unnormalized optimal transport abstract we propose an extension of the computational fluid mechanics approach to the monge kantorovich mass transfer problem which was developed by benamou brenier in 4 our extension allows optimal transfer of unnormalized and unequal masses we obtain a one parameter family of simple modifications of the formulation in 4 this leads us to a new monge ampere type equation and a new kantorovich duality formula these can be solved efficiently by for example the chambolle pock primal dual algorithm 6 this solution to the extended mass transfer problem gives us a simple metric for computing the distance between two unnormalized densities the l 1 version of this metric was shown in 25 which is a precursor of our work here to have desirable properties

strongly order preserving semiflows generated by functional differential equations the aim of this paper is to apply some recently developed improvements in the theory of monotone dynamical systems due to the authors 12 13 j to certain systems of functional differential equations fde s which do not enjoy the quasimonotone property considered by one of us in 9 the present work extends our previous results in this direction for scalar equations in 111 to systems in the fundamental paper s m w hirsch establishes that most orbits of a strongly monotone scmiflow on a strongly ordered space x tend to the set e of equilibria employing ideas of hirsch and of h matano 7 the authors have extended the theory to obtain some improvements in the results of hirsch and matano under matano s weaker assumption that the semiflow is strongly order preserving recently in 13 we gave sufficient conditions for most orbits of a strongly order preserving semiflow to converge to an equilibrium these conditions require additional smoothness of the semiflow and a certain strong monotonicity condition for the linearized flow determined by the variational equation about each equilibrium p policik s has also given sufficient conditions for most orbits to converge for semilinear parabolic evolution equations possessing certain strong monotonicity properties in this paper wc apply the convergence results developed in 131 to nonquasimonotone fde s in this introduction wc briefly review the application of monotone dynamical systems theory to fde s enjoying the quasimonotone property this research appeared in 9 133 then we state one of the main results of this paper which applies when the quasimonotone property fails but when another property m holds some simple examples are given which

optimal mass transportation in the heisenberg group in this paper we consider the problem of optimal transportation of absolutely continuous masses in the heisenberg group hn in the case when the cost function is either the square of the carnot caratheodory distance or the square of the koranyi norm in both cases we show existence and uniqueness of an optimal transport map in the former case the proof requires a delicate analysis of minimizing geodesics of the group and of the differentiability properties of the squared distance function in the latter case the proof requires some fine properties of bv functions in the heisenberg group

an elementary proof of the polar factorization of vector valued functions we present an elementary proof of an important result of y brenier br1 br2 namely that vector fields in d satisfying a nondegeneracy condition admit the polar factorization u x s x where is a convex function and s is a measure preserving mapping brenier solves a minimization problem using monge kantorovich theory whereas we turn our attention to a dual problem whose euler lagrange equation turns out to be

on the blow up rate and the blow up set of breaking waves for a shallow water equation we consider the problem of the development of singularities for classical solutions to a new periodic shallow water equation blow up can occur only in the form of wave breaking i e the solution remains bounded but its slope becomes unbounded in finite time a quite detailed description of the wave breaking phenomenon is given there is at least a point in general depending on time where the slope becomes infinite exactly at breaking time the precise blow up rate is established and for a large class of initial data we also determine the blow up set

an efficient numerical method for the solution of the l_2 optimal mass transfer problem in this paper we present a new computationally efficient numerical scheme for the minimizing flow approach for the computation of the optimal l 2 mass transport mapping in contrast to the integration of a time dependent partial differential equation proposed in s angenent s haker and a tannenbaum siam j math anal 35 2003 pp 61 97 we employ in the present work a direct variational method the efficacy of the approach is demonstrated on both real and synthetic data

local in space criteria for blowup in shallow water and dispersive rod equations we unify a few of the best known results on wave breaking for the camassa holm equation by r camassa a constantin j escher l holm j hyman and others in a single theorem a sufficient condition for the breakdown is that u 0 u 0 is strictly negative in at least one point x 0 in mathbb r such blowup criterion looks more natural than the previous ones as the condition on the initial data is purely local in the space variable our method relies on the introduction of two families of lyapunov functions contrary to mckean s necessary and sufficient condition for blowup our approach applies to other equations that are not integrable we illustrate this fact by establishing new local in space blowup criteria for an equation modeling nonlinear dispersive waves in elastic rods

polar factorization and pseudo rearrangements applications to polya szego type inequalities abstract we are interested in the polar factorization of a function f defined in an open bounded set r n it is well known that there exists a measure preserving map such that f f where f is the decreasing rearrangement of f we prove that under suitable assumptions besides the classical polar factorization of f we have f f u where f u is a pseudo rearrangement of f with respect to the measurable function u and is the measure preserving map such that u u as an application we characterize those functions that realize equality in the polya szego inequality

