a remark on regularity criterion for the dissipative quasi geostrophic equations abstract this paper concerns with a regularity criterion of solutions to the 2d dissipative quasi geostrophic equations based on a logarithmic sobolev inequality in besov spaces the absence of singularities of in 0 t is derived for a solution on the interval 0 t satisfying the condition l r 0 t b p 0 for 2 p r 4 p this is an extension of earlier regularity results in the serrin s type space l r 0 t l p

global solutions to the two dimensional quasi geostrophic equation with critical or super critical dissipation the two dimensional quasi geostrophic 2d qg equation with critical and super critical dissipation is studied in sobolev space h s 2 for critical case open image in new window existence of global large solutions in h s is proved for s open image in new window when open image in new window is small this generalizes and improves the results of constantin d cordoba and wu 4 for s 1 2 and the result of a cordoba and d cordoba 8 for s open image in new window for s 1 these solutions are also unique the improvement for pushing s down from 1 to open image in new window is somewhat surprising and unexpected for super critical case 0 open image in new window existence and uniqueness of global large solution in h s is proved when the product open image in new window is small for suitable s 2 2 p 1 and 0 1

generalized surface quasi geostrophic equations with singular velocities this paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals the first family is a generalized surface quasigeostrophic sqg equation with the velocity field u related to the scalar by where and is the zygmund operator the borderline case 1 corresponds to the sqg equation and the situation is more singular for 1 we obtain the local existence and uniqueness of classical solutions the global existence of weak solutions and the local existence of patch type solutions the second family is a dissipative active scalar equation with which is at least logarithmically more singular than the velocity in the first family we prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data this result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by k ohkitani 2012 wiley periodicals inc

a numerical ode solver that preserves the fixed points and their stability in this paper we provide a one step predictor corrector method for numerically solving first order differential initial value problems with two fixed points the method preserves the stability behaviour of the fixed points which results in an efficient integrator for this kind of problem some numerical examples are provided to show the good performance of the method

worst case evaluation complexity for unconstrained nonlinear optimization using high order regularized models the worst case evaluation complexity for smooth possibly nonconvex unconstrained optimization is considered it is shown that if one is willing to use derivatives of the objective function up to order p for p ge 1 p 1 and to assume lipschitz continuity of the p th derivative then an epsilon approximate first order critical point can be computed in at most o epsilon p 1 p o p 1 p evaluations of the problem s objective function and its derivatives this generalizes and subsumes results known for p 1 p 1 and p 2 p 2

well posedness and decay of solutions for three dimensional generalized navier stokes equations abstract in this paper we study the generalized incompressible navier stokes equations in r 3 based on the energy estimates and regularization of the initial data with the heat semigroup we prove the well posedness of solutions in h 5 4 2 r 3 provided that the h 5 4 2 r 3 norm of initial data is sufficiently small in addition in contrast to the generalized heat equation the upper bound of the time decay rate of solutions to the generalized navier stokes equations is also established

on global regularity of 2d generalized magnetohydrodynamic equations abstract in this article we study the global regularity of 2d generalized magnetohydrodynamic equations 2d gmhd in which the dissipation terms are u and b we show that smooth solutions are global in the following three cases 1 2 1 0 1 2 2 2 2 0 we also show that in the inviscid case 0 if 1 then smooth solutions are global as long as the direction of the magnetic field remains smooth enough

variable stepsize implementation of multistep methods for y f x y y the falkner method is a multistep scheme intended for the numerical solution of second order initial value problems where the first derivative does appear explicitly in this paper we develop a procedure to obtain k step falkner methods explicit and implicit in their variable step size versions providing recurrence formulas to compute the coefficients efficiently considering a pair of explicit and implicit formulae these may be implemented in predictor corrector mode

asymptotic convergence rate of supercritical surface quasi geostrophic equation in morrey space abstract in this paper the asymptotic convergence rate of the supercritical surface quasi geostrophic equation in morrey space is investigated it is shown that if the global solution x t of the supercritical surface quasi geostrophic equation satisfies the growth condition in critical morrey space 0 t m p q r 2 r d t for r 2 p 2 p then every perturbed weak solution x t of the perturbed quasi geostrophic equation converges asymptotically to x t with the optimal rate t t l 2 o 1 t 1 as t in particular the initial perturbation is allowed large the findings are based on the spectral decomposition methods of fractional laplace and morrey space technique

