global weak solution to the inhomogeneous navier stokes vlasov equations abstract the inhomogeneous navier stokes vlasov equations for fluid particle flows are considered in the three dimensional space the coupling in the fluid particle system arises from the drag force in the fluid equations and the acceleration in the vlasov equation an initial boundary value problem is studied in a bounded domain with large initial data the existence of global weak solution is established through an approximation scheme a fixed point argument energy estimates and a weak convergence method

iterative bregman projections for regularized transportation problems this article details a general numerical framework to approximate so lutions to linear programs related to optimal transport the general idea is to introduce an entropic regularization of the initial linear program this regularized problem corresponds to a kullback leibler bregman di vergence projection of a vector representing some initial joint distribu tion on the polytope of constraints we show that for many problems related to optimal transport the set of linear constraints can be split in an intersection of a few simple constraints for which the projections can be computed in closed form this allows us to make use of iterative bregman projections when there are only equality constraints or more generally bregman dykstra iterations when inequality constraints are in volved we illustrate the usefulness of this approach to several variational problems related to optimal transport barycenters for the optimal trans port metric tomographic reconstruction multi marginal optimal trans port and in particular its application to brenier s relaxed solutions of in compressible euler equations partial un balanced optimal transport and optimal transport with capacity constraints

unstable manifolds of euler equations we consider a steady state v0 of the euler equation in a fixed bounded domain in n suppose the linearized euler equation has an exponential dichotomy of unstable and center stable subspaces by rewriting the euler equation as an ode on an infinite dimensional manifold of volume preserving maps in wk q k 1 n q the unstable and stable manifolds of v0 are constructed under a certain spectral gap condition that is satisfied for both two dimensional and three dimensional examples in particular when the unstable subspace is finite dimensional this implies the nonlinear instability of v0 in the sense that arbitrarily small wk q perturbations can lead to l2 growth of the nonlinear solutions 2013 wiley periodicals inc

on the complexity of synthesizing a minimum weighted supervisor under partial observation abstract with full observation the problem of synthesizing a minimum weighted supervisor has been shown polynomial time solvable with partial observation the problem becomes np hard in this paper we will show that the decision version of the problem becomes np complete when the natural projection is a natural observer this np completeness result is unexpected considering that the logic supervisor synthesis problem under the same assumption becomes polynomial time solvable

quantitative supervisory control game for discrete event systems we formulate an optimal supervisory control problem for quantitative non terminating discrete event systems dess modeled by finite weighted automata the control performance of a supervisor is evaluated by the worst case limit average weight of the infinite sequences generated by the supervised des an optimal supervisor is a supervisor that avoids deadlocks and maximizes the control performance we propose a game theoretical design method for an optimal supervisor using a two player turn based mean payoff game automaton as the first player the objective of the supervisor is to maximize the worst case limit average weight of the generated sequences as the second player the des aims to minimize it we show that an optimal supervisor can be computed from an optimal strategy of the first player for this game then we propose an algorithm to compute an f minimally restrictive optimal supervisor which is a finite memory optimal supervisor that enables as many sequences as possible and can be represented by an optimal strategy for a finite version of the two player game

dynamic theory of suspensions with brownian effects we consider a suspension of particles in a fluid settling under the influence of gravity and dispersing by brownian motion a mathematical description is provided by the stokes equations and a fokker planck equation for the one particle phase space density this is a nonlinear system that depends on a number of parametric functions of the spatial concentration of the particles these functions are known only empirically or for dilute suspensions we analyze the system its stability its asymptotic behavior under different scalings and its validity from more microscopic description we summarize our conclusions at the end

generalized incompressible flows multi marginal transport and sinkhorn algorithm starting from brenier s relaxed formulation of the incompress ible euler equation in terms of geodesics in the group of measure preserving diffeomorphisms we propose a numerical method based on sinkhorn s algorithm for the entropic regularization of optimal transport we also make a detailed comparison of this entropic regular ization with the so called bredinger entropic interpolation problem see 1 numerical results in dimension one and two illustrate the feasibility of the method