robust stability and stabilizing controller design of fuzzy systems with discrete and distributed delays in this paper we consider delay dependent stability conditions of takagi sugeno fuzzy systems with discrete and distributed delays although many kinds of stability conditions for fuzzy systems with discrete delays have already been obtained almost no stability condition for fuzzy systems with distributed delays has appeared in the literature this is also true in case of the robust stability for uncertain fuzzy systems with distributed delays here we employ a generalized lyapunov functional to obtain delay dependent stability conditions of fuzzy systems with discrete and distributed delays we introduce some free weighting matrices to such a lyapunov functional in order to reduce the conservatism in stability conditions these techniques lead to generalized and less conservative stability conditions we also consider the robust stability of fuzzy time delay systems with uncertain parameters applying the same techniques made on the stability conditions we obtain delay dependent sufficient conditions for the robust stability of uncertain fuzzy systems with discrete and distributed delays moreover we consider the state feedback stabilization based on stability and robust stability conditions we obtain conditions for the state feedback controller to stabilize the fuzzy time delay systems finally we give two examples to illustrate our results delay dependent stability conditions obtained here are shown to guarantee a wide stability region

tri hamiltonian duality between solitons and solitary wave solutions having compact support a simple scaling argument shows that most integrable evolutionary systems which are known to admit a bi hamiltonian structure are in fact governed by a compatible trio of hamiltonian structures we demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons solitary wave solutions having compact support or cusped and or peaked solitons a general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified korteweg de vries equation the nonlinear schr odinger equation the integrable boussinesq system used to model the two way propagation of shallow water waves and the ito system of coupled nonlinear wave equations these hierarchies include a remarkable variety of interesting integrable nonlinear differential equations textcopyright 1996 the american physical society

a fast approach to optimal transport the back and forth method we present a method to efficiently solve the optimal transportation problem for a general class of strictly convex costs given two probability measures supported on a discrete grid with n points we compute the optimal transport map between them in o n log n operations and o n storage space our approach allows us to solve optimal transportation problems on spatial grids as large as 4096 times 4096 and 384 times 384 times 384 in a matter of minutes

on a class of physically important integrable equations abstract a methodology introduced by fuchssteiner and the author is used to derive a class of physically important integrable evolution equations among these equations are integrable generalizations of the korteweg devries kdv of the modified kdv of the nonlinear schrodinger nls and of the sine gordon equations the modeling of water waves as well as general asymptotic considerations are used to illustrate the occurrence of the generalized modified kdv and nls equations respectively

fast slow diffusion and collapsing sandpiles we regard the limit as p of the flow governed by the p laplacian as providing a simplistic model for the collapse of an initially unstable sandpile upon rescaling to stretch out the initial layer we obtain some simple dynamics and provide fairly explicit solutions in certain cases involving nonlocal geometric curvature motion in particular we note that such models entail instantaneous mass transfer governed by monge kantorovich theory

a variational approach to the macroscopic electrodynamics of anisotropic hard superconductors we consider bean s critical state model for anisotropic superconductors a variational problem solved by the quasi static evolution of the internal magnetic field is obtained as the limit of functionals arising from maxwell s equations combined with a power law for the dissipation moreover the quasi static approximation of the internal electric field is recovered using a first order necessary condition if the sample is a long cylinder subjected to an axial uniform external field the macroscopic electrodynamics is explicitly determined

on the singularity formation for a class of periodic higher order camassa holm equations abstract the primary goal of this paper is to understand how higher order nonlinearities affect the dispersive dynamics as a prototype we study a class of higher order camassa holm equations which can be viewed as a generalization of the camassa holm equation we develop a delicate analysis to investigate the formation of singularities and provide sufficient conditions on initial data that lead to the finite time blow up of the second order derivative of the solution

long time stability for solutions of a β plane equation we prove stability for arbitrarily long times of the zero solution for the so called plane equation which describes the motion of a two dimensional inviscid ideal fluid under the influence of the coriolis effect the coriolis force introduces a linear dispersive operator into the two dimensional incompressible euler equations thus making this problem amenable to an analysis from the point of view of nonlinear dispersive equations the dispersive operator l1 1 2 exhibits good decay but has numerous unfavorable properties chief among which are its anisotropy and its behavior at small frequencies 2016 wiley periodicals inc

