master equation for a quantum system driven by a strong periodic field in the quasienergy representation abstract the evolution of a quantum system weakly coupled with a thermal reservoir and influenced by an external periodic field is formulated in the quasienergy representation where the strong external periodic field disappears and the original model is transformed to a time dependent system weakly coupled with thermal reservoir based on this time dependent subdynamics we derive a master equation of the reduced density operator for the driven quantum system this equation is valid in the weak coupling limit our method can be useful for obtaining the master equation of any system coupled with a thermal reservoir and driven by strong external periodic fields

spiral waves in reaction diffusion equations we consider the reaction diffusion system begin gathered r t nabla 2 r r left 1 r 2 vec nabla theta cdot vec nabla theta right hfill r theta t r nabla 2 theta 2 vec nabla r cdot vec nabla theta qr 3 hfill end gathered this system governs the solutions of reaction diffusion systems near a hopf bifurcation point in two spatial dimensions we use formal asymptotic expansions to contruct one armed and multi armed archimedean spiral waves for small values of the parameter q we then show that the one armed spiral waves are probably stable and the multi armed ones are unstable for q small next by numerical continuation methods we construct spiral waves for all q these calculations show that the one armed spiral waves are unstable when q 1 397 cdots we also find the explicit one dimensional analogues of spiral waves for all q and show that they are unstable for q small

dynamics of spiral rings in the three dimensional ginzburg landau equation in this paper we show that the spiral rings of the 3d ginzburg landau equation shrink out and disappear in finite time in contrast with the 2d case in which the interaction between two spirals of opposite charge is exponential we show analytically and numerically that the spiral rings collapse with a power law due to the continuity of the defect line

unbalanced optimal transport dynamic and kantorovich formulations abstract this article presents a new class of distances between arbitrary nonnegative radon measures inspired by optimal transport these distances are defined by two equivalent alternative formulations i a dynamic formulation defining the distance as a geodesic distance over the space of measures ii a static kantorovich formulation where the distance is the minimum of an optimization problem over pairs of couplings describing the transfer transport creation and destruction of mass between two measures both formulations are convex optimization problems and the ability to switch from one to the other depending on the targeted application is a crucial property of our models of particular interest is the wasserstein fisher rao metric recently introduced independently by 7 15 defined initially through a dynamic formulation it belongs to this class of metrics and hence automatically benefits from a static kantorovich formulation

kinetic equation non perturbative approach and decoherence free subspace for quantum open system a schrodinger lionville type of equation for an quantum open system is presented the equation has a correlated part and many master equations can be derived from it as special cases most significantly it can be applied to construct a decoherence free subspace for quantum computing the original schrodinger lionville equation for the total system is related to it by a non unitary similarity transformation which enables us to propose a non perturbative method for solving the eigenvalue problem for the total hamiltonian in addition it also enables one to uncover a simple procedure to treat the eigenvalue problem of an open system under strong interaction the correlated part of the equation is not necessarily self adjoint so that there exists a complex spectrum for the corresponding hamiltonian liouvillian which enables the time evolution of states to be asymmetric this then exposes just the correlation required to produce evolution which coincides with the second law of thermodynamics c 2002 published by elsevier science b v

geometric properties of cones with applications on the hellinger kantorovich space and a new distance on the space of probability measures abstract we study general geometric properties of cone spaces and we apply them on the hellinger kantorovich space m x hk we exploit a two parameter scaling property of the hellinger kantorovich metric hk and we prove the existence of a distance shk on the space of probability measures that turns the hellinger kantorovich space m x hk into a cone space over the space of probabilities measures p x shk we provide a two parameter rescaling of geodesics in m x hk and for p x shk we obtain a full characterization of the geodesics we finally prove finer geometric properties including local angle condition and partial k semiconcavity of the squared distances that will be used in a future paper to prove existence of gradient flows on both spaces

on the time dependent formulation of analytical continuation in non equilibrium statistical mechanics abstract on the basis of the dynamics of correlations approach to the solution of the liouville von neumann equation we develop within the familiar classification of correlations scheme a time dependent formalism which leads naturally to an analytical continuation procedure thereby allowing explicit evaluation of each term in the formal series solution this procedure throws new light on the so called i rule and opens the way to the treatment of the full correlation subdynamics for time dependent phenomena which will be discussed in future publications

free boundaries in optimal transport and monge ampere obstacle problems given compactly supported 0 f g 2 l 1 r n the problem of trans porting a fraction m min kfkl1 kgkl1 of the mass of f onto g as cheaply as possible is considered where cost per unit mass transported is given by a cost function c typically quadratic c x y x y 2 2 this question is shown to be equivalent to a double obstacle problem for the monge ampere equation for which sucient conditions are given to guarantee uniqueness of the solution such as f vanishing on sptg in the quadratic case the part of f to be transported increases monotonically with m and if sptf and sptg are separated by a hyperplane h then this part will separated from the balance of f by a semiconcave lipschitz graph over the hyperplane if f f and g g are bounded away from zero and infinity on separated strictly convex domains r n for the quadratic cost this graph is shown to be a c 1 loc hypersurface in whose normal coincides with the direction transported the optimal map between f and g is shown to be holder continuous up to this free bound ary and to those parts of the fixed boundary which map to locally convex parts of the path connected target region