least action principle and the incompressible euler equations with variable density in this article we study a variational formulation proposed by v i arnold and by y brenier for the incompressible euler equations with variable density we consider the problem of minimizing an action functional the integral in time of the kinetic energy among fluid motions considered as trajectories in the group of volume preserving diffeomorphisms beginning at the identity and ending at some fixed diffeomorphism at a given time we show that a relaxed version of this variational problem always has a solution and we derive an euler lagrange system for the relaxed minimization problem which we call the relaxed euler equations finally we prove consistency between the relaxed euler equations and the classical euler system showing that weak solutions of the relaxed euler equations with the appropriate geometric structure give rise to classical euler solutions and that classical solutions of the euler system induce weak solutions of the relaxed euler equations the first consistency result is new even in the constant density case the remainder of our analysis is an extension of the work of y brenier 1999 to the variable density case

generalized solutions and hydrostatic approximation of the euler equations abstract solutions to the euler equations on a 3d domain d 3 typically the unit cube or the periodic unit cube can be formally obtained by minimizing the action of an incompressible fluid moving inside d 3 between two given configurations when these two configurations are very close to each other classical solutions do exist as shown by ebin and marsden however shnirelman found a class of data essentially 2d in the sense that they trivially depend on the vertical coordinate for which there cannot be any classical minimizer for such data generalized solutions can be shown to exist as a substitute for classical solutions these generalized solutions have unusual features that look highly unphysical in particular different fluid parcels can cross at the same point and at the same time but the pressure field which does not depend on the vertical coordinate is well and uniquely defined in the present paper we show that these generalized solutions are actually quite conventional in the sense they obey up to a suitable change of variable a well known variant widely used for geophysical flows of the 3d euler equations for which the vertical acceleration is neglected according to the so called hydrostatic approximation

the reconstruction problem for the euler poisson system in cosmology the motion of a continuum of matter subject to gravitational interaction is classically described by the euler poisson system prescribing the density of matter at initial and final times we are able to obtain weak solutions for this equation by minimizing the action of the lagrangian which is a convex functional through this variational formulation the reconstruction problem becomes very similar to an optimal transportation problem then we see that such minimizing solutions are consistent with smooth solutions of the euler poisson system and enjoy some special regularity properties

hydrodynamic limit for the inhomogeneous incompressible navier stokes vlasov fokker planck equations abstract in this paper we study the hydrodynamic limit for the inhomogeneous incompressible navier stokes vlasov fokker planck equations in a two or three dimensional bounded domain when the initial density is bounded away from zero the proof relies on the relative entropy argument to obtain the strong convergence of macroscopic density of the particles n in l 0 t l 1 which extends the works of goudon jabin vasseur 15 and mellt vasseur 26 to inhomogeneous incompressible navier stokes vlasov fokker planck equations precisely the relative entropy estimates in 15 and 26 give the strong convergence of u and n and n respectively however we only obtain the strong convergence of n and u from the relative entropy estimate and we use another way to obtain the strong convergence of via the convergence of u furthermore when the initial density may vanish taking advantage of compactness result l m h 1 of orlicz spaces in 2d we obtain the convergence of n in l 0 t h 1 which is used to obtain the relative entropy estimate thus we also show the hydrodynamic limit for 2d inhomogeneous incompressible navier stokes vlasov fokker planck equations when there is initial vacuum

the three dimensional euler equations where do we stand the three dimensional euler equations have stood for a quarter of a millenium as a challenge to mathematicians and physicists while much has been discovered the nature of solutions is still largely a mystery this paper surveys some of the issues such as singularity formation that have cost so much effort in the last 25 years in this light we review the beale kato majda theorem and its consequences and then list some of the results of numerical experiments that have been attempted a different line of endeavour focuses on work concerning the pressure hessian and how it may be used and modelled the euler equations are finally discussed in terms of their membership of a class of general lagrangian evolution equations using hamilton s quaternions these are reformulated in an elegant manner to describe the motion and rotation of fluid particles c 2007 elsevier b v all rights reserved

convergence of the viscosity method for isentropic gas dynamics a convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

an optimal control theory for discrete event systems in certain discrete event applications it may be desirable to find a particular controller within the set of acceptable controllers which optimizes some quantitative performance measure in this paper we propose a theory of optimal control to meet such design requirements for deterministic systems the discrete event system des is modeled by a formal language event and cost functions are defined which induce costs on controlled system behavior the event costs associated with the system behavior can be reduced in general only by increasing the control costs thus it is nontrivial to find the optimal amount of control to use and the formulation captures the fundamental tradeoff motivating classical optimal control results on the existence of minimally restrictive optimal solutions are presented communication protocols are analyzed to motivate the formulation and demonstrate optimal controller synthesis algorithms for the computation of optimal controllers are developed for the special case of des modeled by regular languages it is shown that this framework generalizes some of the existing literature