evolution of microstructure in unstable porous media flow a relaxational approach we study the flow of two immiscible fluids of different density and mobility in a porous medium if the heavier phase lies above the lighter one the interface is observed to be unstable the two phases start to mix on a mesoscopic scale and the mixing zone grows in time an example of evolution of microstructure a simple set of assumptions on the physics of this two phase flow in a porous medium leads to a mathematically ill posed problem when used to establish a continuum free boundary problem we propose and motivate a relaxation of this nonconvex constraint of a phase distribution with a sharp interface on a macroscopic scale we prove that this approach leads to a mathematically well posed problem that predicts shape and evolution of the mixing profile as a function of the density difference and mobility quotient 1999 john wiley sons inc

monotone semiflows in scalar non quasi monotone functional differential equations abstract by introducing a stronger than pointwise ordering conditions are found under which scalar functional differential equations generate monotone semiflows even if they are not quasi monotone typically the maximum delay must be the smaller the more quasi monotonicity is violated the theory of monotone semiflows is used to show that most solutions converge to equilibrium and that stability of equilibria is essentially the same as for ordinary differential equations

on permanent and breaking waves in hyperelastic rods and rings we prove that the only global strong solution of the periodic rod equation vanishing in at least one point t0 x0 r s1 is the identically zero solution such conclusion holds provided the physical parameter of the model related to the finger deformation tensor is outside some neighborhood of the origin and applies in particular for the camassa holm equation corresponding to 1 we also establish the analogue of this unique continuation result in the case of non periodic solutions defined on the whole real line with vanishing boundary conditions at infinity our analysis relies on the application of new local in space blowup criteria and involves the computation of several best constants in convolution estimates and weighted poincare inequalities

analysis on the blow up of solutions to a class of integrable peakon equations abstract we investigate the blow up mechanism of solutions to a class of quasilinear integrable equations which could possess peakons the dynamics of the blow up quantity along the characteristics is established by the riccati type differential inequality which involves the interaction among three parts a local nonlinearity a nonlocal term and a term stemming from the weak linear dispersion to analyse the interplay among these quantities we provide two different approaches the first one is designed for the case when the equations do not exhibit a weak linear dispersion and hence focuses on the interplay between the first two parts the method is based on a refined analysis on either evolution of the solution u and its gradient u x that is c u u x or the growth rate of the relative ratio u x u the second one handles the general situation when all of three parts are present the idea is to extract the truly blow up component from the riccati type differential inequality and utilizes the morawetz type identity or higher order conservation laws to show that such a component blows up in finite time before the other component degenerates

analysis of a model representing stage structured population growth with state dependent time delay a stage structured model of population growth is proposed where the time to ma turity is itself state dependent it is shown that under appropriate assumptions all solutions are positive and bounded criteria for the existence of positive equilibria and further conditions for the uniqueness of the equilibria are given the stability of the equilibria are also discussed in addition an attracting region is determined for solutions such that this region collapses to the unique positive equilibrium in the state independent case

advection modes by optimal mass transfer classical model reduction techniques approximate the solution of a physical model by a limited number of global modes these modes are usually determined by variants of principal component analysis global modes can lead to reduced models that perform well in terms of stability and accuracy however when the physics of the model is mainly characterized by advection the non local representation of the solution by global modes essentially reduces to a fourier expansion in this paper we describe a method to determine a low order representation of advection this method is based on the solution of monge kantorovich mass transfer problems examples of application to point vortex scattering korteweg de vries equation von k arm an wake and hurricane dean advection are discussed

a lagrangian scheme for the solution of the optimal mass transfer problem a lagrangian method to numerically solve the l 2 optimal mass transfer problem is presented the initial and final density distributions are approximated by finite mass particles having a gaussian kernel mass conservation and the hamilton jacobi equation for the potential are identically satisfied by constant mass transport along straight lines the scheme is described in the context of existing methods to solve the problem and a set of numerical examples including applications to medical imagery are presented

generalizations of the camassa holm equation we classify generalized camassa holm type equations which possess infinite hierarchies of higher symmetries we show that the obtained equations can be treated as negative flows of integrable quasi linear scalar evolution equations of orders 2 3 and 5 we present the corresponding lax representations or linearization transformations for these equations some of the obtained equations seem to be new

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

oscillation induced blow up to the modified camassa holm equation with linear dispersion abstract in this paper we provide a blow up mechanism to the modified camassa holm equation with varying linear dispersion we first consider the case when linear dispersion is absent and derive a finite time blow up result with an initial data having a region of mild oscillation a key feature of the analysis is the development of the burgers type inequalities with focusing property on characteristics which can be deduced from tracing the ratio between solution and its gradient using the continuity and monotonicity of the solutions we then extend this blow up criterion to the case of negative linear dispersion and determine that the finite time blow up can still occur if the initial momentum density is bounded below by the magnitude of the linear dispersion and the initial datum has a local mild oscillation region finally we demonstrate that in the case of non negative linear dispersion the formation of singularities can be induced by an initial datum with a sufficiently steep profile in contrast to the camassa holm equation with linear dispersion the effect of linear dispersion of the modified camassa holm equation on the blow up phenomena is rather delicate