efficient preconditioner updates for shifted linear systems we present a technique for building effective and low cost preconditioners for sequences of shifted linear systems a alpha i x alpha b where a is symmetric positive definite and alpha 0 this technique updates a preconditioner for a available in the form of an ldl t factorization by modifying only the nonzero entries of the l factor in such a way that the resulting preconditioner mimics the diagonal of the shifted matrix and reproduces its overall behavior this approach is supported by a theoretical analysis as well as by numerical experiments showing that it works efficiently for a broad range of values of alpha

a new bernstein s inequality and the 2d dissipative quasi geostrophic equation we show a new bernstein s inequality which generalizes the results of cannone planchon danchin and lemarie rieusset as an application of this inequality we prove the global well posedness of the 2d quasi geostrophic equation with the critical and super critical dissipation for the small initial data in the critical besov space and local well posedness for the large initial data

on the evaluation complexity of cubic regularization methods for potentially rank deficient nonlinear least squares problems and its relevance to constrained nonlinear optimization we propose a new termination criterion suitable for potentially singular zero or nonzero residual least squares problems with which cubic regularization variants take at most mathcal o epsilon 3 2 residual and jacobian evaluations to drive either the euclidean norm of the residual or its gradient below epsilon this is the best known bound for potentially rank deficient nonlinear least squares problems we then apply the new optimality measure and cubic regularization steps to a family of least squares merit functions in the context of a target following algorithm for nonlinear equality constrained problems this approach yields the first evaluation complexity bound of order epsilon 3 2 for nonconvexly constrained problems when higher accuracy is required for primal feasibility than for dual first order criticality

global well posedness for the 2d inviscid benard system with fractional diffusivity and yudovichʼs type data abstract in this paper we consider the cauchy problem of the two dimensional inviscid benard system with fractional diffusivity we show that there is a global unique solution to this system with yudovich s type data

global well posedness in the super critical dissipative quasi geostrophic equations we consider the quasi geostrophic equation with the dissipation term in the case constantin cordoba wu 6 proved the global existence of strong solution in h 1 and h 2 under the assumption of small l norm of initial data in this paper we prove the global existence in the scale invariant besov space b 2 2 2 1 for initial data small in the b 2 2 2 1 norm we also prove a global stability result in b 1 2 1

a family of dissipative active scalar equations with singular velocity and measure initial data this paper considers a family of generalized active scalar equations with fractional dissipation whose velocity fields are more singular than riesz transform we prove global well posedness results for small initial data belonging to besov morrey spaces which contain strongly singular functions and measures concentrated at points diracs and on smooth curves self similar solutions are obtained for initial data and coupling velocity operator with correct homogeneities we also show an asymptotic behavior result and obtain a class of asymptotically self similar solutions

on the choice of the frequency in trigonometrically fitted methods for periodic problems in this paper the use of a trigonometrically fitted method to obtain the approximate solutions of some nonlinear periodic oscillators is presented a great number of different approaches have been considered to obtain analytical approximations for this kind of problems a generalized decomposition method gdm a linearized harmonic balance procedure lhb the homotopy perturbation method hpm the harmonic balanced method hbm the adomian decomposition method etc from those approaches analytical approximations to the frequency of oscillation and periodic solutions are obtained which are valid for a large range of amplitudes of oscillation however these techniques have been limited to obtain only one or two iterates because of the great amount of algebra involved we use a trigonometrically adapted method to obtain numerical approximations to the solutions yielding very acceptable results on the basis that the approximation of the frequency of the method is done with great accuracy there are a lot of trigonometrically fitted methods in the literature but there is not a definite way to obtain the optimal value of the frequency we present a strategy for the choice of the parameter value in the adapted method based on the minimization of the total energy some examples solved numerically confirm the good performance of the adopted strategy

symmetric multistep methods with zero phase lag for periodic initial value problems of second order differential equations we present in this paper two step and four step symmetric multistep methods involving a parameter p to solve a special class of initial value problems associated with second order ordinary differential equations in which the first derivative does not appear explicitly it is shown that the methods have zero phase lag when p is chosen as 2 times the frequency of the given initial value problem the periodicity intervals are given in terms of expressions involving the parameter p as p increases the periodicity intervals increase and for large p the methods are almost p stable

