necessary and sufficient conditions for near optimal harvesting control problem of stochastic age dependent system in this paper we consider a near optimal harvesting control problems for stochastic age structured population system we establish necessary and sufficient conditions for near optimality these conditions are described by adjoint equations and a nearly maximum condition on the hamiltonian the proof of the main result is based on ekeland s variational principle and the adjoint processes with respect to the control variable as is well known optimal controls may fail to exist even in simple cases this justifies the use of near optimal controls which exist under minimal assumptions and are sufficient in most practical cases at last an example is given for illustrating our results

on the regularity of the magnetic field in a diffusive plasma it is known that magnetic fields in ideal chaotic plasmas tend to become extremely irregular and to concentrate in a fractal set and it is assumed that the presence of a positive resistivity will have a smoothing effect here we try to quantify this effect by proving new inequalities which on the one hand relate the local and global size of velocity and magnetic field with the gradient of this field and on the other provide a bound of the area of generalized level surfaces

optimal birth control for an age dependent n dimensional food chain model abstract in this paper we investigate optimal policies for an age dependent n dimensional food chain model which is controlled by fertility by using dubovitskii milyutin s general theory the maximum principles are obtained for problems with free terminal states infinite horizon and target sets respectively

continuous dependence results for a general model of size dependent population dynamics we are concerned with a general model of size structured population dynamics with the growth rate depending on the individual s size and time in this paper we shall study the continuous dependence of the solution on all given data such as aging and birth functions growth rate functions and initial data

analytic results related to magneto hydrodynamic turbulence abstract analytic results are provided for topics related to the magneto hydrodynamic turbulence scaling exponents of general structure functions geometric criteria for singularity formation and dissipative solutions

dynamic programming for optimal control problems with delays in the control variable we study a class of optimal control problems with state constraint where the state equation is a differential equation with delays in the control variable this class of problems arises in some economic applications in particular in optimal advertising problems the optimal control problem is embedded in a suitable hilbert space and the associated hamilton jacobi bellman hjb equation is considered in this space it is proved that the value function is continuous with respect to a weak norm and that it solves in the viscosity sense the associated hjb equation the main results are the proof of a directional c 1 regularity for the value function and the feedback characterization of optimal controls

on the evolution of nearly circular vortex patches recently the classical problem of the evolution of patches of constant vorticity was reformulated as an evolution equation for the boundary of the patch we study this equation in the neighborhood of the circular vortex patch and introduce a hierarchy of area preserving nonlinear approximate equations the first of these equations is shown to have a rich rigid structure it possesses an exhaustive increasing sequence of linear invariant manifolds of arbitrarily large finite dimensions on each of these manifolds the equation can be written as an explicit finite system of ordinary differential equations solutions of these odes starting from arbitrarily small neighborhoods of the circular vortex patch are shown to blow up

localization of analytic regularity criteria on the vorticity and balance between the vorticity magnitude and coherence of the vorticity direction in the 3d nse the first part of the paper provides spatio temporal localization of a family of analytic regularity classes for the 3d nse obtained by beirao da veiga space time integrability of the gradient of the velocity on mathbb r 3 times 0 t which is out of the range of the sobolev embedding theorem reduction to the classical foias ladyzhenskaya prodi serrin space time integrability conditions on the velocity as well as the localization of the beale kato majda regularity criterion time integrability of the l norm of the vorticity the second part introduces a family of local scaling invariant hybrid geometric analytic classes in which coherence of the vorticity direction serves as a weight in the local spatio temporal integrability of the vorticity magnitude

geometric regularity criteria for incompressible navier stokes equations with navier boundary conditions abstract we study the regularity criteria for weak solutions to the 3d incompressible navier stokes equations in terms of the direction of vorticity taking into account the boundary conditions a boundary regularity theorem is proved on regular curvilinear domains with a family of oblique derivative boundary conditions provided that the directions of vorticity are coherently aligned up to the boundary as an application we establish the boundary regularity for weak solutions to navier stokes equations in round balls half spaces and right circular cylindrical ducts subject to the classical navier and kinematic boundary conditions

yield trajectory tracking for hyperbolic age structured population systems abstract for population systems modeled by age structured hyperbolic partial differential equations pdes that are bilinear in the input and evolve with a positive valued infinite dimensional state global stabilization of constant yield set points was achieved in prior work seasonal demands in biotechnological production processes give rise to time varying yield references for the proposed control objective aiming at a global attractivity of desired yield trajectories multiple non standard features have to be considered a non local boundary condition a pde state restricted to the positive orthant of the function space and arbitrary restrictive but physically meaningful input constraints moreover we provide control lyapunov functionals ensuring an exponentially fast attraction of adequate reference trajectories to achieve this goal we make use of the relation between first order hyperbolic pdes and integral delay equations leading to a decoupling of the input dependent dynamics and the infinite dimensional internal one furthermore the dynamic control structure does not necessitate exact knowledge of the model parameters or online measurements of the age profile with a galerkin based numerical simulation scheme using the key ideas of the karhunen loeve decomposition we demonstrate the controller s performance

