gaussian approximation it is very different from the herman kluk formula which is not a correct semiclassical limit we point out errors in two derivations of the latter finally we show how the semiclassical coherent state propagators lead to wkb type quantization rules and to approximations for the husimi distributions of stationary states

a posteriori error estimates for discontinuous galerkin time stepping method for optimal control problems governed by parabolic equations in this paper we examine the discontinuous galerkin dg finite element approximation to convex distributed optimal control problems governed by linear parabolic equations where the discontinuous finite element method is used for the time discretization and the conforming finite element method is used for the space discretization we derive a posteriori error estimates for both the state and the control approximation assuming only that the underlying mesh in space is nondegenerate for problems with control constraints of obstacle type which are the kind most frequently met in applications further improved error estimates are obtained

a posteriori error estimates for control problems governed by stokes equations in this paper we derive a posteriori error estimates for the finite element approximation of distributed optimal control problems governed by the stokes equations we obtain a posteriori error estimators for both the state and the control approximation in the l2 norm and the h1 norm these estimates can be used to construct reliable adaptive finite element approximation for the control problems

on the affine heat equation for non convex curves 1998 american mathematical society first published in journal of the american mathematical society vol 11 no 3 july 1998 published by the american mathematical society

analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls an optimal control problem for impressed cathodic systems in electrochemistry is studied the control in this problem is the current density on the anode a matching objective functional is considered we first demonstrate the existence and uniqueness of solutions for the governing partial differential equation with a nonlinear boundary condition we then prove the existence of an optimal solution next we derive a necessary condition of optimality and establish an optimality system of equations finally we define a finite element algorithm and derive optimal error estimates

new approach to semiclassical analysis in mechanics a new method is proposed for constructing approximate solutions to the schrodinger equation in place of the wave function its gaussian windowed fourier transform is used as the fundamental entity this allows an intuitively attractive connection to be made with a family of classical trajectories and at all times the wave function is inferred from the present state of these trajectories the fact that the connection between the wave function and the classical trajectories is consistently constructed in phase space allows this method to be free of the limitations of other methods

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

a priori error estimates for space time finite element discretization of parabolic optimal control problems part i problems without control constraints in this paper we develop a priori error analysis for galerkin finite element discretizations of optimal control problems governed by linear parabolic equations the space discretization of the state variable is done using usual conforming finite elements whereas the time discretization is based on discontinuous galerkin methods for different types of control discretizations we provide error estimates of optimal order with respect to both space and time discretization parameters the paper is divided into two parts in the first part we develop some stability and error estimates for space time discretization of the state equation and provide error estimates for optimal control problems without control constraints in the second part of the paper the techniques and results of the first part are used to develop a priori error analysis for optimal control problems with pointwise inequality constraints on the control variable

path integral based transition probability evaluation in a quantum system with a diagonal density matrix of the final state the variant of transition probability evaluation in a quantum system based on transition probability representation by a path integral using a density matrix formalism is described to estimate an integral s value a set of variables is selected and the monte carlo method is used for integration over these variables the stationary phase method is applied to integration over the remaining variables in the case of the final state with a diagonal density matrix this scheme leads to a simple expression for transition probability

a stable manifold theorem for the curve shortening equation on presente une famille de solutions homothetiques de l equation pour une courbe planaire x kn et on demontre l existence de varietes non lineaires stables et instables autour de telles solutions

a posteriori error estimates for optimal control problems governed by parabolic equations in this paper we derive a posteriori error estimates for the finite element approximation of quadratic optimal control problem governed by linear parabolic equation we obtain a posteriori error estimates for both the state and the control approximation such estimates which are apparently not available in the literature are an important step towards developing reliable adaptive finite element approximation schemes for the control problem

a maximum principle applied to quasi geostrophic equations we study the initial value problem for dissipative 2d quasi geostrophic equations proving local existence global results for small initial data in the super critical case decay of lp norms and asymptotic behavior of viscosity solution in the critical case our proofs are based on a maximum principle valid for more general flows

multiphase thermomechanics with interfacial structure 2 evolution of an isothermal interface paper 1 1988 g 1 of this series began an investigation whose goal is a thermomechanics of two phase continua based on gibbs s notion of a sharp phase interface endowed with thermomechanical structure in that paper a balance law balance of capillary forces was introduced and then applied in conjunction with suitable statements of the first two laws of thermodynamics the chief results are thermodynamic restrictions on constitutive equations exact and approximate free boundary conditions at the interface and a hierarchy of free boundary problems the simplest versions of these problems the mullins sekerka problems are essentially the classical stefan problem with the free boundary condition u 0 for the temperature replaced by the condition u h k where k is the curvature of the free boundary and h 0 is a material constant this dependence on curvature renders the problem difficult and apart from numerical studies involving linearization stability there are almost no supporting theoretical results

an extension of pontryagin s principle for state constrained optimal control of semilinear elliptic equations and variational inequalities this paper deals with state constrained optimal control problems governed by semilinear elliptic equations or variational inequalities by using ekeland s principle a minimum principle of pontryagin s type under some stability conditions of the optimal cost with respect to the state constraints is derived