existence and regularity of weak solutions to the quasi geostrophic equations in the spaces lp or dot h 1 2 in this paper we study the 2d quasi geostrophic equation with and without dissipation we give global existence results of weak solutions for an initial data in the space lp or dot h 1 2 in the dissipative case when the initial data is in lp p 2 we give a regularity result of these solutions

symmetry of components and liouville type theorems for semilinear elliptic systems involving the fractional laplacian abstract in this paper we study the symmetry of components of positive solutions for the fractional elliptic systems in the whole space r n as an application we prove the uniqueness of positive solutions for fractional schrodinger type systems with critical exponents we also prove the symmetry of components and some liouville type theorems for a class of fractional noncooperative systems which contains the systems arising in bose einstein condensation and in biological models of lotka volterra type

on the a priori estimates for the euler the navier stokes and the quasi geostrophic equations abstract we prove new a priori estimates for the 3d euler the 3d navier stokes and the 2d quasi geostrophic equations by the method of similarity transforms

on the convergence and worst case complexity of trust region and regularization methods for unconstrained optimization a nonlinear stepsize control framework for unconstrained optimization was recently proposed by toint optim methods softw 28 82 95 2013 providing a unified setting in which the global convergence can be proved for trust region algorithms and regularization schemes the original analysis assumes that the hessians of the models are uniformly bounded in this paper the global convergence of the nonlinear stepsize control algorithm is proved under the assumption that the norm of the hessians can grow by a constant amount at each iteration the worst case complexity is also investigated the results obtained for unconstrained smooth optimization are extended to some algorithms for composite nonsmooth optimization and unconstrained multiobjective optimization as well

a new class of two step p stable tfpl methods for the numerical solution of second order ivps with oscillating solutions abstract a new class of two step linear symmetric methods is introduced for the numerical solution of second order initial value problems that have highly oscillatory solutions in this class for the first time in the literature we calculate the coefficients of the method by a combination of a trigonometrically fitted tf method and a vanishing of the phase lag and some of its derivatives vsdpl method and we construct a new class of methods that has the properties of tf methods and vsdpl methods which we call the tfpl method this method is of algebraic order 8 and has an important p stability property the main structure of the method is multiderivative and the combined phases were applied to expand the stability interval and to achieve p stability the advantage of the method in comparison with similar methods in terms of efficiency accuracy and stability is shown by its implementation in some important problems the undamped duffing equation etc

existence of solutions for an inhomogeneous fractional semilinear heat equation abstract we obtain necessary conditions and sufficient conditions on the existence of solutions to the cauchy problem for a fractional semilinear heat equation with an inhomogeneous term we identify the strongest spatial singularity of the inhomogeneous term for the solvability of the cauchy problem

evaluation complexity for nonlinear constrained optimization using unscaled kkt conditions and high order models the evaluation complexity of general nonlinear possibly nonconvex constrained optimization is analyzed it is shown that under suitable smoothness conditions an epsilon approximate first order critical point of the problem can be computed in order o epsilon 1 2 p 1 p evaluations of the problem s functions and their first p derivatives this is achieved by using a two phase algorithm inspired by cartis gould and toint siam j optim 21 2011 pp 1721 1739 siam j optim 23 2013 pp 1553 1574 it is also shown that strong guarantees in terms of handling degeneracies on the possible limit points of the sequence of iterates generated by this algorithm can be obtained at the cost of increased complexity at variance with previous results the epsilon approximate first order criticality is defined by satisfying a version of the kkt conditions with an accuracy that does not depend on the size of the lagrange multipliers

on the well posedness of various one dimensional model equations for fluid motion abstract we consider 1d equations with nonlocal velocity of the form w t u w x u x w w where the nonlocal velocity u is given by 1 u 1 x x w 0 or 2 u h w h is the hilbert transform in this paper we address several local well posedness results with blow up criteria for smooth initial data we then establish the global well posedness by using the blow up criteria

cubic regularization of newton method and its global performance in this paper we provide theoretical analysis for a cubic regularization of newton method as applied to unconstrained minimization problem for this scheme we prove general local convergence results however the main contribution of the paper is related to global worst case complexity bounds for different problem classes including some nonconvex cases it is shown that the search direction can be computed by standard linear algebra technique