symmetries of gaussian measures and operator colligations consider an infinite dimensional linear space equipped with a gaussian measure and the group glo of linear transformations that send the measure to equivalent one limit points of glo can be regarded as spreading maps polymorphisms we show that the closure of glo in the semigroup of polymorphisms contains a certain semigroup of operator colligations and write explicit formulas for action of operator colligations by polymorphisms of the space with gaussian measure

oscillations and concentrations in weak solutions of the incompressible fluid equations the authors introduce a new concept of measure valued solution for the 3 d incompressible euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations one application of the concepts developed here is the following important result a sequence of leray hopf weak solutions of the navier stokes equations converges in the high reynolds number limit to a measure valued solution of 3 d euler defined for all positive times the authors present several explicit examples of solution sequences for 3 d incompressible euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations an extensions of the concept of young measure is developed to incorporate these complex phenomena in the measure valued solutions constructed here

a priori estimates for fluid interface problems we consider the regularity of an interface between two incompressible and inviscid fluid flows in the presence of surface tension we obtain local in time estimates on the interface in h 3 2 k 1 and the velocity fields in h 3 2 k these estimates are obtained using geometric considerations which show that the kelvin helmholtz instabilities are a consequence of a curvature calculation 2007 wiley periodicals inc

sub riemannian interpolation inequalities we prove that ideal sub riemannian manifolds i e admitting no non trivial abnormal minimizers support interpolation inequalities for optimal transport a key role is played by sub riemannian jacobi fields and distortion coefficients whose properties are remarkably different with respect to the riemannian case as a byproduct we characterize the cut locus as the set of points where the squared sub riemannian distance fails to be semiconvex answering to a question raised by figalli and rifford geom funct anal 20 1 124 159 2010 as an application we deduce sharp and intrinsic borell brascamp lieb and geodesic brunn minkowski inequalities in the aforementioned setting for the case of the heisenberg group we recover in an intrinsic way the results recently obtained by balogh et al calc var part differ equ 57 2 61 2018 and we extend them to the class of generalized h type carnot groups our results do not require the distribution to have constant rank yielding for the particular case of the grushin plane a sharp measure contraction property and a sharp brunn minkowski inequality

quantum information entropies and orthogonal polynomials this is a survey of the present knowledge on the analytical determination of the shannon information entropies for simple quantum systems single particle systems in central potentials emphasis is made on d dimensional harmonic oscillator and coulombian potentials in both position and momentum spaces first of all these quantities are explicitly shown to be controlled by the entropic integrals of some classical orthogonal polynomials hermite laguerre and gegenbauer then the connection of these integrals with more common mathematical objects such as the logarithmic potential energy and l p norms of orthogonal polynomials is brie y described third its asymptotic behaviour is discussed for both general and varying weights the explicit computation of these integrals is carried out for the chebyshev and gegenbauer polynomials which have a bounded orthogonality interval as well as for hermite polynomials to illustrate the di culties encountered when the interval is unbounded these results have allowed us to nd the position and momentum entropies of the ground and excited states of the physical systems mentioned above c 2001 elsevier science b v all rights reserved

asymptotic preserving schemes for kinetic fluid modeling of disperse two phase flows abstract we consider a system coupling the incompressible navier stokes equations to the vlasov fokker planck equation such a problem arises in the description of particulate flows we design a numerical scheme to simulate the behavior of the system this scheme is asymptotic preserving thus efficient in both the kinetic and hydrodynamic regimes it has a numerical stability condition controlled by the non stiff convection operator with an implicit treatment of the stiff drag term and the fokker planck operator yet consistent to a standard asymptotic preserving fokker planck solver or an incompressible navier stokes solver only the conjugate gradient method and fast poisson and helmholtz solvers are needed numerical experiments are presented to demonstrate the accuracy and asymptotic behavior of the scheme with several interesting applications

the generation of modulated wavetrains in the solution of the korteweg de vries equation on demontre une relation entre la methode des ondes modulees de whitham et la methode de lax et levermore contenant la limite de petite dispersion

levy flights in inhomogeneous environments abstract we study the long time asymptotics of probability density functions pdfs of levy flights in confining potentials that originate from inhomogeneities of the environment in which the flights take place to this end we employ two model patterns of dynamical behavior langevin driven and levy schrodinger semigroup driven dynamics it turns out that the semigroup modeling provides much stronger confining properties than the standard langevin one for computational and visualization purposes our observations are exemplified for the cauchy driver and its response to external polynomial potentials referring to levy oscillators with respect to both dynamical mechanisms we discuss the links of the levy semigroup motion scenario with that of random searches in spatially inhomogeneous media