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

dynamic active contours for visual tracking visual tracking using active contours is usually set in a static framework the active contour tracks the object of interest in a given frame of an image sequence a subsequent prediction step ensures good initial placement for the next frame this approach is unnatural the curve evolution gets decoupled from the actual dynamics of the objects to be tracked true dynamical approaches exist all being marker particle based and thus prone to the shortcomings of such particle based implementations in particular topological changes are not handled naturally in this framework the now classical level set approach is tailored for evolutions of manifolds of codimension one however dynamic curve evolution is at least a codimension two problem we propose an efficient level set based approach for dynamic curve evolution which addresses the artificial separation of segmentation and prediction while retaining all the desirable properties of the level set formulation it is based on a new energy minimization functional which for the first time puts dynamics into the geodesic active contour framework

superconvergence of mixed finite element methods for optimal control problems t in this paper we investigate the superconvergence property of the numerical solution of a quadratic convex optimal control problem by using rectangular mixed finite element methods the state and co state variables are approximated by the lowest order raviart thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions some realistic regularity assumptions are presented and applied to error estimation by using an operator interpolation technique we derive l 2 superconvergence properties for the flux functions along the gauss lines and for the scalar functions at the gauss points via mixed projections moreover global l 2 superconvergence results are obtained by virtue of an interpolation postprocessing technique thus based on these superconvergence estimates some asymptotic exactness a posteriori error estimators are presented for the mixed finite element methods finally some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

l infty estimates for approximated optimal control problems an optimal control problem for a two dimensional elliptic equation is investigated with pointwise control constraints this paper is concerned with discretization of the control by piecewise linear functions the state and the adjoint state are discretized by linear finite elements approximation of order h in the l infty norm is proved in the main result

curve shortening and the topology of closed geodesics on surfaces we study flat knot types of geodesics on compact surfaces m2 for every flat knot type and any riemannian metric g we introduce a conley index associated with the curve shortening flow on the space of immersed curves on m2 we conclude existence of closed geodesics with prescribed flat knot types provided the associated conley index is nontrivial

parabolic equations for curves on surfaces part i curves with p integrable curvature this is the first of a two part paper in which we develop a theory of parabolic equations for curves on surfaces which can be applied to the so called curve shortening of flow by mean curvature problem as well as to a number of models for phase transitions in two dimensions we introduce a class of equations for which the initial value problem is solvable for initial data with p integrable curvature and we also give estimates for the rate at which the p norms of the curvature must blow up if the curve becomes singular in finite time a detailed discussion of the way in which solutions can become singular and a method for continuing the solution through a singularity will be the subject of the second part

superconvergence of quadratic optimal control problems by triangular mixed finite element methods the aim of this work is to investigate the discretization of a quadratic convex optimal control problem using the mixed finite element method the state and co state are approximated by the order k 1 raviart thomas mixed finite element spaces and the control is approximated by piecewise constant functions we construct an interpolation of the exact control and a projection of the discrete scalar co state to be the approximated solution of the control variable for the continuous optimal control problem as a result it can be proved that the difference between the interpolation and the piecewise constant approximation has superconvergence property for the control of order h 3 2 for k 0 and of order h 2 for k 1 moreover only for the order k 1 raviart thomas mixed finite element approximation does the postprocessing technique possess the superconvergence property of order h 2 finally some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence

an h 1 galerkin mixed finite element method for parabolic partial differential equations in this paper an h1 galerkin mixed finite element method is proposed and analyzed for parabolic partial differential equations with nonselfadjoint elliptic parts compared to the standard h1 galerkin procedure c1 continuity for the approximating finite dimensional subspaces can be relaxed for the proposed method moreover it is shown that the finite element approximations have the same rates of convergence as in the classical mixed method but without lbb consistency condition and quasiuniformity requirement on the finite element mesh finally a better rate of convergence for the flux in l2 norm is derived using a modified h1 galerkin mixed method in two and three space dimensions which confirms the findings in a single space variable and also improves upon the order of convergence of the classical mixed method under extra regularity assumptions on the exact solution