generalized compressible flows and solutions of the h div geodesic problem we study the geodesic problem on the group of diffeomorphism of a domain m rd equipped with the h div metric the geodesic equations coincide with the camassa holm equation when d 1 and represent one of its possible multi dimensional generalizations when d 1 we propose a relaxation a la brenier of this problem in which solutions are represented as probability measures on the space of continuous paths on the cone over m we use this relaxation to prove that smooth h div geodesics are globally length minimizing for short times we also prove that there exists a unique pressure field associated to solutions of our relaxation finally we propose a numerical scheme to construct generalized solutions on the cone and present some numerical results illustrating the relation between the generalized camassa holm and incompressible euler solutions

convex sobolev inequalities related to unbalanced optimal transport abstract we study the behavior of various lyapunov functionals relative entropies along the solutions of a family of nonlinear drift diffusion reaction equations coming from statistical mechanics and population dynamics these equations can be viewed as gradient flows over the space of radon measures equipped with the hellinger kantorovich distance the driving functionals of the gradient flows are not assumed to be geodesically convex or semi convex we prove new isoperimetric type functional inequalities allowing us to control the relative entropies by their productions which yields the exponential decay of the relative entropies

a variational approach to the mean field planning problem abstract we investigate a first order mean field planning problem of the form t u h x d u f x m in 0 t r d t m m h p x d u 0 in 0 t r d m 0 m 0 m t m t in r d associated to a convex hamiltonian h with quadratic growth and a monotone interaction term f with polynomial growth we exploit the variational structure of the system which encodes the first order optimality condition of a convex dynamic optimal entropy transport problem with respect to the unknown density m and of its dual involving the maximization of an integral functional among all the subsolutions u of an hamilton jacobi equation combining ideas from optimal transport convex analysis and renormalized solutions to the continuity equation we will prove existence and at least partial uniqueness of a weak solution m u a crucial step of our approach relies on a careful analysis of distributional subsolutions to hamilton jacobi equations of the form t u h x d u under minimal summability conditions on and to a measure theoretic description of the optimality via a suitable contact defect measure finally using the superposition principle we are able to describe the solution to the system by means of a measure on the path space encoding the local behavior of the players

dynamics of phase singularities in two dimensional oscillating systems abstract we study the dynamics of vortices that arise in two dimensional nonequilibrium systems above a hopf bifurcation to uniform temporal oscillations evolution equations for the vortex positions and for a global phase field are derived and used to study vortex interactions we suggest that the interactions need not be purely attractive or repulsive bound vortex pairs can exist which either precess around fixed points in the plane or drift in directions perpendicular to the inter vortex axes

stationary self propagating fronts in potential flow abstract we analyze the problem of stationary self propagating fronts in potential flow the issues of local existence and uniqueness for solutions of the ode describing stationary fronts multiplicity of solutions and linearized stability of a stationary front as a solution of the hyperbolic evolution equation are addressed the results are illustrated in the case of the dual source system which is a simple model of a combustion system in which local extinction may arise model extensions for combustion applications are presented

two results on entire solutions of ginzburg landau system in higher dimensions in this short article we prove two results on the ginzburg landau system of equations u u u 2 1 where u c2 rn rm n m 1 first we prove a liouville type theorem which asserts that every solution u satisfying rn u 2 1 2 is constant and of unit norm provided n 4 here m 1 in our second result we give an answer to a question raised by brezis open problem 3 of proceedings of the symposium on pure mathematics vol 65 american mathematical society providence ri 1999 about the symmetry for the ginzburg landau system in the case n m 3 we also formulate three open problems concerning the classification of entire solutions of the ginzburg landau system in any dimension

dynamics of defects in the complex ginzburg landau equation abstract we review the derivation of basic equations for the spiral dynamics proposed by us before these coupled equations involving the trajectories and a global phase are solved here in various situations of physical interest

time periodic spatial chaos in the complex ginzburg landau equation the phenomenon of time periodic evolution of spatial chaos is investigated in the frames of one and two dimensional complex ginzburg landau equations it is found that there exists a region of the parameters in which disordered spatial distribution of the field behaves periodically in time the boundaries of this region are determined the transition to the regime of spatiotemporal chaos is investigated and the possibility of describing spatial disorder by a system of ordinary differential equations is analyzed the effect of the size of the system on the shape and period of oscillations is investigated it is found that in the two dimensional case the regime of time periodic spatial disorder arises only in a narrow strip the critical width of which is estimated the phenomenon investigated in this paper indicates that a family of limit cycles with finite basins exists in the functional phase space of the complex ginzburg landau equation in finite regions of the parameters