universality of probability distributions among two dimensional turbulent flows we study statistical properties of two dimensional turbulent flows three systems are considered the navier stokes equation surface quasigeostrophic flow and a model equation for thermal convection in the earth s mantle direct numerical simulations are used to determine one point fluctuation properties comparative study shows universality of probability density functions pdfs across different types of flow for instance the pdfs for derivatives of the advected quantity are the same for the three flows once normalized by the average size of fluctuations the single point statistics is surprisingly robust with respect to the nature of the nonlinearity

generalized mhd equations abstract solutions of the d dimensional generalized mhd gmhd equations t u u u p b b u t b u b b u b are studied in this paper we pay special attention to the impact of the parameters and on the regularity of solutions our investigation is divided into three major cases 1 0 and 0 2 0 and 0 and 3 0 and 0 when 0 and 0 the gmhd equations with any 0 and 0 possess a global weak solution corresponding to any l2 initial data furthermore weak solutions associated with 1 2 d 4 and 1 2 d 4 are actually global classical solutions when their initial data are sufficiently smooth as a special consequence smooth solutions of the 3d gmhd equations with 5 4 and 5 4 do not develop finite time singularities the study of the gmhd equations with 0 and 0 is motivated by their potential applications in magnetic reconnection a local existence result of classical solutions and several global regularity conditions are established for this case these conditions are imposed on either the vorticity u or the current density j b but not both and are weaker than some of current existing ones when 0 and 0 the gmhd equations reduce to the ideal mhd equations it is shown here that the ideal mhd equations admit a unique local solution when the prescribed initial data is in a holder space cr with r 1

on a distributed control problem arising in dynamic optimization of a fixed size population the practical motivation for this paper is provided by the recruitment problem faced by many organizations of fixed size to keep the average age young and thus keep innovation and productivity high while at the same time keeping levels of recruitment high a typical example is an academy of sciences the problem is formalized by an infinite horizon optimal control model for a first order pde with nonlocal dynamics a mckendrick type equation based on the nonstandard necessary optimality condition proved in the paper the following results are established i stationarity of the optimal recruitment density ii strong ergodicity of the optimal solution iii principle of bipolar recruitment in the case where the productivity of the organization is measured by the average age of the members the analysis involves a new type of transversality condition for the costate system the stability with respect to perturbations of the optimal solution of a noncoercive bang bang type problem boundedness and stability of the solution of a specific volterra integral equation of the second kind

singularities of euler flow not out of the blue does three dimensional incompressible euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time this blowup problem is still open after briefly reviewing what is known and pointing out some of the difficulties we propose to tackle this issue for the class of flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations we present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup

nonlinear transfer and spectral distribution of energy in α turbulence abstract two dimensional turbulence governed by the so called turbulence equations which include the surface quasi geostrophic equation 1 the navier stokes system 2 and the governing equation for a shallow flow on a rotating domain driven by a uniform internal heating 3 is studied here in both the unbounded and doubly periodic domains this family of equations conserves two inviscid invariants energy and enstrophy in the navier stokes case the dynamics of which are believed to undergo a dual cascade it is shown that an inverse cascade can exist in the absence of a direct cascade and that the latter is possible only when the inverse transfer rate of the inverse cascading quantity approaches its own injection rate constraints on the spectral exponents in the wavenumber ranges lower and higher than the injection range are derived for navier stokes turbulence with moderate reynolds numbers the realization of an inverse energy cascade in the complete absence of a direct enstrophy cascade is confirmed by numerical simulations

3d euler equations and ideal mhd mapped to regular systems probing the finite time blowup hypothesis abstract we prove by an explicit construction that solutions to incompressible 3d euler equations defined in the periodic cube 0 l 3 can be mapped bijectively to a new system of equations whose solutions are globally regular we establish that the usual beale kato majda criterion for finite time singularity or blowup of a solution to the 3d euler system is equivalent to a condition on the corresponding regular solution of the new system in the hypothetical case of euler finite time singularity we provide an explicit formula for the blowup time in terms of the regular solution of the new system the new system is amenable to being integrated numerically using similar methods as in euler equations we propose a method to simulate numerically the new regular system and describe how to use this to draw robust and reliable conclusions on the finite time singularity problem of euler equations based on the conservation of quantities directly related to energy and circulation the method of mapping to a regular system can be extended to any fluid equation that admits a beale kato majda type of theorem e g 3d navier stokes 2d and 3d magnetohydrodynamics and 1d inviscid burgers we discuss briefly the case of 2d ideal magnetohydrodynamics in order to illustrate the usefulness of the mapping we provide a thorough comparison of the analytical solution versus the numerical solution in the case of 1d inviscid burgers equation

on tangential transversality abstract this is the first of two closely related papers on transversality here we introduce the notion of tangential transversality of two closed subsets of a banach space it is an intermediate property between transversality and subtransversality using it we obtain a variety of known results and some new ones in a unified way our proofs do not use variational principles and we concentrate mainly on tangential conditions in the primal space