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

well posedness of the epdiff equation with a pseudo differential inertia operator abstract in this article we study the class of right invariant fractional order sobolev type metrics on groups of diffeomorphisms of a compact manifold m our main result concerns well posedness properties for the corresponding euler arnold equations also called the epdiff equations which are of importance in mathematical physics and in the field of shape analysis and template registration depending on the order of the metric we will prove both local and global well posedness results for these equations as a result of our analysis we will also obtain new commutator estimates for elliptic pseudo differential operators

first order mean field games with density constraints pressure equals price in this paper we study mean field game systems under density constraints as optimality conditions of two optimization problems in duality a weak solution of the system contains an extra term an additional price imposed on the saturated zones we show that this price corresponds to the pressure field from the models of incompressible euler equations a la brenier by this observation we manage to obtain a minimal regularity which allows us to write optimality conditions at the level of single agent trajectories and to define a weak notion of nash equilibrium for our model

quantitative linearization results for the monge ampere equation this paper is about quantitative linearization results for the monge ampere equation with rough data we develop a large scale regularity theory and prove that if a measure is close to the lebesgue measure in wasserstein distance at all scales then the displacement of the macroscopic optimal coupling is quantitatively close at all scales to the gradient of the solution of the corresponding poisson equation the main ingredient we use is a harmonic approximation result for the optimal transport plan between arbitrary measures this is used in a campanato iteration which transfers the information through the scales

structure preserving discretization of incompressible fluids the geometric nature of euler fluids has been clearly identified and extensively studied over the years culminating with lagrangian and hamiltonian descriptions of fluid dynamics where the configuration space is defined as the volume preserving diffeomorphisms and kelvin s circulation theorem is viewed as a consequence of noether s theorem associated with the particle relabeling symmetry of fluid mechanics however computational approaches to fluid mechanics have been largely derived from a numerical analytic point of view and are rarely designed with structure preservation in mind and often suffer from spurious numerical artifacts such as energy and circulation drift in contrast this paper geometrically derives discrete equations of motion for fluid dynamics from first principles in a purely eulerian form our approach approximates the group of volume preserving diffeomorphisms using a finite dimensional lie group and associated discrete euler equations are derived from a variational principle with non holonomic constraints the resulting discrete equations of motion yield a structure preserving time integrator with good long term energy behavior and for which an exact discrete kelvin s circulation theorem holds

fisher information of special functions and second order differential equations we investigate a basic question of analytic information theory namely the evaluation of the fisher information and the relative fisher information with respect to a non negative function for the probability distributions obtained by squaring the special functions of mathematical physics which are solutions of second order differential equations we obtain explicit expressions for these information theoretic properties via the expectation values of the coefficients of the differential equation we illustrate our approach for various nonrelativistic d dimensional wavefunctions and some special functions of physicomathematical interest emphasis is made in the nikiforov uvarov hypergeometric type functions which include and generalize the hermite functions and the gauss and kummer hypergeometric functions among others

brief on optimal control of a class of partially observed discrete event systems we are interested in a new class of optimal control problems for discrete event systems we adopt the formalism of supervisory control theory proc ieee 77 1 1989 81 and model the system as a finite state machine fsm our control problem is characterized by the presence of uncontrollable as well as unobservable events the notion of occurrence and control costs for events and a worst case objective function we first derive an observer for the partially unobservable fsm which allows us to construct an approximation of the unobservable trajectory costs we then define the performance measure on this observer rather than on the original fsm itself we then use the algorithm presented in sengupta and lafortune siam j control optim 36 2 1998 to synthesize an optimal submachine of the c observer this submachine leads to the desired supervisor for the system