wave breaking and peakons for a modified camassa holm equation in this paper we investigate the formation of singularities and the existence of peaked traveling wave solutions for a modified camassa holm equation with cubic nonlinearity the equation is known to be integrable and is shown to admit a single peaked soliton and multi peakon solutions of a different character than those of the camassa holm equation singularities of the solutions can occur only in the form of wave breaking and a new wave breaking mechanism for solutions with certain initial profiles is described in detail

data driven optimal transport the problem of optimal transport between two distributions x and y is extended to situations where the distributions are only known through a finite number of samples xi and yj a weak formulation is proposed based on the dual of the kantorovich formulation with two main modifications replacing the expected values in the objective function by their empirical means over the xi and yj and restricting the dual variables u x and v y to a suitable set of test functions adapted to the local availability of sample points a procedure is proposed and tested for the numerical solution of this problem based on a fluidlike flow in phase space where the sample points play the role of active lagrangian markers 2016 wiley periodicals inc

stage structured models of intra and inter specific competition within age classes abstract in some species larvae and adults experience competition in completely different ways simple stage structured models without larval competition usually yield a single delay equation for the adults using an age structured system incorporating competition among both larvae and adults we derive a system of distributed delay equations for the numbers of larvae and adults the system is neither cooperative nor reducible to a single equation for either variable positivity boundedness and uniform strong persistence are established linear stability analysis of equilibria is difficult due to the strong coupling but results are proved for small delays using monotone systems theory and exponential ordering for small delay we prove a theorem on generic convergence to equilibria which does not directly follow from standard theory but can be proved indirectly using comparison arguments finally we consider an extension to two strain competition and prove theorems on the linear stability of the boundary equilibria

some tricks from the symmetry toolbox for nonlinear equations generalizations of the camassa holm equation abstract the main subject of the paper is to give a survey and to present new methods on how integrability results i e results for symmetry groups inverse scattering formulations action angle transformations and the like can be transferred from one equation to others in case the equations are not related by backlund transformations as a main example the so called camassa holm equation is chosen for which the relevant results are obtained by having a look on the korteweg de vries kdv equation the camassa holm equation turns out to be a different factorization equation of the kdv it describes shallow water waves and reconciles the properties which were known for different orders of shallow water wave approximations we follow here an old method already marginally mentioned in fuchssteiner and fokas 1981 and fuchssteiner 1983 and recently applied by others olver and rosenau 1995 the method allows an immediate recovery of the recursion operator for the camassa holm equation from the invariance structure of the kdv although both equations are not related by backlund transformations however in addition and different from other approaches from there by use of the squared eigenfunction relation for the kdv equation the lax pair formulation for the different factorization equation is derived for the example under consideration it is of course the one obtained in camassa and holm 1993 since the methods proposed here can be transferred to any compatible factorization of recursion operators its application even in the special case which was chosen for illustration leads to a large class of integrable equations among which the camassa holm equation can be found as well as a three parameter family of generalizations of the equation the advantage of the general approach to the lax pair presentation is that direct transformations between action and angle variables are obtained so using this lax pair formulation as a novel result a direct transformation between action and angle variables for the camassa holm equation is derived further novel results in the paper are a hodograph link back from a backlund transformation of the camassa holm equation to a particular member of the kdv hierarchy additional symmetries and the construction of the conformal algebra for the hierarchy of the camassa holm equation the methods involved include hereditary operators bi hamiltonian formulations nilpotent flows and scaling symmetries

a stage structured mathematical model for fish stock with harvesting in this paper we propose a mathematical model for a single species fish stock with a three stage structure juveniles small adults and large adults with two harvesting strategies for mature classes maturity and size selectivities the purpose of the work is to investigate the dynamical behavior of the model and discuss the effect of harvesting we identify the adult reproduction number mathcal r a for the model obtain the local and global stability of the trivial equilibrium when mathcal r a 1 numerical simulations are provided to investigate the influence of harvesting functions discuss the optimal harvesting rates and explore the effect of periodic coefficients on the dynamical system

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

on a system of partial differential equations of monge kantorovich type abstract we consider a system of pdes of monge kantorovich type arising from models in granular matter theory and in electrodynamics of hard superconductors the existence of a solution of such system in a regular open domain r n whose construction is based on an asymmetric minkowski distance from the boundary of was already established in g crasta a malusa the distance function from the boundary in a minkowski space trans amer math soc submitted for publication in this paper we prove that this solution is essentially unique a fundamental tool in our analysis is a new regularity result for an elliptic nonlinear equation in divergence form which is of some interest by itself