optimal birth control of age dependent competitive species abstract this work is concerned with optimal policies for two age structured biological populations in a competing system which is controlled by fertilities the maximum principles for problems with free terminal infinite horizon and target sets are obtained respectively via dubovitskii milyutin s general extremal theory

neumann boundary control of hyperbolic equations with pointwise state constraints we consider optimal control problems for hyperbolic equations with controls in neumann boundary conditions with pointwise constraints on the control and state functions focusing on the multidimensional wave equation with a nonlinear term we derive new necessary optimality conditions in the form of a pointwise pontryagin maximum principle for the state constrained problem under consideration our approach is based on modern methods of variational analysis that allow us to obtain refined necessary optimality conditions with no convexity assumptions on integrands in the minimizing cost functional

on the generalized self similar singularities for the euler and the navier stokes equations in this paper we study blow up rates and the blow up profiles of possible asymptotically self similar singularities of the euler and the navier stokes equations where the sense of convergence and self similarity are considered in various generalized senses we improve substantially in particular the previous nonexistence results of self similar asymptotically self similar singularities generalization of the self similar transforms is also considered and by appropriate choice of the parameterized transform we obtain new a priori estimates for the euler and the navier stokes equations depending on a free parameter

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

filtered viscous fluid equations abstract we analyze an eulerian lagrangian description of filtered incompressible fluid equations a viscous cauchy formula is derived for such equations the models discussed are related to vortex methods and to filtered kuzmin oseledets equations

space time localization of a class of geometric criteria for preventing blow up in the 3d nse a class of local in the space time conditions on the vorticity directions implying local regularity of weak solutions to the 3d navier stokes equations is established in all the preceding results the relevant geometric conditions although being local in nature have been assumed uniformly throughout the spatial regions of high vorticity magnitude and uniformly in time in addition similar results are obtained assuming a less restrictive integral condition on the vorticity directions

optimal birth control of population dynamics abstract we study optimal birth control policies for an age structured population of mckendrick type which is a distributed parameter system involving first order partial differential equations with nonlocal bilinear boundary control the functional analytic approach of dubovitskii and milyutin is adopted in the investigation maximum principles for problems with a free end condition a fixed final horizon are developed and the time optimal control problem the problem with target sets and the infinite planning horizon case are investigated

an exit probability approach to solving high dimensional dirichlet problems we present an approach to solving high dimensional dirichlet problems in bounded domains which is based on a variant of the feynman kac formula that connects solutions to elliptic partial differential equations and functional integration we integrate the resulting system of stochastic differential equations numerically with the euler scheme to correct for possible intermediate excursions of the simulated paths and to find good approximations for first exit times we extend to higher dimensions a strategy which we introduced in f m buchmann j comput phys 202 2005 pp 446 462 for stopping problems in one dimension in addition for dirichlet problems good approximations of the first exit points are needed to evaluate the boundary condition to this end we sample a random point on the tangent hyperplane of the boundary of the domain once a first exit is estimated to detect excursions and adequately compute first exit times we apply the same local half space approximation we therefore assume that the domain is smooth enough such that these approximations are possible numerical experiments in dimensions up to 128 show that resulting approximations are of very high quality in particular we observe that first order convergence behavior of the euler scheme can be maintained

scaling exponents for active scalars we provide bounds for dirichlet quotients and for generalized structure functions for 3d active scalars and navier stokes equations these bounds put constraints on the possible extent of anomalous scaling

nonlinear stability of circular vortex patches this paper proves that circular vortex patches in the plane are stable for the nonlinear dynamical system generated by the euler equations of incompressible fluids this is achieved by establishing a relative variational principle in terms of either energy or angular momentum thus we exploit and extend arnold s idea in 1965 1969 to a nonsmooth setting as well

on the deformations of the incompressible euler equations we consider systems of deformed system of equations which are obtained by some transformations from the system of incompressible euler equations these have similar properties to the original euler equations including the scaling invariance for one form of deformed system we prove that finite time blow up actually occurs for generic initial data while for the other form of the deformed system we prove the global in time regularity for smooth initial data moreover using the explicit functional relations between the solutions of those deformed systems and that of the original euler system we derive the condition of finite time blow up of the euler system in terms of solutions of one of its deformed systems as another application of those relations we deduce a lower estimate of the possible blow up time of the 3d euler equations

computing the principal eigenelements of some linear operators using a branching monte carlo method in earlier work we developed a monte carlo method to compute the principal eigenvalue of linear operators which was based on the simulation of exit times in this paper we generalize this approach by showing how to use a branching method to improve the efficacy of simulating large exit times for the purpose of computing eigenvalues furthermore we show that this new method provides a natural estimation of the first eigenfunction of the adjoint operator numerical examples of this method are given for the laplace operator and an homogeneous neutron transport operator

remarques sur l instabilite du probleme des poches de tourbillon resume nous etudions le probleme des poches de tourbillon pour des contours initiaux poches du cercle unite une approximation quadratique de l equation d evolution se revele instable en temps fini la norme lipschitz du contour devenant arbitrairement grande