existence of global weak solutions to one component vlasov poisson and fokker planck poisson systems in one space dimension with measures as initial data we consider cauchy problems for the 1 d one component vlasov poisson and fokker planck poisson equations with the initial electron density being in the natural space of arbitrary non negative finite measures in particular the initial density can be a dirac measure concentrated on a curve which we refer to as electron sheet initial data these problems resemble both structurally and functional analytically cauchy problems for the 2 d euler and navier stokes equations in vorticity formulation with vortex sheet initial data here we need to define weak solutions more specifically than usual since the product of a finite measure with a function of bounded variation is involved we give a natural definition of the product establish its weak stability and existence of weak solutions follows our concept of weak solutions through the newly defined product is justified since solutions to the fokker planck poisson equation the analogue of navier stokes equation are shown to converge to weak solutions of the vlasov poisson equation as the fokker planck term vanishes the main difficulty is the aforementioned weak stability which we establish through a careful analysis of the explicit structure of these equations this is needed because the problem studied here is beyond the range of applicability of the velocity averaging compactness methods of diperna lions 1994 john wiley sons inc

regularity via duality in calculus of variations and degenerate elliptic pdes abstract a technique based on duality to obtain h 1 or other sobolev regularity results for solutions of convex variational problems is presented this technique first developed in order to study the regularity of the pressure in the variational formulation of the incompressible euler equation has been recently re employed in mean field games here it is shown how to apply it to classical problems in relation with degenerate elliptic pdes of p laplace type this allows to recover many classical results via a different point of view and to have inspiration for new ones the applications include among others variational models for traffic congestion and more general minimization problems under divergence constraints but the most interesting results are obtained in dynamical problems such as mean field games with density constraints or density penalizations

asymptotics of information entropies of some toda like potentials the spreading of the quantum probability density for the highly excited states of a single particle system with an exponential type potential on the positive semiaxis is quantitatively determined in both position and momentum spaces by means of the boltzmann shannon information entropy this problem boils down to the calculation of the asymptotics of the entropy like integrals of the modified bessel function of the second kind also called the mcdonald function or basset function the dependence of the two physical entropies on the large quantum number n is given in detail it is shown that the semiclassical wkb position space entropy grows slower than the corresponding quantity of not only the harmonic oscillator but also the single particle systems with any power type potential of the form v x x2k x r and k n the momentum space entropy calculated with a method based on the properties of the mcdonald function is rigorously found to have a behavior of the form ln ln n in strong contrast with the

reduced hausdorff dimension and concentration cancellation for two dimensional incompressible flow where v t v1 v2 is the fluid velocity p is the scalar pressure dv dt av at v v v and vo is an initial incompressible velocity field i e div vo 0 in this paper we study the detailed limiting behavior of approximate solution sequences for 2 d euler with vortex sheet initial data a sequence of smooth velocity fields v8 x t is an approximate solution sequence for 2 d euler provided that v is incompressible i e div v 0 and satisfies the following properties 1 the velocity fields v6 have uniformly bounded local kinetic energy i e

asymptotics of the information entropy of the airy function the boltzmann shannon information entropy of linear potential wavefunctions is known to be controlled by the information entropy of the airy function ai x here the entropy asymptotics is analysed so that the first two leading terms previously calculated in the wkb approximation as well as the following term already conjectured are derived by using only the specific properties of the airy function

first order asymptotics of matrix integrals a rigorous approach towards the understanding of matrix models we investigate the large n limit of spectral measures of matrices which relate to the gibbs measures of a number of statistical mechanical systems on random graphs these include the ising and potts models on random graphs for most of these models we prove that the spectral measures converge almost surely and describe their limit via solutions to an euler equation for isentropic flow with negative pressure p 3 1 2 3

the camassa holm equation as an incompressible euler equation a geometric point of view the group of diffeomorphisms of a compact manifold endowed with the l 2 metric acting on the space of probability densities gives a unifying framework for the incompressible euler equation and the theory of optimal mass transport recently several authors have extended optimal transport to the space of positive radon measures where the wasserstein fisher rao distance is a natural extension of the classical l 2 wasserstein distance in this paper we show a similar relation between this unbalanced optimal transport problem and the hdiv right invariant metric on the group of diffeomorphisms which corresponds to the camassa holm ch equation in one dimension on the optimal transport side we prove a polar factorization theorem on the automorphism group of half densities geometrically our point of view provides an isometric embedding of the group of diffeomorphisms endowed with this right invariant metric in the automorphisms group of the fiber bundle of half densities endowed with an l 2 type of cone metric this leads to a new formulation of the generalized ch equation as a geodesic equation on an isotropy subgroup of this automorphisms group on s1 solutions to the standard ch thus give particular solutions of the incompressible euler equation on a group of homeomorphisms of r 2 which preserve a radial density that has a singularity at 0 an other application consists in proving that smooth solutions of the euler arnold equation for the hdiv right invariant metric are length minimizing geodesics for sufficiently short times