the structure of spiral domain patterns and shocks in the 2d complex ginzburg landau equation abstract cellular patterns appear spontaneously in a number of nonequilibrium systems governed by the dynamics of a complex field in the case of the complex ginzburg landau equation disordered cells of effectively frozen spirals appear separated by thin walls shocks on a scale much larger than the basic wavelength of the spirals we show that these structures can be understood in very simple terms in particular we show that the walls are to a good approximation segments of hyperbolae and this allows us to construct the wall pattern given the vortex centers and a phase constant for each vortex the fact that the phase is only defined up to an integer multiple of 2 introduces a quantization condition on the sizes of the smallest spiral domains the transverse structure of the walls is analyzed by treating them as heteroclinic connections of a system of ordinary differential equations the structure depends on the angle the wall makes with the local phase contours and the behavior can be either monotonic or oscillatory depending on the parameters

on properties of the generalized wasserstein distance the wasserstein distances wp p geqq 1 defined in terms of a solution to the monge kantorovich problem are known to be a useful tool to investigate transport equations in particular the benamou brenier formula characterizes the square of the wasserstein distance w2 as the infimum of the kinetic energy or action functional of all vector fields transporting one measure to the other another important property of the wasserstein distances is the kantorovich rubinstein duality stating the equality between the distance w1 of two probability measures and the supremum of the integrals in d of lipschitz continuous functions with lipschitz constant bounded by one an intrinsic limitation of wasserstein distances is the fact that they are defined only between measures having the same mass to overcome such a limitation we recently introduced the generalized wasserstein distances w p a b defined in terms of both the classical wasserstein distance wp and the total variation or l1 distance see piccoli and rossi in archive for rational mechanics and analysis 211 1 335 358 2014 here p plays the same role as for the classic wasserstein distance while a and b are weights for the transport and the total variation term in this paper we prove two important properties of the generalized wasserstein distances 1 a generalized benamou brenier formula providing the equality between w 2 a b and the supremum of an action functional which includes a transport term kinetic energy and a source term 2 a duality a la kantorovich rubinstein establishing the equality between w 1 1 1 and the flat metric

poincare s theorem and subdynamics for driven systems abstract large dynamical systems with explicit time dependence will be studied for these systems which we call driven systems there exist resonances between internal frequencies and or between internal and external frequencies as in the time independent case the usual canonical or unitary perturbation theory leads to divergences due to the resonances as a result there exist no trajectories analytic in both the coupling constant and the initial data this is a generalization of poincare s theorem on non integrability and extends the notion of large poincare systems lps i e systems with a continuous spectrum and a continuous set of resonances here the resonances involve external frequencies along the line of the subdynamics theory developed by prigogine and his co workers we study lps within the liouville space formalism we construct projection operators which decompose the equations of motion and are analytic in the coupling constant our approach recovers coveney s theory of time dependent subdynamics but is based on recursion formulas which significantly simplify the construction of the projection operators these projectors are non hermitian and provide a description with broken time symmetry as an application we study the modification of the well known three stages of the decay process from a time dependent perturbation

dynamics of spiral waves in non equilibrium media in this paper the spiral waves in two dimensional nonequilibrium media are studied on the basis of the two dimensional ginzburg landau equation in the general case the perturbation of origin spiral waves leads to the spiral rotation and to the motion of spiral cores motion in media with hard and soft excitation is analysed separately

generalized wasserstein distance and its application to transport equations with source in this article we generalize the wasserstein distance to measures with different masses we study the properties of this distance in particular we show that it metrizes weak convergence for tight sequences we use this generalized wasserstein distance to study a transport equation with a source in which both the vector field and the source depend on the measure itself we prove the existence and uniqueness of the solution to the cauchy problem when the vector field and the source are lipschitzian with respect to the generalized wasserstein distance

sources and sinks of wave patterns abstract we analyze in this paper the behavior of sources and sinks of traveling waves in one dimensional systems it is shown that the presence of these defects induces a selection of the wavenumber in the bulk of the pattern itself this selection mechanism can drive the pattern into the benjamin fier eckhaus unstable regime generating spatio temporal chaos with space time dislocations in the convectively unstable regime the source of group perturbation becomes a noise amplifier