simulating diffusions with piecewise constant coefficients using a kinetic approximation using a kinetic approximation of a linear diffusion operator we propose an algorithm that allows one to deal with the simulation of a multi dimensional stochastic process in a media which is locally isotropic except on some surface where the diffusion coefficient presents some discontinuities numerical examples are given in dimensions one to three on pdes or stochastic pdes with or without source terms

alternatives to the median absolute deviation abstract in robust estimation one frequently needs an initial or auxiliary estimate of scale for this one usually takes the median absolute deviation mad n 1 4826 med xi med j x j because it has a simple explicit formula needs little computation time and is very robust as witnessed by its bounded influence function and its 50 breakdown point but there is still room for improvement in two areas the fact that mad n is aimed at symmetric distributions and its low 37 gaussian efficiency in this article we set out to construct explicit and 50 breakdown scale estimators that are more efficient we consider the estimator sn 1 1926 med med j xi xj and the estimator qn given by the 25 quantile of the distances xi x j i j note that sn and qn do not need any location estimate both sn and qn can be computed using o n log n time and o n storage the gaussian efficiency of sn is 58 whereas qn attains 82 we study sn and qn by means of their influence functions their b

numerical study of comparison of vorticity and passive vectors in turbulence and inviscid flows the nonlinear vortex stretching in incompressible navier stokes turbulence is compared with a linear stretching process of passive vectors pvs in particular we pay special attention to the difference of these processes under long and short time evolutions for finite time evolution we confirm our previous finding that the stretching effect of vorticity is weaker than that of general passive vectors for a majority of the initial conditions with the same energy spectra the above difference can be explained qualitatively by examining the biot savart formula in order to see to what extent infinitesimal time development explains the above difference we examine the probability density functions pdfs of the stretching rates of the passive vectors in the vicinity of a solution of navier stokes equations it is found that the pdfs are found to have a gaussian distribution suggesting that there are equally many pvs that stretched less and more than the vorticity this suggests the importance of the vorticity strain correlation built up over finite time in turbulence we also discuss the case of euler equations where the dynamics of the jacobian matrix relating the physical and material coordinates is examined numerically a kind of alignment problem associated with the cauchy green tensor is proposed and studied using the results of numerical simulations it is found that vorticity tends to align itself with the most compressing eigenvector of the cauchy green tensor a two dimensional counterpart of active passive comparison is briefly studied there is no essential difference between stretching of vorticity gradients and that of passive scalar gradients and a physical interpretation is given to it

optimal control of heterogeneous systems basic theory a general model of a heterogeneous control system is introduced in the form of a first order distributed system with nonlocal dynamics and exogenous side conditions the heterogeneity is represented by a parameter taking values in an abstract measurable space so that both continuous and discrete heterogeneity as well as probabilistic heterogeneity without density are included a distributed and a lumped controls are involved the latter appearing also in the side conditions an existence theorem is proved for the uncontrolled system and the sensitivity of the solution with respect to the control variables is estimated the main result is an optimality condition in the form of the pontryagin local maximum principle a global maximum principle holds for the distributed control under an additional condition that rules out discrete measurable heterogeneity spaces a number of possible applications are outlined age structured systems size structured systems nonlocal advection reaction equations static parametric heterogeneity in epidemiology and two stage control systems with uncertain switching time

an extension of the maximum principle for a class of optimal control problems in infinite dimensional spaces an extension of the pontryagin maximum principle and of the transversality conditions for a class of optimal control problems for a system of a parabolic equation and an ordinary differential equation in a hilbert space are given in particular the time optimal problem for some of these systems is considered as an application the optimal control of the diffusion of a class of epidemics is studied

on the euler equations of incompressible fluids euler equations of incompressible fluids use and enrich many branches of mathematics from integrable systems to geometric analysis they present important open physical and mathematical problems examples include the stable statistical behavior of ill posed free surface problems such as the rayleightaylor and kelvin helmholtz instabilities the paper describes some of the open problems related to the incompressible euler equations with emphasis on the blowup problem the inviscid limit and anomalous dissipation some of the recent results on the quasigeostrophic model are also mentioned

contour dynamics for the euler equations in two dimensions we present a contour dynamics algorithm for the euler equations of fluid dynamics in two dimensions this is applied to regions of piecewise constant vorticity withinfinite area vortexregions favrs essentially this reduces the dimensionality by one and we are treating the interaction of closed polygonal contours whose nodes are advected by the total fluid motion computed self consistently a leapfrog centered scheme is used for temporal advancement computer simulation results are given for two and four like signed interacting favrs in all cases wavelike surface deformations are observed if the distance between favrs is comparable to their extent diameter these surface deformations are large they play an essential role in the observed coalescence of favrs

new monte carlo schemes for simulating diffusions in discontinuous media we introduce new monte carlo simulation schemes for diffusions in a discontinuous media divided in subdomains with piecewise constant diffusivity these schemes are higher order extensions of the usual schemes and take into account the two dimensional aspects of the diffusion at the interface between subdomains this is achieved using either stochastic process techniques or an approach based on finite differences numerical tests on elliptic parabolic and eigenvalue problems involving an operator in divergence form show the efficiency of these new schemes