on the regularity of the pressure field of relaxed solutions to euler equations with variable density abstract in this paper we improve the regularity in time of the gradient of the pressure field in the solution of relaxed version of variational formulation proposed by v i arnold and by y brenier for the incompressible euler equations with variable density we obtain that the pressure field is not only a measure but a function in l l o c 2 0 t b v l o c d as an extension of the work of ambrosio and figalli 2008 in 1 to the variable density case

global existence and large time behavior of classical solutions to the euler maxwell vlasov fokker planck system abstract we consider global classical solutions to the cauchy problem for an euler maxwell vlasov fokker planck system which is a fluid particle model describe the evolutions of disperse two phase flows in these two phase flows the compressible euler maxwell equations modeling a dense phase fluid and the vlasov fokker planck equation for the dispersed phase particles which is usually considered from the statistical view of point they are coupled through the friction force global existence of classical solutions to the cauchy problem in r 3 is established when initial data is a small perturbation of some given equilibrium the algebraic rate of convergence of solution toward the equilibrium state is also obtained under additional conditions on initial data the proof is based on the classical energy estimates and fourier multiplier technique which are considerably complicated and some new ideas and techniques are thus required

pseudo differential energy estimates of singular perturbations in this paper we study singular limits of hyperbolic systems which exhibit large time oscillations and in particular pseudo differential energy estimates that enable us to obtain uniform existence time and a priori bounds on the solutions 1997 john wiley sons inc

on a weak solution for a transonic flow problem on essaie d etablir l existence de solutions faibles pour un ecoulement stationnaire bidimensionnel irrotationnel compressible non visqueux dans un canal de longueur infinie ou autour d un profil aerodynamique la vitesse de l ecoulement etant donnee a l infini

minimal geodesics on groups of volume preserving maps and generalized solutions of the euler equations the three dimensional motion of an incompressible inviscid fluid is classically described by the euler equations but can also be seen following arnold 1 as a geodesic on a group of volume preserving maps local existence and uniqueness of minimal geodesics have been established by ebin and marsden 16 in the large for a large class of data the existence of minimal geodesics may fail as shown by shnirelman 26 for such data we show that the limits of approximate solutions are solutions of a suitable extension of the euler equations or equivalently are sharp measure valued solutions to the euler equations in the sense of diperna and majda 14 1999 john wiley sons inc

continuity equations and ode flows with non smooth velocity in this paper we review many aspects of the well posedness theory for the cauchy problem for the continuity and transport equations and for the ordinary differential equation ode in this framework we deal with velocity fields that are not smooth but enjoy suitable weak differentiability assumptions we first explore the connection between the partial differential equation pde and the ode in a very general non smooth setting then we address the renormalization property for the pde and prove that such a property holds for sobolev velocity fields and for bounded variation velocity fields finally we present an approach to the ode theory based on quantitative estimates

deriving new evolution equations for microstructures via relaxation of variational incremental problems we study material models for rate independent inelasticity in situations where no internal length scale is given and formation of microstructure for the deformation gradients and the internal variables may occur we develop a rational procedure for deriving consistent macroscopic models which allow for the computation of nontrivial effective quantities without resolving the fine scales the method involves the relaxation of variational incremental problems which are derived from an elastic and a plastic potential we use young measures to describe the microscopic distribution of the internal variables as well as the quasiconvexification of the elastic stored energy density with respect to the deformation gradient the resulting model provides a new rate independent model in terms of the deformation and the young measure the approach is based on a derivative free energetic formulation using one functional for elastic energy storage and one for the dissipation distance the latter is derived from a dissipation potential defining the dissipation distance between internal states and hence the wasserstein distance between young measures this approach is strongly linked to an associated time incremental problem which is a minimization problem of the type used for several years now in the engineering literature the update algorithm for the incremental problem is discussed in detail and two simple examples are given