decay of weak solutions to the 2d dissipative quasi geostrophic equation we address the decay of the norm of weak solutions to the 2d dissipative quasi geostrophic equation when the initial data 0 is in l2 only we prove that the l2 norm tends to zero but with no uniform rate that is there are solutions with arbitrarily slow decay for 0 in lp l2 with 1 p frac 2 2 alpha 1 have uniform decay rates this result allows us to prove decay for the lq norms for q geq frac 2 2 alpha 1 when 0 is in l 2 cap l frac 2 2 alpha 1

nonexistence of simple hyperbolic blow up for the quasi geostrophic equation the problem we are concerned with is whether singularities form in finite time in incompressible fluid flows it is well known that the answer is no in the case of euler and navier stokes equations in dimension two in dimension three it is still an open problem for these equations in this paper we focus on a two dimensional active scalar model for the 3d euler vorticity equation constantin majda and tabak 7 suggested by studying rigorous theorems and detailed numerical experiments a general principle if the level set topology in the temperature field for the 2d quasi geostrophic active scalar in the region of strong scalar gradients does not contain a hyperbolic saddle then no finite time singularity is possible numerical simulations showed evidence of singular behavior when the geometry of the level sets of the active scalar contain a hyperbolic saddle there is a naturally associated notion of simple hyperbolic saddle breakdown the main theorem we present in this paper shows that such breakdown cannot occur in finite time we also show that the angle of the saddle cannot close in finite time and it cannot be faster than a double exponential in time using the same techniques we see that analogous results hold for incompressible 2d and 3d euler these results were announced in 9 but with a slight difference in the definition of a simple hyperbolic saddle the definition given in section 4 generalizes the one given in the announcement see also constantin 4 discussed in section 7 remark 5 below the whole work described in this paper is basically part of the author s thesis i am particularly grateful to my thesis advisor charles fefferman for his attention support guidance and advice i am indebted to d christodoulou and p constantin for helpful corrections and suggestions i wish to thank a majda for suggesting the subject and e tabak for discussions and com

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

a p stable complete in phase obrechkoff trigonometric fitted method for periodic initial value problems in this paper we derive a p stable trigonometric fitted obrechkoff method with phase lag frequency distortion infinity it is easy to see from numerical results presented that the new method is much more accurate than previous methods

p stable symmetric super implicitmethods for periodic initial value problems the idea of super implicit methods requiring not just past and present but also future values was suggested by fukushima recently here we construct p stable super implicit methods for the solution of second order initial value problems the benefit of such methods is realized when using vector or parallel computers

large time decay and growth for solutions of a viscous boussinesq system in this paper we analyze the decay and the growth for large time of weak and strong solutions to the three dimensional viscous boussinesq system we show that generic solutions blow up as t in the sense that the energy and the l norms of the velocity field grow to infinity for large time for 1 p 3 in the case of strong solutions we provide sharp estimates both from above and from below and explicit asymptotic profiles we also show that solutions arising from u0 0 with zero mean for the initial temperature 0 have a special behavior as x or t tends to infinity contrarily to the generic case their energy dissipates to zero for large time

commutator estimate and its application to regularity criteria of the dissipative quasi geostrophic equation abstract a new commutator estimate with respect to a nonlinear convection upper bounded by a single partial derivative component in hilbert spaces is obtained as an application regularity criteria on the supercritical quasi geostrophic equation are obtained provided that solution growth conditions are assumed to involve a single partial derivative component

a new family of explicit linear two step singularly p stable obrechkoff methods for the numerical solution of second order ivps abstract according to lambert and watson 6 theorem 4 says that the linear multistep p stable methods can not be explicit they must be implicit and being implicit is the essential condition to obtain important feature of p stability few explicit p stable methods have been created in which they are nonlinear or at most p stable for the first time in literature in this paper we create a new family of explicit linear two step singularly p stable methods with phase lag of order infinity for the numerical solution of initial value problems of second order ordinary differential equations finally the numerical results obtained by the new family for some well known problems show its superiority in efficiency accuracy convergency and stability

global regularity of the two dimensional incompressible generalized magnetohydrodynamics system abstract in this paper we consider the global regularity of the two dimensional generalized incompressible magnetohydrodynamics system with velocity dissipation exponent 1 4 and magnetic diffusion exponent 1 based on the improved integrability in terms of vorticity this result is a further improvement of very recent works jiu and zhao 2014 and yamazaki 2013 which require 1 3 and 1 therefore our result is more closer to the resolution of the well known global regularity issue on the 2d magnetohydrodynamics system 0 1