necessary second order conditions for a weak local minimum in a problem with endpoint and control constraints abstract one of the first steps towards necessary second order optimality conditions in problems with constraints was taken by dubovitskii and milyutin in 1965 they offered a scheme that was very effective in smooth optimization problems but seemed to be not suitable for applications in problems with pointwise control constraints in this article we consider a modification of the dubovitskii milyutin scheme which allows to derive necessary second order conditions for a weak local minimum in optimal control problems with a finite number of endpoint constraints of equality and inequality type and with pointwise control constraints of inequality type given by smooth functions assuming that the gradients of active control constraints are linearly independent we provide rather straightforward proof of these conditions for a measurable and essentially bounded optimal control

hamilton jacobi equations arising from boundary control problems with state constraints the analysis of a class of infinite dimensional hamilton jacobi bellman hjb equations is undertaken related to linear convex boundary control problems for pdes with constraints on the state a definition of generalized solution namely weak solution of the hjb equation is provided weak is the limit of strong while strong is the limit of classical as defined in s faggian appl math optim 51 2005 pp 123 162 consequently an existence and uniqueness result is provided for weak solutions of the hjb equation as well as existence of an optimal control being the limit of optimal controls of approximating problems the study then describes an economic application to optimal investment with vintage capital having positivity constraints on the capital

multidimensional exponential timestepping with boundary test exponential timestepping algorithms are efficient for exit time problems because a boundary test can be performed at the end of each timestep giving high order convergence in numerical evaluation of mean exit times successive time increments are independent random variables with an exponential distribution we show how to perform exact timestepping for brownian motion in more than one dimension and consider hitting times of curved surfaces taking as examples the exit times from a circle from a ball from an ellipse and from the region bounded by the two lines of a hyperbola we report the results of numerical experiments performed with the algorithms developed in this work

the fefferman stein decomposition for the constantin lax majda equation regularity criteria for inviscid fluid dynamics revisited the celebrated beale kato majda bkm criterion for the 3d euler equations has been updated by kozono and taniuchi by replacing the supremum with the bounded mean oscillation norm we consider this generalized criterion in an attempt to understand it more intuitively by giving an alternative explanation for simplicity we first treat the constantin lax majda clm equation t h for the vorticity in one dimension and identify a mechanism underlying the update of such an estimate we consider a fefferman stein fs decomposition for the initial vorticity 0 h 1 and how it propagates under the dynamics of the clm equation in particular we obtain a set of dynamical equations for it which reads in its simplest case 0 t 0h 0 1h 1 and 1 t 0h 1 1h 0 the equation for the second component 1 responsible for a possible logarithmic blow up is linear and homogeneous hence it remains zero if it is so initially until a stronger blow up takes place this rules out a logarithmic blo

on the conserved quantities for the weak solutions of the euler equations and the quasi geostrophic equations in this paper we obtain sufficient conditions on the regularity of the weak solutions to guarantee conservation of the energy and the helicity for the incompressible euler equations the regularity of the weak solutions are measured in terms of the triebel lizorkin type of norms dot mathcal f s p q and the besov norms dot mathcal b s p q in particular in the besov space case our results refine the previous ones due to constantin e titi energy and the author of this paper helicity where the regularity is measured by a special class of the besov space norm dot mathcal b s p infty dot mathcal n s p which is the nikolskii space we also obtain a sufficient regularity condition for the conservation of the lp norm of the temperature function in the weak solutions of the quasi geostrophic equation

a pontryagin maximum principle for infinite dimensional problems a basic idea of the classical approach for obtaining necessary optimality conditions in optimal control is to construct suitable needle like control variations we use this idea to prove the main result of the present paper a pontryagin maximum principle for infinite dimensional optimal control problems with pointwise terminal constraints in arbitrary banach state space by refining the classical variational technique we are able to replace the differentiability of the norm of the state space guaranteed by the strict convexity of its dual norm which is assumed in the known results by a separation argument we also drop another key assumption which is common in the existing literature on infinite dimensional control problems that the set of variations in the state space of the state trajectory s endpoint resulting from the control variations be finite codimensional instead we require only that it has nonempty interior in its closed affine hull as an application of the abstract result we present an illustrative example an optimal control problem for an age structured system with pointwise terminal state constraints

on the spectral dynamics of the deformation tensor and new a priori estimates for the 3d euler equations in this paper we study the dynamics of eigenvalues of the deformation tensor for solutions of the 3d incompressible euler equations using the evolution equation of the l2 norm of spectra we deduce new a priori estimates of the l2 norm of vorticity as an immediate corollary of the estimate we obtain a new sufficient condition of l2 norm control of vorticity we also obtain decay in time estimates of the ratios of the eigenvalues in the remarks we discuss what these estimates suggest in the study of searching initial data leading to possible finite time singularities we find that the dynamical behaviors of l2 norm of vorticity are controlled completely by the second largest eigenvalue of the deformation tensor