the least action principle and the related concept of generalized flows for incompressible perfect fluids the link between the euler equations of perfect incompressible flows and the least action principle has been known for a long time i solu tions can be considered as geodesic curves along the manifold of volume pre serving mappings here the shortest path problem is investigated given two different volume preserving mappings at two different times find for the inter mediate times an incompressible flow map that minimizes the kinetic energy or more generally the action in its classical formulation this problem has been solved 7 only when the two different mappings are sufficiently close in some very strong sense in this paper a new framework is introduced where gen eralized flows are defined in the spirit of l c young as probability measures on the set of all possible trajectories in the physical space then the minimiza tion problem is generalized as the continuous linear programming problem that is much easier to handle the existence problem is completely solved in this content downloaded from 157 55 39 159 on sun 18 sep 2016 05 02 20 utc all use subject to http about jstor org terms the least action principle 255 the case of the d dimensional torus it is also shown that under natural re strictions a classical solution to the euler equations is the unique optimal flow in the generalized framework finally a link is established with the concept of measure valued solutions to the euler equations 6 and an example is provided where the unique generalized solution can be explicitly computed and turns out to be genuinely probabilistic inria bp 105 78153 le chesnay cedex france this content downloaded from 157 55 39 159 on sun 18 sep 2016 05 02 20 utc all use subject to http about jstor org terms

euler equations on homogeneous spaces and virasoro orbits abstract we show that the following three systems related to various hydrodynamical approximations the korteweg de vries equation the camassa holm equation and the hunter saxton equation have the same symmetry group and similar bihamiltonian structures it turns out that their configuration space is the virasoro group and all three dynamical systems can be regarded as equations of the geodesic flow associated to different right invariant metrics on this group or on appropriate homogeneous spaces in particular we describe how arnold s approach to the euler equations as geodesic flows of one sided invariant metrics extends from lie groups to homogeneous spaces we also show that the above three cases describe all generic bihamiltonian systems which are related to the virasoro group and can be integrated by the translation argument principle they correspond precisely to the three different types of generic virasoro orbits finally we discuss interrelation between the above metrics and kahler structures on virasoro orbits as well as open questions regarding integrable systems corresponding to a finer classification of the orbits

on optimal attraction in discrete event processes abstract the concept of attraction has been shown to play a basic role in the synthesis of supervisors for stabilizing discrete event processes in this paper we define and investigate optimal attraction in the sense that the cost or distance of the convergent path is minimized with respect to some distance measure we find conditions for the existence of supervisors achieving optimal attraction and provide efficient algorithms for their synthesis

analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid we establish the existence of three new subgroups of the group of volumepreserving diffeomorphisms of a compact n dimensional n 2 riemannian manifold which are associated with the dirichlet neumann and mixed type boundary conditions that arise in second order elliptic pdes we prove that when endowed with the hs hilbert class topologies for s n 2 1 these subgroups are c differential manifolds we consider these new diffeomorphism groups with an h1 equivalent right invariant metric and prove the existence of unique smooth geodesics t of this metric as well as existence and uniqueness of the jacobi equations associated to this metric geodesics on these subgroups are in fact the flows of a timedependent velocity vector field u t x so that t t u t t with 0 x x and remarkably the vector field u t x solves the so called lagrangian averaged euler lae equations on m these equations and their viscous counterparts the lagrangian averaged navier stokes lans equations model the motion of a fluid at scales larger than an a priori fixed parameter 0 while averaging or filtering out the small scale motion and this is achieved without the use of artificial viscosity we prove that for divergence free initial data satisfying u 0 on m the lae equations are well posed globally when n 2 we also find the boundary conditions that make the lans equations well posed globally when n 3 and prove that solutions of the lans equations converge when n 2 3 for almost all t in some fixed time interval 0 t in hs s n 2 1 3 to solutions of the lae equations thus confirming the scaling arguments of barenblatt chorin received april 26 2000 1991 mathematics subject classification 35q35 35q53 58b20 58d05

geodesics in the space of measure preserving maps and plans we study brenier s variational models for incompressible euler equations these models give rise to a relaxation of the arnold distance in the space of measure preserving maps and more generally measure preserving plans we analyze the properties of the relaxed distance we show a close link between the lagrangian and the eulerian model and we derive necessary and sufficient optimality conditions for minimizers these conditions take into account a modified lagrangian induced by the pressure field moreover adapting some ideas of shnirelman we show that even for non deterministic final conditions generalized flows can be approximated in energy by flows associated to measure preserving maps