long time dynamics of 2d quasi geostrophic equations with damping in lp abstract the long time behavior of solutions of 2d quasi geostrophic equations with damping on the whole space is studied the existence of the global attractor in l p is proved new difficulties come from nonlocal terms and the banach space l p we overcome them by using commutator estimates and positivity inequalities associated with fractional laplacian

dynamics of a fractional hydrodynamical equation for the heisenberg paramagnet in the present work we study the global solvability and large time dynamics for a fractional generalization of the hydrodynamical equation modeling the soft micromagnetic materials introducing a cancellation property we prove the existence of weak solutions and establish a uniqueness criterion a maximal principle is obtained and the global existence and uniqueness of smooth solutions are proved by some a priori estimates finally we analyze the asymptotic behavior of the solutions within the theory of infinite dimensional dissipative dynamical systems we prove that the problem generates a strongly continuous semigroup on a suitable phase space and show the existence of a maximal global attractor a in this phase space moreover in absence of external force global attractor a converges exponentially to a single equilibrium

a higher order levenberg marquardt method for nonlinear equations in this paper we present a high order levenberg marquardt method for nonlinear equations at every iteration not only a lm step is computed but also two approximate lm steps are computed which use the previous calculated jacobian under the local error bound condition which is weaker than nonsingularity the new method has biquadratic convergence a globally convergent lm algorithm is also given by the trust region technique numerical results show that the new fourth order lm algorithm performs very well and could save many jacobian calculations especially for large scale problems

convergence of a regularized euclidean residual algorithm for nonlinear least squares the convergence properties of the new regularized euclidean residual method for solving general nonlinear least squares and nonlinear equation problems are investigated this method derived from a proposal by nesterov optim methods softw 22 2007 pp 469 483 uses a model of the objective function consisting of the unsquared euclidean linearized residual regularized by a quadratic term at variance with previous analysis its convergence properties are here considered without assuming uniformly nonsingular globally lipschitz continuous jacobians nor an exact subproblem solution it is proved that the method is globally convergent to first order critical points and under stronger assumptions to roots of the underlying system of nonlinear equations the rate of convergence is also shown to be quadratic under stronger assumptions

lower bounds for an integral involving fractional laplacians and the generalized navier stokes equations in besov spaces when estimating solutions of dissipative partial differential equations in l p related spaces we often need lower bounds for an integral involving the dissipative term if the dissipative term is given by the usual laplacian lower bounds can be derived through integration by parts and embedding inequalities however when the laplacian is replaced by the fractional laplacian the approach of integration by parts no longer applies in this paper we obtain lower bounds for the integral involving by combining pointwise inequalities for with bernstein s inequalities for fractional derivatives as an application of these lower bounds we establish the existence and uniqueness of solutions to the generalized navier stokes equations in besov spaces the generalized navier stokes equations are the equations resulting from replacing in the navier stokes equations by

a two dimensional model for quasigeostrophic flow comparison with the two dimensional euler flow abstract a simple two dimensional model for quasigeostrophic flow is contrasted with the two dimensional incompressible euler equations the model arises under the assumptions of fast rotation uniform stratification and uniform potential vorticity it is found that the more loca feed back of the quasigeostrophic model gives rise to strongly nonlinear front formation as opposed to two dimensional euler where the steepening process of mature fronts obeys a nonlocal nearly linear mechanism

regularity criterion of the 2d benard equations with critical and supercritical dissipation abstract in this paper we investigate the cauchy problem for the two dimensional 2d incompressible benard equations on the one hand we prove global in time existence of smooth solutions to the 2d benard equations with critical dissipation on the other hand we establish several regularity criteria involving temperature for 2d benard equations with supercritical dissipation

existence and uniqueness of the solution to the dissipative 2d quasi geostrophic equations in the sobolev space we study the two dimensional dissipative quasi geostrophic equations in the sobolev space open image in new window existence and uniqueness of the solution local in time is proved in h s when s 2 1 existence and uniqueness of the solution global in time is also proved in h s when s 2 1 and the initial data open image in new window is small for the case s 2 1 we also obtain the unique large global solution in h s provided that open image in new window is small enough

a remark on global regularity of 2d generalized magnetohydrodynamic equations abstract in this paper we study the global regularity of the following 2d two dimensional generalized magnetohydrodynamic equations u t u u p b b u b t u b b u b and get global regular solutions when 0 1 2 1 3 2 3 which improves the results in 5 in particular we obtain the global regularity of the 2d generalized mhd when 0 and 3 2