uniformly local l p estimate for 2 d vorticity equation and its application to euler equations with initial vorticity in bmo we shall prove the global existence theorem for the 2 dimensional euler equations in open image in new window with the initial vorticity in bmo containing functions which do not decay at infinity and have logarithmic singularities

global regularity for vortex patches we present a proof of chemin s 4 result which states that the boundary of a vortex patch remains smooth for all time if it is initially smooth

optimal birth control of age dependent competitive species ii free horizon problems we study optimal birth policies for two age dependent populations in a competing system which is controlled by fertilities new results on problems with free final time and integral phase constraints are presented and the approximate controllability of system is discussed

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

on the evolution of sharp fronts for the quasi geostrophic equation we consider the problem of the evolution of sharp fronts for the surface quasi geostrophic qg equation this problem is the analogue to the vortex patch problem for the two dimensional euler equation the special interest of the quasi geostrophic equation lies in its strong similarities with the three dimensional euler equation while being a two dimen sional model in particular an analogue of the problem considered here the evolution of sharp fronts for qg is the evolution of a vortex line for the three dimensional euler equation the rigorous derivation of an equation for the evolution of a vortex line is still an open problem the influence of the singularity appearing in the velocity when using the biot savart law still needs to be understood we present two derivations for the evolution of a periodic sharp front the first one heuristic shows the presence of a logarithmic singularity in the velocity while the second making use of weak solutions obtains a rigorous equation for the evolution explaining the influence of that term in the evolution of the curve finally using a nash moser argument as the main tool we obtain local existence and uniqueness of a solution for the derived equation in the c case 2004 wiley periodicals inc

a general model of size dependent population dynamics with nonlinear growth rate we study a general model of size dependent population dynamics with nonlinear growth rate the existence of a local solution is shown by using schauder s fixed point theorem uniqueness and continuous dependence on initial data are also established under additional conditions

a quaternionic structure in the three dimensional euler and ideal magneto hydrodynamics equations by considering the three dimensional incompressible euler equations a 4 vector is constructed out of a combination of scalar and vector products of the vorticity and the vortex stretching vector u s the evolution equation for can then be cast naturally into a quaternionic riccati equation this is easily transformed into a quaternionic zero eigenvalue schrodinger equation whose potential depends on the hessian matrix of the pressure with minor modifications this system can alternatively be written in complex notation an infinite set of solutions of scalar zero eigenvalue schrodinger equations has been found by adler and moser which are discussed in the context of the present problem similarly when the equations for ideal magneto hydrodynamics mhd are written in elsasser variables a pair of 4 vectors are governed by coupled

analytical behavior of two dimensional incompressible flow in porous media in this paper we study the analytic structure of a two dimensional mass balance equation of an incompressible fluid in a porous medium given by darcy s law we obtain local existence and uniqueness by the particle trajectory method and we present different global existence criterions these analytical results with numerical simulations are used to indicate nonformation of singularities moreover blow up results are shown in a class of solutions with infinite energy

pontryagin s principle for state constrained boundary control problems of semilinear parabolic equations this paper deals with state constrained optimal control problems governed by semilinear parabolic equations we establish a minimum principle of pontryagin s type to deal with the state constraints we introduce a penalty problem by using ekeland s principle the key tool for the proof is the use of a special kind of spike perturbations distributed in the domain where the controls are defined conditions for normality of optimality conditions are given

on the global evolution of vortex filaments blobs and small loops in 3d ideal flows we consider a wide class of approximate models of evolution of singular distributions of vorticity in three dimensional incompressible fluids and we show that they have global smooth solutions the proof exploits the existence of suitable hamiltonian functions the approximate models we analyze essentially discrete and continuous vortex filaments and vortex loops are related to some problem of classical physics concerning turbulence and also to the numerical approximation of flows with very high reynolds number finally we apply our strategy to discrete models for filaments used in numerical methods

nonlinear inviscid incompressible dynamics abstract we describe the mechanisms of geometric depletion of nonlinearity present in the three dimensional incompressible euler equations and in the two dimensional active scalar models

optimal immigration age patterns in populations of fixed size a b s t r a c t due to a low birth rate many countries need immigration to sustain a given population size the resulting model consists of specific non standard age structured control systems on an infinite horizon this paper investigates the question of optimal choice of the immigration age profile within certain bounds the main qualitative result is that under certain conditions the optimal control the immigration age profile is time invariant although a dynamic framework is used this makes it possible to give a detailed description of the optimal immigration policy and to obtain numerical solutions for the austrian case the analysis requires a number of new results obtained in the paper a new type of transversality condition for the adjoint equation involved in the optimality conditions generic conditions for well posedness of the problem and appropriate stability conditions

on depletion of the vortex stretching term in the 3d navier stokes equations certain new cancellation properties in the vortex stretching term are detected leading to new geometric criteria for preventing finite time blow up in the 3d navier stokes equations