small noise limit and convexity for generalized incompressible flows schrodinger problems and optimal transport this paper is concerned with six variational problems and their mutual connections the quadratic monge kantorovich optimal transport the schrodinger problem brenier s relaxed model for incompressible fluids the so called brodinger problem recently introduced by arnaudon et al an entropic interpolation problem for incompressible viscid fluids arxiv preprint arxiv 1704 02126 2017 the multiphase brenier model and the multiphase brodinger problem all of them involve the minimization of a kinetic action and or a relative entropy of some path measures with respect to the reversible brownian motion as the viscosity parameter nu rightarrow 0 we establish gamma convergence relations between the corresponding problems and prove the convergence of the associated pressures arising from the incompressibility constraints we also present new results on the time convexity of the entropy for some of the dynamical interpolations along the way we extend previous results by lavenant calc var partial differ equ 56 6 170 2017 and benamou et al generalized incompressible flows multi marginal transport and sinkhorn algorithm arxiv preprint arxiv 1710 08234 2017

local well posedness for fluid interface problems in this paper we prove the local well posedness of the fluid interface problem with surface tension where the velocity fields are not assumed to be irrotational and the fluid domains are not assumed to be simply connected viewed as a lagrangian system with the configuration space being an infinite dimensional manifold possessing many symmetries this problem is reduced to the evolution of the interface determined by its mean curvature and the evolution of the rotational part of the velocity fields determined by the symmetries this framework also applies to several other fluid surface problems which are outlined in the paper

a framework for state attraction of discrete event systems under partial observation state attraction for discrete event systems des addresses the problem of reaching a desired subset of the plant state space after a bounded number of event occurrences the problem of state attraction arises for example in fault tolerant supervisory control or in the control of reconfigurable manufacturing systems and is also applicable to systems biological problems such as the control of gene regulatory networks state attraction is investigated with the assumption of full event observation in the existing literature this paper extends the concept of state attraction to the case of partial observation the notion of weak attraction under partial observation wapo is introduced and necessary and sufficient conditions for the existence of a supervisor under partial observation that achieves wapo are derived furthermore a solution algorithm is proposed that finds such supervisor whenever it exists it is shown that such supervisor can always be realized as a subautomaton of the observer automaton of the des plant an application example from systems biology illustrates the obtained results

geometry and a priori estimates for free boundary problems of the euler s equation in this paper we derive estimates to the free boundary problem for the euler equation with surface tension and without surface tension provided the rayleigh taylor sign condition holds we prove that as the surface tension tends to 0 when the rayleigh taylor condition is satisfied solutions converge to the euler flow with zero surface tension 2007 wiley periodicals inc

optimal supervisory control of discrete event dynamicalsystems the notion of optimal supervisory control of discrete event dynamical systems dedss is formalized in the framework of ramadge and wonham a deds is modeled as a state machine and is controlled by disabling some of its transitions two types of cost functions are defined a cost of control function corresponding to disabling transitions in the state machine and a penalty of control function corresponding to reaching some undesired states or not reaching some desired states in the controlled system the control objective is to design an optimal control mechanism if it exists so that the net cost is minimized since a deds is represented as a state machine a directed graph network flow techniques are naturally applied for designing optimal supervisors it is also shown that our techniques can be used to solve supervisory control problems under complete as well as partial observation in particular for the first time techniques for computing the supremal controllable and normal sublanguage and the infimal controllable and normal observable superlanguage without having to perform alternate computations of controllable and normal observable languages are obtained

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form

simulation of fluid and particles flows asymptotic preserving schemes for bubbling and flowing regimes in this work we propose asymptotic preserving numerical schemes for the bubbling and flowing regimes of particles immersed in a fluid treated by two phase flow models the description comprises compressible euler equations for the dense phase fluid and a kinetic fokker planck equation for the disperse phase particles coupled through friction terms we show numerical simulations in the relevant case of gravity in the one dimensional case demonstrating the overall behavior of the schemes

formulas for calculating supremal controllable and normal sublanguages abstract supremal controllable and normal sublanguages have been shown to play an important role in supervisor synthesis in this paper we discuss the computation of supremal controllable and normal sublanguages we derive formulas for both supremal controllable sublanguages and supremal normal sublanguages when the languages involved are closed as a result those languages can be computed without applying recursive algorithms we also discuss those aspects of these formulas