a pointwise estimate for fractionary derivatives with applications to partial differential equations this article emphasizes the role played by a remarkable pointwise inequality satisfied by fractionary derivatives in order to obtain maximum principles and lp decay of solutions of several interesting partial differential equations in particular there are applications to quasigeostrophic flows in two space variables with critical viscosity that model the eckman pumping see baroud ch n plapp b b she z s swinney h l 2002 phys rev lett 88 114501 and constantin p 2002 phys rev lett 89 184501

energy decay for a weak solution of the navier stokes equation with slowly varying external forces we consider the navier stokes system with slowly decaying external forces formula we show that the energy norm of a weak solution has non uniform decay u t 2 0 t under suitable conditions on the data f and a which make the energy of solution bounded in time also we show the exact rate of the decay uniform decay of the energy u t 2 c 1 t e for external forces with a given explicit decay rate

l2 stability for solutions of subcritical quasi geostrophic equation with large perturbations abstract this paper concerns with the asymptotic stability for the weak solution of the two dimensional dissipative subcritical quasi geostrophic equation in the class l p 0 t l q r 2 for p 2 q 2 q 1 2 it is proved that although the initial data and external force perturbations from are large every perturbed weak solution converges asymptotically to in l 2 r 2 as t as a byproduct we also obtain the weak strong uniqueness criterion of subcritical quasi geostrophic equation which improves the earlier results

nonexistence of global solutions for a class of two time nonlinear evolution equations nonexistence results of global solutions for a class of ultraparabolic equations and systems involving the fractional laplacian operator are derived the proofs are based on the choice of a suitable test function in the weak formulation of the considered problems the considered equations and systems have their applications in diffusion theory in porous media

an improved trigonometrically fitted p stable obrechkoff method for periodic initial value problems in this paper we present an improved p stable trigonometrically fitted obrechkoff method with phase lag frequency distortion infinity compared with the previous p stable trigonometrically fitted obrechkoff method developed by simos our new method is simpler in structure and more stable in computation we have also improved the accuracy of the first order derivative formula from the numerical illustration presented we can show that the new method is much more accurate than the previous methods

the maximum principle and the global attractor for the dissipative 2d quasi geostrophic equations the long time behavior of the solutions to the two dimensional dissipative quasi geostrophic equations is studied we obtain a new positivity lemma which improves a previous version of a cordoba and d cordoba 10 and 11 as an application of the new positivity lemma we obtain the new maximum principle i e the decay of the solution in l p for any p 2 when f is zero as a second application of the new positivity lemma for the sub critical dissipative case with open image in new window the existence of the global attractor for the solutions in the space h s for any s 2 1 is proved for the case when the time independent f is non zero therefore the global attractor is infinitely smooth if f is this significantly improves the previous result of berselli 2 which proves the existence of an attractor in some weak sense for the case 1 the global attractor exists in h s for any s 0 and the estimate of the hausdorff and fractal dimensions of the global attractor is also available

a first approach in solving initial value problems in odes by elliptic fitting methods exponentially fitted and trigonometrically fitted methods have a long successful history in the solution of initial value problems but other functions might be considered in adapted methods specifically this paper aims at the derivation of a new numerical scheme for approximating initial value problems of ordinary differential equations using elliptic functions the example considered is the undamped duffing equation where the forcing term is of autonomous type affected by a perturbation parameter the new scheme is constructed by considering a suitable approximation to the theoretical solution based on elliptic functions the proposed elliptic fitting procedure has been tested on a variety of problems showing its good performance

a trust region algorithm for equality constrained optimization a trust region algorithm for equality constrained optimization is proposed that employs a differentiable exact penalty function under certain conditions global convergence and local superlinear convergence results are proved

existence and decay of solutions to the dissipative quasi geostrophic equation with delays abstract we consider the dissipative quasi geostrophic equation with forcing and advective terms containing delays we prove the existence of weak solutions and algebraic decay of the energy provided the initial data 0 is in l p l 2 1 p 2 and the forcing term satisfies some mild conditions on its long time behavior