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

existence and uniqueness results for the 2 d dissipative quasi geostrophic equation abstract this paper concerns itself with besov space solutions of the 2 d quasi geostrophic qg equation with dissipation induced by a fractional laplacian the goal is threefold first to extend a previous result on solutions in the inhomogeneous besov space b 2 q r j wu global solutions of the 2d dissipative quasi geostrophic equation in besov spaces siam j math anal 36 2004 2005 1014 1030 to cover the case when r 2 2 second to establish the global existence of solutions in the homogeneous besov space b p q r with general indices p and q and third to determine the uniqueness of solutions in any one of the four spaces b 2 q s b p q r l q 0 t b 2 q s 2 q and l q 0 t b p q r 2 q where s 2 2 and r 1 2 2 p

symmetries and invariant solutions for the geometric heat flows we study lie symmetries and invariant solutions of the geometric heat flows the basic similarity reductions for the ghe are performed reduced equations and exact solutions associated with the symmetries are obtained group invariant solutions and reductions for the affine case are also discussed in a special case

local a posteriori error estimates for convex boundary control problems in this paper we derive some improved a posteriori error estimates for finite element approximation of neumann boundary control problems we first establish local upper a posteriori error estimates for both the state and the control approximation of the general convex boundary control problems we then derive local upper and lower a posteriori error estimates for a class of control problems that frequently appear in applications

on the 2d critical and supercritical dissipative quasi geostrophic equation in besov spaces abstract we prove the local smoothing effect of the 2d critical and supercritical dissipative quasi geostrophic equations in critical besov spaces as an application a global well posedness result is established by adapting a method in kiselev nazarov and volberg 2007 16 and an idea in dong and du 2008 15 with suitable modifications moreover we show that the unique solution obtained in chen miao and zhang 2007 11 is a classical solution these generalize some previous results in dong 2010 13 dong and du 2008 15 the main ingredients of the proofs are two commutator estimates and the preservation of suitable modulus of continuity of the solutions

finite element methods for optimal control problems governed by integral equations and integro differential equations in this paper we analyze finite element galerkin discretizations for a class of constrained optimal control problems that are governed by fredholm integral or integro differential equations the analysis focuses on the derivation of a priori error estimates and a posteriori error estimators for the approximation schemes

finite dimensional approximation of a class of constrained nonlinear optimal control problems an abstract framework for the analysis and approximation of a class of nonlinear optimal control and optimization problems is constructed nonlinearities occur in both the objective functional and the constraints the framework includes an abstract nonlinear optimization problem posed on infinite dimensional spaces an approximate problem posed on finite dimensional spaces together with a number of hypotheses concerning the two problems the framework is used to show that optimal solutions exist to show that lagrange multipliers may be used to enforce the constraints to derive an optimality system from which optimal states and controls may be deduced and to derive existence results and error estimates for solutions of the approximate problem the abstract framework and the results derived from that framework are then applied to three concrete control or optimization problems and their approximation by finite element methods the first involves the von k arm an plate equations of nonlinear elasticity the second the ginzburg landau equations of superconductivity and the third the navier stokes equations for incompressible viscous flows

upper and lower bounds of decay rates for h 12 weak solutions of the surface quasi geostrophic equation abstract this study is concerned with the large time decay estimates for weak solutions of the surface quasi geostrophic equation with h 1 2 initial data due to some special observation together with applying the decay characters to the class of h 1 2 functions we may show the upper and lower bounds of decay rates for the h 1 2 weak solution of the surface quasi strophic equation moreover as a 1 2 order derivative of h 1 2 weak solution we further derive an improved upper and lower bounds of decay rates for l 2 weak solution the findings may be of independent interest and the technique can be also applied to the other related fluid dynamical model with solutions in the negative sobolev spaces

spatial analyticity of the solutions to the subcritical dissipative quasi geostrophic equations we employ a new bilinear estimate to show that solutions to the subcritical dissipative quasi geostrophic equations with initial data in the scaling invariant lebesgue space are analytic in space variables some decay in time estimates for space time derivatives are also obtained

superconvergence properties of optimal control problems an optimal control problem for a two dimensional 2 d elliptic equation is investigated with pointwise control constraints this paper is concerned with discretization of the control by piecewise constant functions the state and the adjoint state are discretized by linear finite elements approximations of the optimal solution of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state it is proved that these approximations have convergence order h2