navier stokes equations and area of interfaces we present new a priori estimates for the vorticity of solutions of the three dimensional navier stokes equations these estimates imply that thel 1 norm of the vorticity is a priori bounded in time and that the time average of the 4 3 e power of thel 4 3 e spatial norm of the gradient of the vorticity is a priori bounded using these bounds we construct global leray weak solutions of the navier stokes equations which satisfy these inequalities in particular it follows that vortex sheet vortex line and even more general vortex structures with arbitrarily large vortex strengths are initial data which give rise to global weak solutions of this type of the navier stokes equations next we apply these inequalities in conjunction with geometric measure theoretical arguments to study the two dimensional hausdorff measure of level sets of the vorticity magnitude we obtain a priori bounds on an average such measure when expressed in terms of the reynolds number and the kolmogorov dissipation length these bounds are left langle mu right rangle leqq frac l 3 eta left 1 operatorname re 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 right 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 the right hand side of this inequality has a simple geometrical interpretation it represents the area of a union of non overlapping spheres of radii which fill a fraction of the spatial domain as the reynolds number increases this fraction decreases we study also the area of level sets of scalars and in particular isotherms in rayleigh benard convection we define a quantity r t x 0 describing an average value of the area of a portion of a level set contained in a small ball of radiusr about the pointx 0 we obtain the inequality left langle mu right rangle r t left x 0 right leqq c kappa 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 r 5 mathord left vphantom 5 2 right kern nulldelimiterspace 2 left langle v left x 0 right right rangle 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 wherek is the diffusivity constant and is the velocity the inequality is valid forr small but larger than a small scale k 1 in the case of turbulent velocities this scale is smaller than the smallest significant physical scale suggesting that 2 5 is a lower bound for the fractal dimension of an ensemble average interface in turbulent flow a fact which agrees with the experimental lower bound of 2 35 we define a similar quantity t representing an average value of the area of a portion of a level set contained in the region of spaced x d dist x d whered is the domain of aspect ratio of order one and diameterl where the convection takes place we obtain the inequality left langle mu right rangle delta t leqq c left l 3 delta 1 l 5 mathord left vphantom 5 2 right kern nulldelimiterspace 2 delta 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 ra 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 right where ra is the rayleigh number

on blow up criteria of smooth solutions to the 3 d euler equations in a bounded domain abstract we show that a smooth solution of the 3 d euler equations in a bounded domain breaks down if and only if a certain norm of vorticity blows up at the same time here this norm is weaker than bmo norm

numerical approximations in optimal control of a class of heterogeneous systems the paper presents a numerical procedure for solving a class of optimal control problems for heterogeneous systems the latter are described by parameterized systems of ordinary differential equations coupled by integrals along the parameter space such problems arise in economics demography epidemiology management of biological resources etc the numerical procedure includes discretization and a gradient projection method for solving the resulting discrete problem a main point of the paper is the performed error analysis which is based on the property of metric regularity of the system of necessary optimality conditions associated with the considered problem

optimal control of age structured systems with mixed state control constraints abstract the paper deals with a general optimal control problem for age structured systems a necessary optimality condition of pontryagin type is obtained where the novelty is in that mixed control state constraints are present the proof uses an abstract lagrange multiplier theorem and the main difficulty is to obtain regularity of the lagrange multipliers in the particular problem at hand

age structured linear state differential games abstract in this paper we search for conditions on age structured differential games to make their analysis more tractable we focus on a class of age structured differential games which show the features of ordinary linear state differential games and we prove that their open loop nash equilibria are sub game perfect by means of a simple age structured advertising problem we provide an application of the theoretical results presented in the paper and we show how to determine an open loop nash equilibrium

necessary conditions for a weak minimum in optimal control problems with integral equations subject to state and mixed constraints the first order necessary optimality conditions for a weak minimum are derived for optimal control problems with volterra type integral equations considered on a fixed time interval subject to endpoint constraints of equality and inequality type mixed state control constraints of inequality and equality type and pure state constraints of inequality type the main assumption is the uniform linear positive independence of the gradients of active mixed constraints with respect to the control the conditions obtained generalize the euler lagrange equation as a stationarity condition for the lagrange problem in the classical calculus of variations with ordinary differential equations

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

optimal harvesting for periodic age dependent population dynamics here we investigate the optimal harvesting problem for some periodic age dependent population dynamics namely we consider the linear lotka mckendrick model with periodic vital rates and a periodic forcing term that sustains oscillations existence and uniqueness of a positive periodic solution are demonstrated and the existence and uniqueness of the optimal control are established we also state necessary optimality conditions a numerical algorithm is developed to approximate the optimal control and the optimal harvest some numerical results are presented