fractional navier stokes equations and a holder type inequality in a sum of singular spaces abstract in this paper we study the local well posedness of the fractional navier stokes system with initial data belonging to a sum of two pseudomeasure type spaces denoted by p m a b p m a p m b the proof requires showing a holder type inequality in p m a b as well as establishing estimates of the semigroup generated by the fractional power of laplacian on these spaces

a preconditioning framework for sequences of diagonally modified linear systems arising in optimization we propose a framework for building preconditioners for sequences of linear systems of the form a delta k x k b k where a is symmetric positive semidefinite and delta k is diagonal positive semidefinite such sequences arise in several optimization methods e g in affine scaling methods for bound constrained convex quadratic programming and bound constrained linear least squares as well as in trust region and overestimation methods for convex unconstrained optimization problems and nonlinear least squares for all the matrices of a sequence the preconditioners are obtained by updating any preconditioner for a available in the ldl t form the preconditioners in the framework satisfy the natural requirement of being effective on slowly varying sequences furthermore under an additional property they are also able to cluster eigenvalues of the preconditioned matrix when some entries of delta k are sufficiently large we present two low cost preconditioners sharing the above mentioned p

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

a fourth order runge kutta method based on bdf type chebyshev approximations in this paper we consider a new fourth order method of bdf type for solving stiff initial value problems based on the interval approximation of the true solution by truncated chebyshev series it is shown that the method may be formulated in an equivalent way as a runge kutta method having stage order four the method thus obtained have good properties relatives to stability including an unbounded stability domain and large a value concerning a a stability a strategy for changing the step size based on a pair of methods in a similar way to the embedding pair in the runge kutta schemes is presented the numerical examples reveals that this method is very promising when it is used for solving stiff initial value problems

asymptotic behavior for the quasi geostrophic equations with fractional dissipation in r2 abstract in this paper the surface quasi geostrophic equations qge with fractional dissipation in r 2 are considered our aim is to study the long time behavior of solutions of qge in the subcritical case to this end we investigate the global well posedness and global attractor for qge in h s r 2 via commutator estimates for nonlinear terms a new iterative technique for estimates of higher order derivatives and with the help of a nonlocal damping term besides by using the fractional lieb thirring inequality estimates of the finite hausdorff and fractal dimensions of the global attractor are found

upper bound of decay rate for solutions to the navier stokes voigt equations in r 3 in this paper we first show the global existence uniqueness and regularity of weak solutions for the navier stokes voigt equations in r 3 then we combine the fourier splitting method of schonbek and the gronwall inequality to prove that the solutions have the following decay rates m u x t 2 m 1 u x t 2 c 1 t 3 2 m for large t when u 0 h m r 3 l 1 r 3 and m 0 1

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely

asymptotic expansion of solutions to the drift diffusion equation with fractional dissipation abstract the initial value problem for the drift diffusion equation arising from the model of semiconductor device simulations is studied the dissipation on this equation is given by the fractional laplacian 2 large time behavior of solutions to the drift diffusion equation with 0 1 is discussed when 1 large time behavior of solutions is known however when 0 1 the perturbation methods used in the preceding works would not work in this paper the asymptotic expansion of solutions with high order is derived

growth of solutions for qg and 2d euler equations the work of constantin majda tabak 1 developed an analogy between the quasi geostrophic and 3d euler equations constantin majda and tabak proposed a candidate for a singularity for the quasi geostrophic equation their numerics showed evidence of a blow up for a particular initial data where the level sets of the temperature contain a hyperbolic saddle the arms of the saddle tend to close in finite time producing a sharp front numerics studies done later by ohkitani yamada 8 and constantin nie schorghofer 2 with the same initial data suggested that instead of a singularity the derivatives of the temperature were increasing as double exponential in time the study of collapse on a curve was first studied in 1 for the quasi geostrophic equation where they considered a simplified ansatz for classical frontogenesis with trivial topology at the time of collapse the scalar is discontinuous across the curve x2 f x1 with different limiting values for the temperature on each side of the front they show that under this topology the directional field remains smooth up to the collapse which contradicts the following theorem proven in 1

solutions of the 2d quasi geostrophic equation in holder spaces abstract the 2d quasi geostrophic equation t u 0 u r is a two dimensional model of the 3d hydrodynamics equations when 1 2 the issue of existence and uniqueness concerning this equation becomes difficult it is shown here that this equation with either 0 or 0 and 0 1 2 has a unique local in time solution corresponding to any initial datum in the space c r l q for r 1 and q 1