the optimal partial transport problem given two densities f and g we consider the problem of transporting a fraction m in 0 min f l 1 g l 1 of the mass of f onto g minimizing a transportation cost if the cost per unit of mass is given by x y 2 we will see that uniqueness of solutions holds for m in f wedge g l 1 min f l 1 g l 1 this extends the result of caffarelli and mccann in ann math in print where the authors consider two densities with disjoint supports the free boundaries of the active regions are shown to be n 1 rectifiable provided the supports of f and g have lipschitz boundaries and under some weak regularity assumptions on the geometry of the supports they are also locally semiconvex moreover assuming f and g supported on two bounded strictly convex sets omega lambda subset mathbb r n and bounded away from zero and infinity on their respective supports c 0 alpha rm loc regularity of the optimal transport map and local c1 regularity of the free boundaries away from omega cap lambda are shown finally the optimal transport map extends to a global homeomorphism between the active regions

synchronized disorder in a 2d complex ginzburg landau equation the phenomenon of spiral pair synchronization by oscillating strips nozaki bekki solution in a 2d cgle is investigated analytically the equations describing the interaction of the strips with one another and with spirals are derived analysis of the equations shows that under certain conditions the strips lead to frequency and phase locking of the spirals in this case the spiral pair dipole is aligned parallel to the strips with the position along the strips being arbitrary thus the interaction with strips may transform the spatio temporal chaos of spirals to the regime of periodically oscillating spatial disorder the dynamics of circular strips is investigated and their lifetime is estimated the behavior of the spirals bounded by circular stript is analysed

on the convergence of augmented lagrangian method for optimal transport between nonnegative densities the dynamical formulation of the optimal transport problem introduced by j d benamou and y brenier corresponds to the time space search of a density and a momentum minimizing a transport energy between two densities in order to solve this problem an algorithm has been proposed to estimate a saddle point of a lagrangian we will study the convergence of this algorithm to a saddle point of the lagrangian in the most general conditions particularly in cases where initial and final densities vanish on some areas of the transportation domain the principal difficulty of our study will consist in the proof under these conditions of the existence of a saddle point and especially in the uniqueness of the density momentum component indeed these conditions imply to have to deal with non regular optimal transportation maps for these reasons a detailed study of the properties of the velocity field associated to an optimal transportation map in quadratic space is required

strong resonances of spatially distributed oscillators a laboratory to study patterns and defects abstract this paper describes phenomena which arise in a continuous system of coupled oscillators subject to a strongly resonant forcing we show that the temporal forcing induces a new type of instability leading to the formation of spatial patterns such as stripes hexagons and spatio temporal intermittent states reminiscent of chemical and magnetical systems we also report the existence of localized states where the instability is confined in the core of a defect leading to an increased mobility and diffusive behaviour of the defect

irreversible liouville equation and nonperturbative approach in quantum computing having analyzed the role of time ordering boundary conditions in deriving irreversible kinetic equation a general irreversible liouville equation is proposed by this equation a variety of irreversible kinetic equations such as a stochastic diffusion equation a irreversible project liouville equation can be constructed we also develop a nonperturbative method to solve the eigenvalue problem for this equation application in controlling decoherence for quantum computing is also studied c 2004 elsevier b v all rights reserved

integral representation of convex functionals on a space of measures abstract in duality pairs such as m b c 0 and w 1 p w 0 1 p a convex integral functional on the space of functions has a polar which admits an integral representation this representation is the sum of a first term involving the absolutely continuous component of the measure and of a second one which is a positively homogeneous function of the singular part the duality is useful in plasticity theory in the sobolev case the study of non parametric integrands is new a description of the sub differential is obtained

alternative formulation of classical and quantum dynamics for non integrable systems we continue our work on large poincare systems lps characterized by a continuous spectrum and a continuous set of resonances poincare s theorem prevents the existence of solutions of the eigenvalue problem associated with the hamiltonian as well as the liouville operator which would be analytic in the coupling constant in simple cases such as the friedrichs model studied in a recent paper poincare s divergences can be avoided using a natural time ordering of the dynamical states in the hilbert space however in general the time ordering has to be performed in the liouville space of density matrices

analytical description of a state dominated by spiral defects in two dimensional extended systems abstract we study the state of a diluted gas of spiral defects observed in a recent simulation of a two dimensional extended system governed by the ginzburg landau equation with complex coefficients we show that this state which is responsible for the disorganization of the system can be described by an approximate solution in which the new variables are the positions of the spirals and a new global phase the dynamics of these new variables and the conditions for the existence of the state are determined

nonlinear fokker planck equations with reaction as gradient flows of the free energy abstract we interpret a class of nonlinear fokker planck equations with reaction as gradient flows over the space of radon measures equipped with the recently introduced hellinger kantorovich distance the driving entropy of the gradient flow is not assumed to be geodesically convex or semi convex we prove new generalized dissipation inequalities which allow us to control the relative entropy by its production we establish the entropic exponential convergence of the trajectories of the flow to the equilibrium along with other applications this result has an ecological interpretation as a trend to the ideal free distribution for a class of fitness driven models of population dynamics our existence theorem for weak solutions under mild assumptions on the nonlinearity is new even in the absence of the reaction term