an algebraic reduction of the scaling gap in the navier stokes regularity problem it is shown within a mathematical framework based on the suitably defined scale of sparseness of the super level sets of the positive and negative parts of the vorticity components and in the context of a blow up type argument that the ever resisting scaling gap that is the scaling distance between a regularity criterion and a corresponding a priori bound shortly a measure of the super criticality of the three dimensional navier stokes regularity problem can be reduced by an algebraic factor since independent fundamental works of ladyzhenskaya prodi and serrin as well as kato and fujita in 1960s all the reductions have been logarithmic in nature regardless of the functional set up utilized more precisely it is shown that it is possible to obtain an a priori bound that is algebraically better than the energy level bound while keeping the corresponding regularity criterion at the same level as all the classical regularity criteria the mathematics presented was inspired by morphology of the regions of intense vorticity velocity gradients observed in computational simulations of turbulent flows as well as by the physics of turbulent cascades and turbulent dissipation

growth rate analysis of scalar gradients in generalized surface quasigeostrophic equations of ideal fluids the growth rates of scalar gradients are studied numerically and analytically in a family of two dimensional 2d incompressible fluid equations related to the surface quasigeostrophic sqg equation the active scalar is related to the stream function by 2 0 2 a notable difference is observed in a comparison of the instantaneous growth rates in l p and in l norms depending on the stage of the time evolution the crux is the phase shift effect of singular integral operators which displaces the peak location of the scalar gradient from that of the strain rate on this basis a method of detecting such a dislocation is proposed in view of the importance of their coalescence needed for a possible blow up moreover it is found in the long time evolution that a solution of the sqg equation whose regularity is not known is less singular than that of the 2d euler equations known to be regular on the time interval covered by this computation this consistently expands an earlier observation by majda and tabak physica d 98 515 1996 in some detail a 1d model problem is discussed to illustrate the present method and extensions to the 3d case are also are briefly discussed

on strong tangential transversality abstract this is the second of two closely related papers on transversality here we introduce the notion of strong tangential transversality of two closed subsets of a banach space which is a natural sufficient condition for tangential transversality some properties of uniform tangent sets are obtained a new sum rule for the clarke subdifferential is proven

optimality conditions for age structured control systems abstract we consider a fairly general model extension of the gurtin maccamy model of population dynamics of an age structured control system with nonlocal dynamics and nonlocal boundary conditions a necessary optimality condition is obtained in the form of pontryagin s maximum principle which is applicable to a number of practically meaningful models where the previously known results fail we discuss such models an epidemic control and a capital accumulation model as illustrations

energy spectra of steady two dimensional turbulent flows more than 30 years ago the energy spectrum of twodimensional turbulence was theoretically predicted yet the evidence for this spectrum has remained weak in some cases classical theory 1 3 predicts the energy spectrum based on a cascade of either energy or enstrophy once the relevant physical quantities have been identified simple dimensional analysis leads to the power spectrum as a function of wave number k however the spectra measured in many numerical simulations decay substantially faster with k than predicted 4 the range of wave numbers larger than the inverse system size and up to the dissipative damping is called inertial range in this paper we assume power laws for the inertial range as a working hypothesis since possible deviations from a power behavior cannot be determined from numerical data with confidence a general review of two dimensional turbulence and the slope of the energy spectrum can be found in refs 5 6 following we summarize recent results on the inertial range spectrum

optimal birth control of population dynamics ii problems with free final time phase constraints and mini max costs abstract we study optimal birth control of population systems of mckendrick type which is a distributed parameter system involving first order partial differential equations with nonlocal bilinear boundary control new results on problems with free final time phase constraints and mini max costs are presented

pontryagin s principle for local solutions of control problems with mixed control state constraints this paper deals with optimal control problems of semilinear parabolic equations with pointwise state constraints and coupled integral state control constraints we obtain necessary optimality conditions in the form of a pontryagin s minimum principle for local solutions in the sense of l p p leq infty

geometric statistics in turbulence the author presents results regarding certain average properties of incompressible fluids derived from the equations of motion the author estimates the average dissipation rate the average dimension of level sets the role played by the field of direction of vorticity in the three dimensional euler and navier stokes equations is discussed and a class of two dimensional equations that are useful models of incompressible dynamics is described the author presents results concerning scaling exponents in turbulence

optimal sampled data controls with running inequality state constraints pontryagin maximum principle and bouncing trajectory phenomenon sampled data control systems have steadily been gaining interest for their applications in automatic engineering where they are implemented as digital controllers and recently results have been obtained in optimal control theory for nonlinear sampled data control systems and certain generalizations in this paper we derive a pontryagin maximum principle for general nonlinear finite dimensional optimal sampled data control problems with running inequality state constraints in particular we obtain a nonpositive averaged hamiltonian gradient condition with the adjoint vector being a function of bounded variations our proof is based on the ekeland variational principle in general optimal control problems with running inequality state constraints are difficult to solve using numerical methods due to the discontinuities the jumps and the singular part of the adjoint vector however in our case we find that under certain general hypotheses the adjoint vector only experiences jumps at most at the sampling times and moreover the trajectory only contacts the running inequality state constraints at most at the sampling times we call this behavior a bouncing trajectory phenomenon and it constitutes the second major focus of this paper finally taking advantage of the bouncing trajectory phenomenon we numerically solve three examples with different kinds of constraints and in several dimensions